#  Normed modules and the categorization of Lebesgue integration

Yu-Zhe Liu  School of Mathematics and Statistics, Guizhou University, Guiyang
550025, Guizhou, P. R. China<EMAIL_ADDRESS>/<EMAIL_ADDRESS>,  Shengda Liu
The State Key Laboratory of Multimodal Artificial Intelligence Systems,
Institute of Automation, Chinese Academy of Sciences, Beijing 100190, P. R.
China<EMAIL_ADDRESS>,  Zhaoyong Huang∗ Department of Mathematics,
Nanjing University, Nanjing 210093, Jiangsu, P. R. China<EMAIL_ADDRESS>and  Panyue Zhou  School of Mathematics and Statistics, Changsha University of
Science and Technology, Changsha 410114, Hunan, P. R. China
<EMAIL_ADDRESS>
###### Abstract.

We explore the assignment of norms to $\Lambda$-modules over a finite-
dimensional algebra $\Lambda$, resulting in the establishment of normed
$\Lambda$-modules. Our primary contribution lies in constructing a new
category $\mathscr{N}\\!\\!or^{p}$ related to normed modules along with its
full subcategory $\mathscr{A}^{p}$. By examining the objects and morphisms in
these categories, we establish a framework for understanding the
categorization of Lebesgue integration.

MSC2020: 16G10; 46B99; 46M40.

Keywords: Categorization; Lebesgue integration; Normed modules, Finite-
dimensional algebras; Banach spaces

∗Corresponding author.

###### Contents

  1. 1  Introduction
  2. 2  Preliminaries
    1. 2.1  Categories and limits
    2. 2.2  $\mathds{k}$-algebras and their completions
      1. 2.2.1  $\mathds{k}$-algebras
      2. 2.2.2  Topologies on $\mathds{k}$-algebras
      3. 2.2.3  Completions induced by $\mathcal{J}$-topologies
    3. 2.3  The total order of $\mathds{k}$-algebras
  3. 3  Normed $\mathds{k}$-algebras
    1. 3.1  Norms of $\mathds{k}$-algebras
    2. 3.2  Completions of normed $\mathds{k}$-algebras
    3. 3.3  Elementary simple functions
  4. 4  Normed modules over $\mathds{k}$-algebras
    1. 4.1  Norms of $\mathit{\Lambda}$-modules
    2. 4.2  Completions of normed $\mathit{\Lambda}$-modules
    3. 4.3  $\sigma$-algebras and the elementary simple function set $\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$
  5. 5  Two categories
    1. 5.1  The categories $\mathscr{N}\\!\\!or^{p}$ and $\mathscr{A}^{p}$
    2. 5.2  The triple $(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})$
    3. 5.3  $\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$ is a direct limit
  6. 6  The $\mathscr{A}^{p}$-initial object in $\mathscr{N}\\!\\!or^{p}$
    1. 6.1 The existence of morphism from $(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})$
    2. 6.2 The uniqueness of morphism from $(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})$
    3. 6.3 The $\mathscr{A}^{p}$-initial object in $\mathscr{N}\\!\\!or^{p}$
  7. 7  The categorization of Lebesgue integration
  8. 8  Applications
    1. 8.1  Lebesgue integration
    2. 8.2  Cauchy-Schwarz inequality

##  1\.  Introduction

Lebesgue integration, introduced by Henri Lebesgue [15], is fundamentally
pivotal in the field of mathematical analysis. The process of understanding
the Lebesgue’s integrability and its application to the real number line
typically involves a series of methodical and incremental steps. This journey
begins with defining measurable sets and null sets, followed by an exploration
of convergence in measure. It then advances through the concepts of step
functions and simple functions, along with their convergence sequences,
culminating in the meticulous construction of spaces for integrable functions
and the validation of consistent integration methods. While this path is
comprehensive, it serves as an elaborate gateway to fully grasp the essence of
Lebesgue integration, see [7, 12] and so on. Indeed, building upon the
foundational methods for defining integrals previously mentioned, our
exploration extends well beyond traditional boundaries. The versatility and
adaptability of these principles lay the groundwork for deriving more
specialized forms of integration, designed to address the complex requirements
of various fields. This notably includes the development of the Bochner
integral [6], which is particularly effective in handling vector-valued
functions and proves invaluable in the realm of functional analysis. In a
similar vein, this framework also leads to the emergence of the Ito integral
[14], a fundamental element in stochastic calculus that provides deep insights
into the complex behavior of stochastic processes. These advancements are not
merely extensions; they are crucial in bridging the theoretical concepts of
integration with their practical applications across diverse domains,
reflecting the dynamic interplay between theoretical constructs and their
real-world implications.

As the landscape of integration theory expands, so too does the exploration
into its algebraic facets, marking a significant evolution in the approach to
integration. Algebraic approaches to integration can be traced back at least
to Segal’s work [20]. Building upon the foundational works of Escardó-Simpson
[10] and Freyd [11], Leinster [16] constructed a special category
$\mathscr{A}^{p}$, where $p$ is a real number at least $1$. In this category,
objects are triples consisting of a Banach space $V$, an element $v$ in $V$
with $|v|\leq 1$, and a $\mathds{k}$-linear map $\delta:V\oplus_{p}V\to V$
that satisfies $\delta(v,v)=v$. Here, the notation “$V_{1}\oplus_{p}V_{2}$”
represents the direct sum of two normed spaces $V_{1}$ and $V_{2}$, where the
norm is defined as
$|(v_{1},v_{2})|=\left(\frac{1}{2}(|v_{1}|^{p}+|v_{2}|^{p})\right)^{1/p}$.
Furthermore, Leinster established three significant results as follows:

  * (1)

$(L_{p}([0,1]),1,\gamma)$ is the initial object in $\mathscr{A}^{p}$, where
$\gamma$ is a special $\mathds{k}$-linear map from
$L_{p}([0,1])\oplus_{p}L_{p}([0,1])$ to $L_{p}([0,1])$ (indeed, $\gamma$ is
the map $\gamma_{\frac{1}{2}}$ given in Corollary 8.1);

  * (2)

$(\mathbb{R},1,m)$ is an object in $\mathscr{A}^{1}$, where
$m:\mathbb{R}\oplus_{1}\mathbb{R}\to\mathbb{R}$ sends $(x,y)$ to
$\frac{1}{2}(x+y)$;

  * (3)

there exists a unique morphism

$H:(L_{1}([0,1]),1,\gamma)\to(\mathbb{R},1,m)$

in $\mathscr{A}^{1}$,

see [16, Theorem 2.1 and Proposition 2.2]. The map $H$ is a
$\mathds{k}$-linear map from $L_{1}([0,1])$ to $\mathds{k}$ that adheres to
specific criteria enabling its interpretation as a morphism in the category
$\mathscr{A}^{1}$. Significantly, $H$ establishes a fundamental link between
Lebesgue integration on $\mathbb{R}$ and the aforementioned category
$\mathscr{A}^{p}$. Explicitly, for any function $f$ in $L_{1}([0,1])$, the map
is defined as

$H(f)=\int_{0}^{1}f\,\mathrm{d}\mu,$

where $\mu$ denotes the Lebesgue measure on $\mathbb{R}$. This profound
relationship illustrates Lebesgue integrability and integration are not merely
abstract constructs; rather, they naturally emerge from the foundational
principles of Banach spaces. Consequently, it can be logically inferred that
the categorization of Lebesgue integration is inherently connected to, and can
be derived from, the categorization of Banach spaces.

Some authors have been trying to characterize calculus by using the category
theory, including differential algebras/categories [4, 5, 17, 1, 13, 18] and
integral algebras/categories [8, 9]. In this paper, we depart from the
conventional trajectory and propose a novel approach. We extend the domain of
Lebesgue integration beyond the real numbers to the broader framework of
normed modules over normed finite-dimensional $\mathds{k}$-algebras. At the
core of our approach lies the aim to provide a categorical interpretation of
traditional analytical methods, thus paving a novel categorical route to the
underlying principles of Lebesgue integration. To establish this extended
framework, we revisit pivotal results in the category theory and
representation theory. These foundational elements enable us to elegantly
circumvent traditional methodologies, offering a more direct and algebraically
inclined understanding of integrable function spaces and the integration
operator. Our exploration requires a foundational grasp of key concepts and
conclusions in the category theory, representation theory, and the
groundbreaking work of Leinster [16].

Firstly, we introduce functions defined on a finite-dimensional algebra
$\mathit{\Lambda}$, along with the norm defined on $\mathit{\Lambda}$ and any
$\mathit{\Lambda}$-module $M$. It is pertinent to note that all
$\mathit{\Lambda}$-modules considered in this paper are left
$\mathit{\Lambda}$-modules. The specifics of these structures are elaborated
in Subsections 3.1 and 4.1, respectively. A pivotal motivation for us to
introduce normed modules is the pursuit of an integration definition that
transcends the conventional reliance on $L_{p}$ spaces. This approach is
rooted in the understanding that an equivalent definition of $L_{p}$ spaces
can emerge through the integration itself. However, as highlighted by
Leinster, the notion of Lebesgue integrals is intrinsically linked to Banach
spaces. Consequently, our investigation also necessitates considering the
completions of normed finite-dimensional algebras and normed modules, see
Subsections 3.2 and 4.2.

Secondly, for a special subset, denoted $\mathbb{I}_{\mathit{\Lambda}}$, of
$\mathit{\Lambda}$, we construct the category $\mathscr{N}\\!\\!or^{p}$ in
Subsection 5.1. Its object has the form $(N,v,\delta)$, where $N$ is a normed
$\mathit{\Lambda}$-module, $v$ is an element in $V$ satisfying
$|v|\leq\mu(\mathbb{I}_{\mathit{\Lambda}})$, and
$\delta:V^{\oplus_{p}2^{n}}\to V$ is a $\mathit{\Lambda}$-homomorphism sending
$(v,\ldots,v)$ to $v$. The morphism
$h:(N,v,\delta)\to(N^{\prime},v^{\prime},\delta^{\prime})$ is induced by a
special $\mathit{\Lambda}$-homomorphism $V\to V^{\prime}$ satisfying
$h\delta=\delta^{\prime}(h^{\oplus_{p}2^{n}})$. Furthermore, we consider the
full subcategory $\mathscr{A}^{p}$ of $\mathscr{N}\\!\\!or^{p}$ where each
object $(N,v,\delta)$ consists of a Banach $\mathit{\Lambda}$-module $N$, an
element $v\in N$, and a $\mathit{\Lambda}$-homomorphism
$\delta:N^{\oplus_{p}2^{n}}\to N$.

Thirdly, we investigate the set
$\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$ of elementary simple
functions (a special step function defined on $\mathit{\Lambda}$), where
$\tau$ is a homomorphism between two $\mathds{k}$-algebras. We demonstrate its
structure as a $\mathit{\Lambda}$-module (Lemma 4.8). Consequently, we obtain
an object
$(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})$
(Lemma 5.5) in $\mathscr{N}\\!\\!or^{p}$ and an object
$(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})$
in $\mathscr{A}^{p}$, where
$\widehat{\mathbf{S}{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$ is the completion
of $\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$ and
$\widehat{\gamma}_{\xi}$ is induced by $\gamma_{\xi}$.

Fourthly, we prove our main result in Section 6.

######  Theorem 1.1.

(Theorem 6.3 and Remark 6.4) The triple
$(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})$
is an object in $\mathscr{N}\\!\\!or^{p}$. For any object $(N,v,\delta)$ in
$\mathscr{A}^{p}$, there exists a unique morphism

$h\in\mathrm{Hom}_{\mathscr{N}\\!\\!or^{p}}((\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi}),(N,v,\delta))$

such that the diagram

$\textstyle{(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\subseteq}$$\textstyle{(N,v,\delta)}$$\textstyle{(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{h}}$

commutes, where $\widehat{h}$ is given by the completion of
$\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$.

Furthermore, we construct an object
$(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m)$ in $\mathscr{A}^{p}$,
where $m:\mathds{k}^{\oplus_{p}2^{n}}\to\mathds{k}$ is a
$\mathit{\Lambda}$-homomorphism whose definition is given in Section 7. Take
$(N,v,\delta)=(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m)$ in Theorem
1.1, we obtain the following result.

######  Theorem 1.2.

(Theorem 7.6) If $\mathds{k}=(\mathds{k},|\cdot|,\preceq)$ is an extension of
$\mathbb{R}$, then there exists a unique $\mathit{\Lambda}$-homomorphism
$T:\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})\to\mathds{k}$ such that

$T:(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})\to(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m)$

is a morphism in
$\mathrm{Hom}_{\mathscr{N}\\!\\!or^{p}}((\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi}),(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m))$
and the diagram

$\textstyle{(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{\subseteq}$$\textstyle{(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m)}$$\textstyle{(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{T}}$

commutes, where $\widehat{T}$ is the unique morphism lying in
$\mathrm{Hom}_{\mathscr{A}^{p}}((\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi}),(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m))$.
Furthermore, we have the following three properties of $\widehat{T}$ by the
direct limits
$\underrightarrow{\lim}T_{i}:\widehat{T}=\underrightarrow{\lim}E_{i}\to\mathds{k}$
(The definitions of $E_{i}$ and $T_{i}$ are given in Notation 5.3 and Section
7, respectively):

  * (1)

(The formula (7.1)) $\widehat{T}(1)=\mu(\mathbb{I}_{\mathit{\Lambda}})$;

  * (2)

(Lemma 7.1)
$\widehat{T}:\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})\to\mathds{k}$ is
a homomorphism of $\mathit{\Lambda}$-modules;

  * (3)

(Proposition 7.5) $\widehat{T}(|f|)\leq|\widehat{T}(f)|$.

The morphism $\widehat{T}$ provides the categorization for integration, that
is,

$\displaystyle\int_{\mathbb{I}_{\mathit{\Lambda}}}f\mathrm{d}\mu:=\widehat{T}(f).$
(1.1)

The above (1), (2) and (3) show that

$\int_{\mathbb{I}_{\mathit{\Lambda}}}1\mathrm{d}\mu=\mu(\mathbb{I}_{\mathit{\Lambda}}),$
$\displaystyle\int_{\mathbb{I}_{\mathit{\Lambda}}}(\lambda_{1}\cdot
f_{1}+\lambda_{2}\cdot
f_{2})\mu=\lambda_{1}\cdot\int_{\mathbb{I}_{\mathit{\Lambda}}}f_{1}\mu+\lambda_{2}\cdot\int_{\mathbb{I}_{\mathit{\Lambda}}}f_{2}\mu\
\ (\lambda_{1},\lambda_{2}\in\mathit{\Lambda}),$ (1.2)

and

$\bigg{|}\int_{\mathbb{I}_{\mathit{\Lambda}}}f\mathrm{d}\mu\bigg{|}\leq\int_{\mathbb{I}_{\mathit{\Lambda}}}|f|\mathrm{d}\mu,$

respectively.

Finally, we provide two applications for our main results in Section 8. In
Subsection 8.1, we assume $\mathds{k}=\mathbb{R}$,
$(\mathit{\Lambda},\prec,\|\cdot\|_{\mathit{\Lambda}})=(\mathbb{R},\leq,|\cdot|)$,
$B_{\mathbb{R}}=\\{1\\}$,
$\mathfrak{n}:B_{\mathbb{R}}\to\\{1\\}\subseteq\mathbb{R}^{\geq 0}$,
$\mathbb{I}_{\mathbb{R}}=[0,1]$, $\xi=\frac{1}{2}$,
$\kappa_{0}(x)=\frac{x}{2}$, $\kappa_{1}(x)=\frac{x+1}{2}$ and
$\tau=\text{id}_{\mathbb{R}}:\mathbb{R}\to\mathbb{R}$, and let $\mu$ be the
Lebesgue measure. Then (1.1) is a Lebesgue integration

$\int_{\mathbb{I}_{\mathbb{R}}=[0,1]}f\mathrm{d}\mu=\int_{0}^{1}f\mathrm{d}\mu,$

and (1.2) shows that Lebesgue integration is $\mathbb{R}$-linear. This result
provides a categorization of Lebesgue integration. In Subsection 8.2, we show
that the functor $\widehat{T}$ satisfies the Cauchy-Schwarz inequality.

##  2\.  Preliminaries

In this section we recall some concepts in the category theory and
representation theory of algebras. These concepts are familiar to algebraists,
but may not be as familiar to those in the field of analysts.

###  2.1.  Categories and limits

Recall that a category $\mathcal{C}$ consists of three ingredients: a class of
objects, a set $\mathrm{Hom}_{\mathcal{C}}(X,Y)$ of morphisms for any objects
$X$ and $Y$ in $\mathcal{C}$, and the composition
$\mathrm{Hom}_{\mathcal{C}}(X,Y)\times\mathrm{Hom}_{\mathcal{C}}(Y,Z)\to\mathrm{Hom}_{\mathcal{C}}(X,Z)$,
denoted by

$(f:X\to Y,g:Y\to Z)\mapsto gf:X\to Z,$

for any objects $X$, $Y$ and $Z$ in $\mathcal{C}$. These ingredients are
subject to the following axioms:

  * (1)

the $\mathrm{Hom}$ sets are pairwise disjoint;

  * (2)

for any object $X$, the identity morphism $1_{X}:X\to X$ in
$\mathrm{Hom}_{\mathcal{C}}(X,X)$ exists;

  * (3)

the composition is associative: given morphisms
$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{X,}$
we have

$h(gf)=(hg)f.$

Next, we review the limits in the category theory.

######  Definition 2.1 (c.f. [19, Chapter 5, Section 5.2]).

Let $\mathfrak{I}=(\mathfrak{I},\preceq)$ be a partially ordered set, and let
$\mathcal{C}$ be a category. A direct system in $\mathcal{C}$ over
$\mathfrak{I}$ is an ordered pair
$((M_{i})_{i\in\mathfrak{I}},(\varphi_{ij})_{i\prec j})$, where
$(M_{i})_{i\in\mathfrak{I}}$ is an indexed family of objects in $\mathcal{C}$
and $(\varphi_{ij}:M_{i}\to M_{j})_{i\prec j}$ is an indexed family of
morphisms for which $\varphi_{ii}=1_{M_{i}}$ for all $i$, such that the
following diagram

$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{ij}}$$\scriptstyle{\varphi_{ik}}$$\textstyle{M_{k}}$$\textstyle{M_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{jk}}$

commutes whenever $i\prec j\prec k$. Furthermore, for the above direct system
$((M_{i})_{i\in\mathfrak{I}},(\varphi_{ij})_{i\prec j})$, the direct limit
(also called inductive limit or colimit) is an object, say
$\underrightarrow{\lim}M_{i}$, and insertion morphisms
$(\alpha_{i}:M_{i}\to\underrightarrow{\lim}M_{i})_{i\in\mathfrak{I}}$ such
that

  * (1)

$\alpha_{j}\varphi_{ij}=\alpha_{i}$ whenever $i\preceq j$;

  * (2)

for any object $X$ in $\mathcal{C}$ such that there are given morphisms
$f_{i}:M_{i}\to X$ satisfying $f_{j}\varphi_{ij}=f_{i}$ for all $i\preceq j$,
there exists a unique morphism $\theta:\underrightarrow{\lim}M_{i}\to X$
making the following diagram

$\textstyle{\underrightarrow{\lim}M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{(\exists!)}$$\textstyle{X}$$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{f_{i}}$$\scriptstyle{\varphi_{ij}}$$\scriptstyle{(i\preceq
j)}$$\textstyle{M_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{j}}$$\scriptstyle{f_{j}}$

commutes.

######  Example 2.2.

Let $\\{x_{n}\\}_{n\in\mathbb{N}^{+}}$ be a monotonically increasing sequence
of real numbers, and let $\mathbb{R}$ be the partially ordered category
$(\mathbb{R},\leq)$, in which the elements are real numbers and the morphisms
are of the form $\leq_{r_{2}r_{1}}:r_{1}\to r_{2}$ ($r_{2}\leq r_{1}$). If
$\\{x_{n}\\}_{n\in\mathbb{N}^{+}}$ has the limit $x$ in analysis, i.e., for
any $\epsilon>0$, there exists $N\in\mathbb{N}^{+}$ such that
$|x_{n}-x|<\epsilon$ holds for all $n>N$, then
$x=\underrightarrow{\lim}x_{n}$. Indeed, for any $x^{\prime}\in\mathbb{R}$
such that the morphisms $(\alpha_{i}=\leq_{x_{i}x^{\prime}}:x_{i}\to
x^{\prime})_{i\in\mathbb{N}^{+}}$ exist, there is a morphism
$\theta=\leq_{xx^{\prime}}:x\to x^{\prime}$ such that the following diagram

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta=\leq_{xx^{\prime}}}$$\textstyle{x^{\prime}}$$\textstyle{x_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leq_{x_{i}x}}$$\scriptstyle{\leq_{x_{i}x^{\prime}}}$$\scriptstyle{\leq_{x_{i}x_{j}}}$$\textstyle{x_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leq_{x_{j}x}}$$\scriptstyle{\leq_{x_{j}x^{\prime}}}$

commutes. It is clear that the morphism $\theta$ is unique in this example.
Furthermore, $x\leq x^{\prime}$ holds because if $x^{\prime}<x$ then we can
find some $x_{t}$ such that $x_{t}>x^{\prime}$, i.e.,
$\alpha_{t}\in\mathrm{Hom}_{(\mathbb{R},\leq)}(x^{\prime},x_{t})=\varnothing$,
this is a contradiction.

######  Definition 2.3 (c.f. [19, Chapter 5, Section 5.2]).

Let $\mathfrak{I}=(\mathfrak{I},\preceq)$ be a partially ordered set, and let
$\mathcal{C}$ be a category in this subsection. An inverse system in
$\mathcal{C}$ over $\mathfrak{I}$ is an ordered pair
$((M_{i})_{i\in\mathfrak{I}},(\psi_{ij})_{j\prec i})$, where
$(M_{i})_{i\in\mathfrak{I}}$ is an indexed family of objects in $\mathcal{C}$
and $(\psi_{ij}:M_{j}\to M_{i})_{j\prec i}$ is an indexed family of morphisms
for which $\psi_{ii}=1_{M_{i}}$ for all $i$, such that the following diagram

$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{ij}}$$\scriptstyle{\psi_{ik}}$$\textstyle{M_{k}}$$\textstyle{M_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{jk}}$

commutes whenever $i\prec j\prec k$. Furthermore, for the above direct system
$((M_{i})_{i\in\mathfrak{I}},(\psi_{ij})_{j\prec i})$, the inverse limit (also
called projective limit or limit) is an object, say
$\underleftarrow{\lim}M_{i}$, and projects morphisms
$(\alpha_{i}:\underleftarrow{\lim}M_{i}\to M_{i})_{i\in\mathfrak{I}}$ such
that

  * (1)

$\psi_{ji}\alpha_{j}=\alpha_{i}$ whenever $i\preceq j$;

  * (2)

for any object $X$ in $\mathcal{C}$ such that there are given morphisms
$f_{i}:X\to M_{i}$ satisfying $\psi_{ji}f_{j}=f_{i}$ for all $i\preceq j$,
there exists a unique morphism $\vartheta:X\to\underrightarrow{\lim}M_{i}$
making the following diagram

$\textstyle{\underleftarrow{\lim}M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta}$$\scriptstyle{(\exists!)}$$\textstyle{X}$$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{f_{i}}$$\scriptstyle{\psi_{ij}}$$\scriptstyle{(i\preceq
j)}$$\textstyle{M_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{j}}$$\scriptstyle{f_{j}}$

commutes.

######  Example 2.4.

Let $\\{x_{n}\\}_{n\in\mathbb{N}^{+}}$ be a monotonically decreasing sequence
of real numbers, and let $\mathbb{R}$ be the partially ordered category
$(\mathbb{R},\leq)$. If $\\{x_{n}\\}_{n\in\mathbb{N}^{+}}$ has the limit $x$
in analysis, then we have $x=\underleftarrow{\lim}x_{n}$ by a way similar to
that in Example 2.3.

###  2.2.  $\mathds{k}$-algebras and their completions

Let $\mathds{k}$ be a field. In this subsection we recall the definitions of
$\mathds{k}$-algebras and the completions of $\mathds{k}$-algebras. All
concepts in this subsection are parallel to those in [3, Chapter 10, Section
10.1] which extracts some important results about the completions of Abelian
groups.

####  2.2.1.  $\mathds{k}$-algebras

######  Definition 2.5.

A $\mathds{k}$-algebra $A$ defined over $\mathds{k}$ is both a ring and a
$\mathds{k}$-linear space such that

$k(aa^{\prime})=(ka)a^{\prime}=a(ka^{\prime}).$

Let $e_{1}$, $\ldots$, $e_{t}$ be the complete set of primitive orthogonal
idempotents, i.e., any $e_{i}$ is a primitive idempotent and $e_{i}e_{j}=0$
holds for all $i\neq j$. Then $A$ has a decomposition
$A=\bigoplus_{i=1}^{t}Ae_{i}$, where each direct summand $Ae_{i}$ is an
indecomposable left $A$-module. We say $A$ is basic if $Ae_{i}\not\cong
Ae_{j}$ for all $1\leq i\neq j\leq t$.

######  Example 2.6.

The set $\mathbf{M}_{n}(\mathds{k})$ of all $n\times n$ matrices over
$\mathds{k}$, the polynomial ring $\mathds{k}[x_{1},\cdots,x_{n}]$, and the
field $\mathds{k}$ itself are $\mathds{k}$-algebras. Aa $\mathds{k}$-algebra
$\mathit{\Lambda}$ is called finite-dimensional if its $\mathds{k}$-dimension
$\dim_{\mathds{k}}\mathit{\Lambda}$, i.e., the dimension of $\mathit{\Lambda}$
as a $\mathds{k}$-linear space, is finite.

Recall that a quiver is a quadruple
$\mathcal{Q}=(\mathcal{Q}_{0},\mathcal{Q}_{1},\mathfrak{s},\mathfrak{t})$
where $\mathcal{Q}_{0}$ is the set of vertices, $\mathcal{Q}_{1}$ is the set
of arrows, and $\mathfrak{s},\mathfrak{t}:\mathcal{Q}_{1}\to\mathcal{Q}_{0}$
are functions respectively sending each arrow to its starting point and ending
point. Then any vertex $v\in\mathcal{Q}_{0}$ can be seen as a path on
$\mathcal{Q}$ whose length is zero, and any arrow $\alpha\in\mathcal{Q}_{1}$
can be seen as a path on $\mathcal{Q}$ whose length is one. A path $\wp$ of
length $l$, denoted $\ell(\wp)$, is the composition
$\alpha_{l}\cdots\alpha_{2}\alpha_{1}$ of arrows $\alpha_{1}$, $\ldots$,
$\alpha_{l}$, where $\mathfrak{t}(\alpha_{i})=\mathfrak{s}(\alpha_{i+1})$ for
all $1\leq i<l$. Then, naturally, we define the composition of two paths
$\wp_{1}=\alpha_{l}\cdots\alpha_{1}$ and $\wp_{2}=\beta_{\ell}\cdots\beta_{1}$
as:

$\wp_{2}\wp_{1}=\beta_{\ell}\cdots\beta_{1}\alpha_{l}\cdots\alpha_{1}$

provided that the ending point $\mathfrak{t}(\wp_{1})$ of $\wp_{1}$ coincides
with the starting point $\mathfrak{s}(\wp_{2})$ of $\wp_{2}$, otherwise (i.e.,
$\mathfrak{t}(\wp_{1})\neq\mathfrak{s}(\wp_{2})$), then the composition is
defined to be zero. Consequently, let $\mathcal{Q}_{l}$ be the set of all
paths of length $l$. Then
$\mathds{k}\mathcal{Q}:=\mathrm{span}_{\mathds{k}}(\bigcup_{l\geq
0}\mathcal{Q}_{l})$, known as the path algebra of $\mathcal{Q}$, is a
$\mathds{k}$-algebra whose multiplication defined as follows:

$\mathds{k}\mathcal{Q}\times\mathds{k}\mathcal{Q}\to\mathds{k}\mathcal{Q}\
\text{via}\
(k_{1}\wp_{1},k_{2}\wp_{2})\mapsto\begin{cases}k_{1}k_{2}\cdot\wp_{2}\wp_{1},&\text{
if }\mathfrak{t}(\wp_{1})=\mathfrak{s}(\wp_{2});\\\ \ \ \ \ \ \ 0,&\text{
otherwise.}\end{cases}$

The following result shows that we can describe all finite-dimensional
$\mathds{k}$-algebras using quivers.

######  Theorem 2.7 (Gabriel).

For any finite-dimensional $\mathds{k}$-algebra $A$, there is a finite quiver
$\mathcal{Q}$, i.e., the vertex set and arrow set are finite sets, and an
admissible ideal111An admissible ideal $\mathcal{I}$ of
$\mathds{k}\mathcal{Q}$ is an ideal such that
$R_{\mathcal{Q}}^{m}\subseteq\mathcal{I}\subseteq R_{\mathcal{Q}}^{2}$ holds
for some $m\geq 2$, see [2, Chapter II, Section II.1, page 53], where
$R_{\mathcal{Q}}^{t}$ is the ideal of $\mathds{k}\mathcal{Q}$ generated by all
paths of length $\geq t$. $\mathcal{I}$ of $\mathds{k}\mathcal{Q}$ such that
the module category of $A$ is equivalent to that of
$\mathds{k}\mathcal{Q}/\mathcal{I}$. Furthermore, if $A$ is basic, we have
$A\cong\mathds{k}\mathcal{Q}/\mathcal{I}$.

######  Remark 2.8.

We provide a remark for the isomorphism
$A\cong\mathds{k}\mathcal{Q}/\mathcal{I}$ given in Theorem 2.7 here: the
existence of the quiver $\mathcal{Q}$ is unique if $A$ is basic and
$\mathcal{I}$ is admissible; the definition of admissible can be found in [2,
Chapter I, Section I.6].

####  2.2.2.  Topologies on $\mathds{k}$-algebras

Now we recall the topologies of $\mathds{k}$-algebras $A$ (not necessarily
basic or finite-dimensional). Let $\mathfrak{i}(A)$ be the set of all ideals
of $A$, which forms a partially ordered set
$\mathfrak{i}(A)=(\mathfrak{i}(A),\preceq)$ with the partial order defined by
the inclusion. That is, for any $A_{1},A_{2}\in\mathfrak{i}(A)$, we have

$A_{1}\preceq A_{2}\text{ if and only if }A_{1}\subseteq A_{2}.$

Naturally, we have at least one descending chain, denoted by $\mathcal{J}$, of
ideals

$A_{0}=A\succeq A_{1}\succeq A_{2}\succeq\cdots$

We say a subset $U$ of $A$ satisfies the N-condition, if it meets the
following criteria:

  * (N1)

$U$ contains the zero of $A$;

  * (N2)

there exists some $j\in\mathbb{N}$ such that $U\supseteq A_{j}$.

Furthermore, we denote by $\mathfrak{U}_{A}(0)$ the set of all subsets
satisfying the $N$-condition, which forms a partially ordered set with the
partial order “$\preceq$” given by “$\subseteq$”.

######  Lemma 2.9.

The set $\mathfrak{U}_{A}(0)$ is a topology defined on $A$, in other words, it
satisfies the following four conditions.

  * (1)

For any $U\in\mathfrak{U}_{A}(0)$, we have $0\in U$.

  * (2)

$\mathfrak{U}_{A}(0)$ is closed under finite intersection, that is, for any
$U_{1},\ldots,U_{t}\in\mathfrak{U}_{A}(0)$, we have $\bigcap_{1\leq j\leq
t}U_{j}\in\mathfrak{U}_{A}(0)$.

  * (3)

If $U\in\mathfrak{U}_{A}(0)$ and $U\subseteq V\subseteq A$, then
$V\in\mathfrak{U}_{A}(0)$.

  * (4)

If $U\in\mathfrak{U}_{A}(0)$, then there is a set $V\in\mathfrak{U}_{A}(0)$
such that $V\subseteq U$ and $U-y:=\\{u-y\mid u\in U\\}\in\mathfrak{U}_{A}(0)$
for all $y\in V$.

###### Proof.

First, (1) is trivial by the condition (N1).

Second, for arbitrary two subset $U_{1}$ and $U_{2}$, there are $A_{j_{1}}$
and $A_{j_{2}}$ such that $U_{1}\supseteq A_{j_{1}}$ and $U_{2}\supset
A_{j_{2}}$. Then $U_{1}\cap U_{2}\supseteq A_{j_{1}}\cap A_{j_{2}}$. By the
definition of $A_{j}$, we have $A_{j_{1}}\cap
A_{j_{2}}=A_{\min\\{j_{1},j_{2}\\}}$, that is,

$U_{1}\cap U_{2}\supseteq A_{\min\\{j_{1},j_{2}\\}}.$

Since $0\in U_{1}\cap U_{2}$ trivially, we have $U_{1}\cap
U_{2}\in\mathfrak{U}_{A}(0)$. By induction, we obtain (2).

Third, assume $U\in\mathfrak{U}_{A}(0)$ and $U\subseteq V\subseteq A$. By the
definition of $\mathfrak{U}_{A}(0)$, we have $0\in U$ and $U\supseteq A_{j}$
for some $j$. Thus, $0\in V$ and $V\supseteq A_{j}$, so we obtain (3).

Finally, for each $U\in\mathfrak{U}_{A}(0)$, we can find $V$ in the following
way. There exists an index $\jmath$ such that $U\not\supseteq A_{\jmath-1}$
and $U\supseteq A_{\jmath}\supseteq A_{\jmath+1}\supseteq\cdots$. Take
$V=\bigcap_{j\leq\jmath}A_{j}$ ($=A_{\jmath}\subseteq U$). For any $y\in V$,
we have (N1), that is, $0=y-y\in U-y=\\{u-y\mid u\in U\\}$ by $y\in V\subseteq
U$; and have (N2) since $a=(a+y)-y$ holds for any $a\in V$ and $a+y\in V$.
Then we obtain $U-y\in\mathfrak{U}_{A}(0)$, that is, (4) holds. ∎

######  Definition 2.10.

The set $\mathfrak{U}_{A}(0)$ is called the $\mathcal{J}$-topology of $A$.
Furthermore, we can define open sets on $A$.

  * (1)

The subset in $\mathfrak{U}_{A}(0)$ is called a neighborhood of $0$. For any
$U\in\mathfrak{U}_{A}(0)$, the union $\bigcup_{V}V$ of all subsets $V$ given
in Lemma 2.9 (4) is called the interior of $U$ and denote $\bigcup_{V}V$ by
$U^{\circ}$.

  * (2)

A neighborhood $U$ is called open if $U=U^{\circ}$. An open set $O$ defined on
$A$ is one of the following cases:

    * (a)

$O$ equals either $A$ or $\varnothing$;

    * (b)

$O$ is the intersection of a finite number of open neighborhoods;

    * (c)

$O$ is the union of any number of open neighborhoods.

It induces the definitions of continuous homomorphisms of
$\mathds{k}$-algebras.

######  Definition 2.11.

Let $A_{1}$ and $A_{2}$ be two $\mathds{k}$-algebras, and let
$\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ be two descending chains of ideals in
$A_{1}$ and $A_{2}$, respectively. Let $\mathfrak{U}_{A_{1}}(0)$ and
$\mathfrak{U}_{A_{2}}(0)$ be the $\mathcal{J}_{1}$-topology
$\mathcal{J}_{2}$-topology given by $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$,
respectively. A homomorphism $h:A_{1}\to A_{2}$ of $\mathds{k}$-algebras is
called continuous if the preimage of arbitrary open set on $A_{2}$ is an open
set on $A_{1}$.

######  Lemma 2.12.

Let $A$ be a $\mathds{k}$-algebra with a $\mathcal{J}$-topology. Then the
addition $+:A\times A\to A$ and each $\mathds{k}$-linear transformation
$h_{\lambda}:A\to A$ defined by $a\mapsto\lambda a$ $(\lambda\in A)$ are
continuous.

###### Proof.

It is obvious that $\text{id}_{A}=h_{1}:A\to A$ via $a\mapsto a$ is
continuous. The continuity of $h_{\lambda}$ can be given by $\text{id}_{A}$.

Let $\mathcal{J}=$

$A=A_{0}\succeq A_{1}\succeq A_{2}\succeq\cdots.$

For any open neighborhood $U$ of $0$, its preimage is

$+^{-1}(U)=\\{(x_{1},x_{2})\mid x_{1}+x_{2}\in U\\}=:\widetilde{U}.$

We need show that $\widetilde{U}\in\mathfrak{U}_{A\times A}((0,0))$ and
$\widetilde{U}^{\circ}=\widetilde{U}$ in the case for $A\times A$ being a
$\mathds{k}$-algebra, where the descending chain, say $\mathcal{J}_{A\times
A}$, of $A\times A$ is induced by $\mathcal{J}$ as follows.

$A\times A=A_{0}\times A_{0}\succeq A_{1}\times A_{1}\succeq A_{2}\times
A_{2}\succeq\cdots.$

First of all, the zero element of $A\times A$ is $(0,0)$ which satisfies that
$0\in U$ and $0+0=0\in U$, then $(0,0)\in\widetilde{U}$.

Secondly, since $U$ is a neighborhood of $0$, there exists an ideal $A_{j}$ of
$\mathcal{J}$ such that $U\supseteq A_{j}$. Then for any $x_{1},x_{2}\in
A_{j}$, we have $x_{1}+x_{2}\in A_{j}\subseteq U$, that is,
$(x_{1},x_{2})\in\widetilde{U}$. It follows that $A_{j}\times
A_{j}\subseteq\widetilde{U}$. We obtain $\widetilde{U}\in\mathfrak{U}_{A\times
A}((0,0))$.

Thirdly, for any $(y_{1},y_{2})\in\widetilde{U}$, we have $y_{1}+y_{2}\in U$
by the definition of $\widetilde{U}$, then,

$(0,0)=(y_{1}-y_{1},y_{2}-y_{2})\in\widetilde{U}-(y_{1},y_{2})=\\{(x_{1}-y_{1},x_{2}-y_{2})\mid
x_{1}+x_{2}\in U\\},$

that is, (N1) holds. On the other hand, for any $(z_{1},z_{2})\in A_{j}\times
A_{j}$, we have

$(z_{1},z_{2})=((z_{1}+y_{1})-y_{1},(z_{2}+y_{2})-y_{2}).$

Note that $z_{1}+y_{1}+z_{2}+y_{2}=(y_{1}+y_{2})+(z_{1}+z_{2})$ is an element
lying in $U+(z_{1}+z_{2})$. Since $U$ is open, we have

$U+(z_{1}+z_{2})=U^{\circ}-(-(z_{1}+z_{2}))=\\{u+(z_{1}+z_{2})\mid u\in
U\\}\in\mathfrak{U}_{A}(0)$

by Lemma 2.9 (4) and Definition 2.10, that is, $U+(z_{1}+z_{2})$ is a set
satisfying Lemma 2.9 (4). Then

$U^{\circ}=\bigcup_{{V\subseteq U,\ V\text{ satisfies}}\atop{\text{Lemma
\ref{lemm:top} (4)}}}V\supseteq U+(z_{1}+z_{2}),$

and so, we obtain $(y_{1}+y_{2})+(z_{1}+z_{2})\in U+(z_{1}+z_{2})\subseteq
U^{\circ}$, that is, $(y_{1}+y_{2})+(z_{1}+z_{2})\in U$. Thus,
$(z_{1},z_{2})\in\widetilde{U}$. It follows that $A_{j}\times
A_{j}\subseteq\widetilde{U}-(y_{1},y_{2})$, and thus (N2) holds. Therefore,
$\widetilde{U}-(y_{1},y_{2})\in\mathfrak{U}_{A\times A}((0,0))$. In summary,
we have that $\widetilde{U}$ satisfies Lemma 2.9 (4), and so, by Definition
2.10, it is clear that $\widetilde{U}^{\circ}=\widetilde{U}$. ∎

######  Definition 2.13 (c.f. [3, Chapter 10, page 101]).

A topology $\mathds{k}$-algebra is a $\mathds{k}$-algebra equipped with a
topology such that the addition $+:A\times A\to A$ and each
$\mathds{k}$-linear transformation $-h_{1}:A\to A$ via $a\mapsto-a$ are
continuous.

The following result is a consequence of Lemma 2.12.

######  Proposition 2.14.

Given an arbitrary $\mathds{k}$-algebra $A$ and its descending chain
$\mathcal{J}$ of ideals. Then $A$ becomes a topology $\mathds{k}$-algebra with
the $\mathcal{J}$-topology $\mathfrak{U}_{A}(0)$.

In this paper, we refer to $A$ as a $\mathcal{J}$-topological
$\mathds{k}$-algebra.

####  2.2.3.  Completions induced by $\mathcal{J}$-topologies

Assume that $|\cdot|:\mathds{k}\to\mathbb{R}^{\geq 0}$ be a norm defined on
the field $\mathds{k}$ in this subsection, that is, $|\cdot|$ is the map
satisfying

  * (1)

$|k|=0$ if and only if $k=0$;

  * (2)

$|k_{1}k_{2}|=|k_{1}||k_{2}|$ holds for all $k_{1},k_{2}\in\mathds{k}$;

  * (3)

and the triangle inequality $|k_{1}+k_{2}|\leq|k_{1}|+|k_{2}|$ holds for all
$k_{1},k_{2}\in\mathds{k}$.

Then $\\{\mathfrak{B}_{r}=\\{a\in A\mid|a|<r\\}\mid r\in\mathbb{R}^{+}\\}$
induces a standard topology $\mathfrak{U}_{\mathds{k}}(0)$ on $\mathds{k}$
whose element is called the neighborhood of $0\in\mathds{k}$.

Let $A$ be a $\mathcal{J}$-topological $\mathds{k}$-algebra whose dimension is
finite and let $B_{A}=\\{b_{1},\ldots,b_{n}\\}$ be a basis of $A$. Then,
naturally, we can define the Cauchy sequence by the $\mathcal{J}$-topology.
More precisely, a sequence $\\{x_{i}\\}_{i\in\mathbb{N}}$ in $A$ is called a
$\mathcal{J}$-Cauchy sequence if for any $U$, lying in $\mathfrak{U}_{A}(0)$,
containing some subset $\sum_{i=1}^{n}\mathfrak{u}_{i}b_{i}$ of $A$ with
$\mathfrak{u}_{i}\in\mathfrak{U}_{\mathds{k}}(0)$ ($1\leq i\leq n$), there is
$n\in\mathbb{N}$ such that $x_{s}-x_{t}\in U$ holds for all $s,t\geq n$. Two
$\mathcal{J}$-Cauchy sequences $\\{x_{i}\\}_{i\in\mathbb{N}}$ and
$\\{y_{i}\\}_{i\in\mathbb{N}}$ are called equivalent, denoted by
$\\{x_{i}\\}_{i\in\mathbb{N}}$ $\sim$ $\\{y_{i}\\}_{i\in\mathbb{N}}$, if for
any $U\in\mathfrak{U}_{A}(0)$, there is an integer $n\in\mathbb{N}$ such that
$x_{i}-y_{i}\in U$ holds for all $i\geq N$. It is easy to see that “$\sim$” is
an equivalence relation. We use $[\\{x_{i}\\}_{i\in\mathbb{N}}]$ to denote the
equivalence class containing $\\{x_{i}\\}_{i\in\mathbb{N}}$, and use
$\mathfrak{C}_{\mathcal{J}}(A)$ to denote the set of all equivalence classes
of $\mathcal{J}$-Cauchy sequences. Then we have three families of
$A$-homomorphisms:

  * (1)

$(\varphi_{ji}:A/A_{j}\to A/A_{i})_{j\geq i}$, where all $\varphi_{ji}$ are
naturally induced by $A_{i}\supseteq A_{j}$;

  * (2)

$(p_{i}:\mathfrak{C}_{\mathcal{J}}(A)\to A/A_{i})_{i\in\mathbb{N}}$, where
$p_{i}(x_{0},\ldots,x_{i-1},x_{i},x_{i+1},\ldots)=x_{i}$ ($p_{i}$ is called
the $i$-th projection);

  * (3)

$(u_{i}:A/A_{i}\to\mathfrak{C}_{\mathcal{J}}(A))_{i\in\mathbb{N}},$ where
$u_{i}(a+A_{i})=(0,\ldots,\mathop{0}\limits^{i-1},a,\mathop{0}\limits^{i+1},0\ldots)$.

Let $\mathcal{X}$ be the category whose object set is $\\{A/A_{i}\mid
i\in\mathbb{N}\\}\cup\\{\mathfrak{C}_{\mathcal{J}}(A)\\}$ and morphism set is
the collection of all $A$-homomorphisms as above. Then we obtain the following
commutative diagram

$\textstyle{\mathfrak{C}_{\mathcal{J}}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{h}}$$\scriptstyle{(\exists!)}$$\textstyle{A/A_{h}}$$\textstyle{A/A_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{i}}$$\scriptstyle{\varphi_{hi}}$$\scriptstyle{\varphi_{ji}}$$\scriptstyle{(i\leq
j)}$$\textstyle{A/A_{j}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{j}}$$\scriptstyle{\varphi_{hj}}$

It follows from the above construction that the following proposition holds.

######  Proposition 2.15 (c.f. [3, Chapter 10, page 103]).

Using the notations as above, we have

$\underleftarrow{\lim}A/A_{i}\cong\mathfrak{C}_{\mathcal{J}}(A).$

We write $\widehat{A}:=\mathfrak{C}_{\mathcal{J}}(A)$ and call it the
completion of $A$. In particular, if $A=\mathds{k}$, then the descending chain
$\mathcal{J}:$

$A_{0}=\mathds{k}\succeq A_{1}=0$

induces a $\mathcal{J}$-topology

$\mathfrak{U}_{A}(0)=\\{\text{the neighborhood of }0\\}$

of $A$. In this case, the $\mathcal{J}$-Cauchy sequence coincides with the
usual Cauchy sequence.

######  Proposition 2.16.

Let $A$ be a basic finite-dimensional $\mathds{k}$-algebra and let
$\mathcal{J}$ be the descending chain

$A_{0}=A=\mathrm{rad}^{0}A\succeq A_{1}=\mathrm{rad}A\succeq
A_{2}=\mathrm{rad}^{2}A\succeq\cdots$

Then $A$ is complete (in the sense of $\mathcal{J}$-topology) if and only if
$\mathds{k}$ is complete.

###### Proof.

Let $A$ be a basic finite-dimensional $\mathds{k}$-algebra. Then, by Theorem
2.7, there is a finite quiver $\mathcal{Q}$ and an ideal $\mathcal{I}$ of
$\mathds{k}\mathcal{Q}$ such that

$A\cong\mathds{k}\mathcal{Q}/\mathcal{I}=\bigoplus_{l\in\mathbb{N}}\mathds{k}\mathcal{Q}_{l}.$

Thus, up to isomorphism, each element $a\in A$ can be written as
$\sum_{j=1}^{n}k_{j}\wp_{j}$, where $n$ is the dimension of $A$,
$k_{u}\in\mathds{k}$ and $\wp_{u}$ is a path on $\mathcal{Q}$.

Assume that $\mathds{k}$ is complete. Since $A$ is finite-dimensional, we have
$\mathrm{rad}^{l}A=\mathrm{span}_{\mathds{k}}\\{\mathcal{Q}_{i}\mid i\geq
l\\}$. Thus, $\mathrm{rad}^{L+1}A=0$, where $L=\max_{\wp\in\mathcal{Q}_{\geq
0}}\ell(\wp)$, that is,

$\mathcal{J}=$
$A\succeq\mathrm{rad}A\succeq\mathrm{rad}^{2}A\succeq\cdots\mathrm{rad}^{L}A\succeq
0\succeq 0\succeq\cdots$.

Let $\\{x_{i}=\sum_{j=1}^{n}k_{ij}\wp_{j}\\}_{i\in\mathbb{N}}$ be a
$\mathcal{J}$-Cauchy sequence in $A$. Take

$U=\bigg{\\{}\sum\nolimits_{\ell(\wp)=L}k_{\wp}\wp\mid k_{\wp}\text{ lie in
some neighborhood in }\mathfrak{U}_{\mathds{k}}(0)\bigg{\\}}\ \
(\supsetneq\mathrm{rad}^{L+1}A=0).$

Then, there is $N(U)\in\mathbb{N}$ such that

$x_{s}-x_{t}=\sum_{j=1}^{n}(k_{sj}-k_{tj})\wp_{j}\in\mathrm{rad}^{L}A\text{
holds for all }s,t\geq N(U).$

Thus, $k_{sj}-k_{tj}$ lies in some neighborhood in
$\mathfrak{U}_{\mathds{k}}(0)$, and so, for all $i$,
$\\{k_{ij}\\}_{i\in\mathbb{N}}$ is a Cauchy sequence in $\mathds{k}$. Then it
is clear that $A$ is complete.

Conversely, if $A$ is complete, we assume that $\mathds{k}$ is not complete,
and $\widehat{\mathds{k}}$ be the completion of $\mathds{k}$. Then we have a
natural $\mathds{k}$-linear embedding
$\mathfrak{e}:\mathds{k}\to\widehat{\mathds{k}}$ sending $k\in\mathds{k}$ to
$\\{k_{i}\\}_{i\in\mathbb{N}}$, where $k_{1}=k_{2}=\cdots=k$. Then there is a
Cauchy sequence
$\\{x_{i}\\}_{i\in\mathbb{N}}\in\widehat{\mathds{k}}\backslash\mathfrak{e}(\mathds{k})$.
Consider the sequence $\\{x_{i}\cdot\wp\\}_{i\in\mathbb{N}}$ in $A$, where
$\wp\in\mathrm{rad}^{L}A$ is a path of length $L$. Then
$\\{x_{i}\cdot\wp\\}_{i\in\mathbb{N}}$ is a $\mathcal{J}$-Cauchy sequence in
$A$. However, we have
$\\{x_{i}\cdot\wp\\}_{i\in\mathbb{N}}\in\widehat{A}\backslash A$ in this case,
which contradicts that $A$ is complete. ∎

###  2.3.  The total order of $\mathds{k}$-algebras

Recall that a field $\mathds{k}$ equipped with a total order “$\preceq$” is a
ordered field if it satisfies the following four conditions:

  * (1)

for any $a,b\in\mathds{k}$, either $a\preceq b$, $b\preceq a$ or $a=b$ holds;

  * (2)

if $a\preceq b$, $b\preceq c$, then $a\preceq c$;

  * (3)

if $a\preceq b$, then $a+c\preceq b+c$ for all $c\in\mathds{k}$;

  * (4)

if $a\preceq b$ and $0\preceq c$, then $ac\preceq bc$.

In order to give the definition of integration defined on finite-dimensional
$\mathds{k}$-algebra $\mathit{\Lambda}$, we need to assume that $\mathds{k}$
is a field with the total order “$\preceq$”. However, it is well-known that
$\mathds{k}$ might not always be an ordered field, as the case for
$\mathds{k}$ being the complex field $\mathbb{C}$. Interestingly, for our
purposes, the existence of such a total order is not a prerequisite. We only
require that the finite-dimensional $\mathds{k}$-algebra involved in our
study, encompasses certain partially ordered subsets. Specifically, the subset
$\mathbb{I}_{\mathit{\Lambda}}$ outlined in Subsection 3.3 is sufficient. For
the sake of simplicity, we assume that $\mathds{k}$ is fully ordered, although
this assumption does not sacrifice generality. This simplification aids in our
definition of integration within the context of category theory.

######  Remark 2.17.

We provide a remark to show that if $\mathds{k}$ is total ordered, then any
finite-dimensional $\mathds{k}$-algebra $\mathit{\Lambda}$ can be endowed with
a total order. Let $B_{\mathit{\Lambda}}=\\{b_{i}\mid 1\leq i\leq n\\}$ be a
$\mathds{k}$-basis of $\mathit{\Lambda}$. If $B_{\mathit{\Lambda}}$ is totally
ordered (assuming $b_{i}\preceq b_{j}$ if and only if $i\leq j$), then we can
define a total order for $\mathit{\Lambda}$ as follows.

Step 1. For any two arbitrary elements $a,a^{\prime}\in\mathit{\Lambda}$, we
define $a\prec_{p}a^{\prime}$ if and only if
$\varphi(a)\prec_{p}\varphi(a^{\prime})$, where $\varphi$ is a map from
$\mathit{\Lambda}$ to $\mathbb{R}^{\geq 0}$ (for example, $\varphi$ is the
norm $\|\cdot\|_{p}$ defined in Section 3).

Step 2. Assume $a=\sum_{i=1}^{m}k_{i}b_{i}$ and
$a^{\prime}=\sum_{i=1}^{m}k_{i}^{\prime}b_{i}$ ($0\leq m\leq n$) such that
$k_{i}=k_{i}^{\prime}$ holds for all $i<m$. If
$\varphi(a)=\varphi(a^{\prime})$, then we define $a\preceq_{p}a^{\prime}$ if
and only if $k_{m}\preceq k_{m}^{\prime}$.

##  3\.  Normed $\mathds{k}$-algebras

In the sequel, let $\mathit{\Lambda}$ be a finite-dimensional
$\mathds{k}$-algebra with a $\mathds{k}$-basis
$B_{\mathit{\Lambda}}=\\{b_{i}\mid 1\leq i\leq n\\}$. Then any element
$a\in\mathit{\Lambda}$ is of the form $a=\sum_{i=1}^{n}k_{i}b_{i}$. In this
section, we define some algebraic structure for $\mathit{\Lambda}$.

###  3.1.  Norms of $\mathds{k}$-algebras

Take $\mathfrak{n}:B_{\mathit{\Lambda}}\to\mathbb{R}^{+}$ a map from
$\mathit{\Lambda}$ to $\mathbb{R}^{+}$ and, for any $p>1$,
$||\cdot||_{p}:\mathit{\Lambda}\to\mathbb{R}^{\geq 0}$ is the function defined
by

$\displaystyle\|a\|_{p}=\Big{\|}\sum_{i=1}^{n}k_{i}b_{i}\Big{\|}_{p}:=\big{(}(|k_{1}|\mathfrak{n}(b_{1}))^{p}+\cdots+(|k_{n}|\mathfrak{n}(b_{n}))^{p}\big{)}^{\frac{1}{p}}.$
(3.1)

######  Proposition 3.1.

Any triple $(\mathit{\Lambda},\mathfrak{n},\|\cdot\|_{p})$
(=$\mathit{\Lambda}$ for short) is a normed $\mathds{k}$-linear space.

###### Proof.

First of all, for any $a=\sum_{i=1}^{n}k_{i}b_{i}\in\mathit{\Lambda}$, we have
$\|a\|_{p}\geq 0$ because $\mathfrak{n}(b_{i})>0$ and $|k_{i}|\geq 0$ ($1\leq
i\leq n$). In particular, if $\|a\|_{p}=0$, then

$(|k_{1}|\mathfrak{n}(b_{1}))^{p}+\cdots+(|k_{n}|\mathfrak{n}(b_{n}))^{p}=0$.

Since $|k_{i}|\mathfrak{n}(b_{i})\geq 0$ and $\mathfrak{n}(b_{i})>0$ hold for
all $1\leq i\leq n$, we obtain $|k_{i}|\mathfrak{n}(b_{i})=0$, and so
$k_{i}=0$. Thus, $a=\sum_{i=1}^{n}0b_{i}=0$. Then it is easy to see that
$\|a\|_{p}=0$ if and only if $a=0$.

Next, for any $k\in\mathds{k}$ and
$a=\sum_{i=1}^{n}k_{i}b_{i}\in\mathit{\Lambda}$, we have

$\displaystyle\|ka\|_{p}$
$\displaystyle=\|k(k_{1}b_{1}+\cdots+k_{n}b_{n})\|_{p}$
$\displaystyle=\Big{(}\sum_{i=1}^{n}(|kk_{i}|\mathfrak{n}(b_{i}))^{p}\Big{)}^{\frac{1}{p}}=\Big{(}\sum_{i=1}^{n}|k|^{p}(|k_{i}|\mathfrak{n}(b_{i}))^{p}\Big{)}^{\frac{1}{p}}$
$\displaystyle=|k|\Big{(}\sum_{i=1}^{n}(|k_{i}|\mathfrak{n}(b_{i}))^{p}\Big{)}^{\frac{1}{p}}=|k|\cdot\|a\|_{p}.$

Finally, we prove the triangle inequality
$\|a+a^{\prime}\|_{p}\leq\|a\|_{p}+\|a^{\prime}\|_{p}$ for arbitrary two
elements $a=\sum_{i=1}^{n}k_{i}b_{i}$ and
$a^{\prime}=\sum_{i=1}^{n}k_{i}^{\prime}b_{i}$. It can be induced by the
discrete Minkowski inequality
$(\sum_{i=1}^{n}x_{i}^{p})^{\frac{1}{p}}+(\sum_{i=1}^{n}y_{i}^{p})^{\frac{1}{p}}\geq(\sum_{i=1}^{n}(x_{i}+y_{i})^{p})^{\frac{1}{p}}$
as follows:

$\displaystyle\|a\|_{p}+\|a^{\prime}\|_{p}$
$\displaystyle=\Big{(}\sum_{i=1}^{n}(|k_{i}|\mathfrak{n}(b_{i}))^{p}\Big{)}^{\frac{1}{p}}+\Big{(}\sum_{i=1}^{n}(|k_{i}^{\prime}|\mathfrak{n}(b_{i}))^{p}\Big{)}^{\frac{1}{p}}$
$\displaystyle\geq\Big{(}\sum_{i=1}^{n}(|k_{i}|\mathfrak{n}(b_{i})+k_{i}^{\prime}\mathfrak{n}(b_{i}))^{p}\Big{)}^{\frac{1}{p}}$
$\displaystyle=\Big{(}\sum_{i=1}^{n}(|k_{i}+k_{i}^{\prime}|\mathfrak{n}(b_{i}))^{p}\Big{)}^{\frac{1}{p}}=\|a+a^{\prime}\|_{p}.$

Therefore, $(\mathit{\Lambda},\mathfrak{n},\|\cdot\|_{p})$ is a normed space.
∎

######  Definition 3.2.

A normed $\mathds{k}$-algebra is a triple
$(\mathit{\Lambda},\mathfrak{n},\|\cdot\|_{p})$, where
$\mathfrak{n}:B_{\mathit{\Lambda}}\to\mathbb{R}^{+}$ and
$\|\cdot\|_{p}:\mathit{\Lambda}\to\mathbb{R}^{\geq 0}$ are called the normed
basis function and norm of $\mathit{\Lambda}$, respectively.

###  3.2.  Completions of normed $\mathds{k}$-algebras

We can define open neighborhoods $B(0,r)$ of $0$ for any normed
$\mathds{k}$-algebra $(\mathit{\Lambda},\mathfrak{n},\|\cdot\|_{p})$ by

$B(0,r):=\\{a\in\mathit{\Lambda}\mid\|a\|_{p}<r\\}.$

Let $\mathfrak{U}_{\mathit{\Lambda}}^{B}(0)$ be the class of all subsets $U$
of $\mathit{\Lambda}$ satisfying the following conditions.

  * (1)

$U$ is the intersection of a finite number of $B(0,r)$;

  * (2)

$U$ is the union of any number of $B(0,r)$.

Then $\mathfrak{U}_{\mathit{\Lambda}}^{B}(0)$ is a topology, say
$\|\cdot\|_{p}$-topology, defined on $\mathit{\Lambda}$, and we can define the
Cauchy sequence, say $\|\cdot\|_{p}$-Cauchy sequence, by the above topology.

Recall that $\mathit{\Lambda}$ has a $\mathcal{J}$-topology
$\mathfrak{U}_{\mathit{\Lambda}}(0)$ given by the descending chain

$\mathit{\Lambda}=\mathrm{rad}^{0}\mathit{\Lambda}\succeq\mathrm{rad}^{1}\mathit{\Lambda}\succeq\mathrm{rad}^{2}\mathit{\Lambda}\succeq\cdots.$

Thus, we obtain two completions $\widehat{\mathit{\Lambda}}{}^{B}$ and
$\widehat{\mathit{\Lambda}}$ by $\|\cdot\|_{p}$-topology and
$\mathcal{J}$-topology, respectively. The following lemma establishes the
relation between $\widehat{\mathit{\Lambda}}{}^{B}$ and
$\widehat{\mathit{\Lambda}}$.

######  Proposition 3.3.

Let $\mathit{\Lambda}=(\mathit{\Lambda},\mathfrak{n},\|\cdot\|_{p})$ be an
$n$-dimensional normed $\mathds{k}$-algebra with the $\mathcal{J}$-topology
$\mathfrak{U}_{\mathit{\Lambda}}(0)$ given by
$\mathit{\Lambda}=\mathrm{rad}^{0}\mathit{\Lambda}\succeq\mathrm{rad}^{1}\mathit{\Lambda}\succeq\mathrm{rad}^{2}\mathit{\Lambda}\succeq\cdots$
($\|\cdot\|_{p}$ is a norm defined on $\mathit{\Lambda}$ given in Proposition
3.1). Then $\widehat{\mathit{\Lambda}}{}^{B}=\widehat{\mathit{\Lambda}}$.

###### Proof.

Similar to Proposition 2.16 we can show that
$\widehat{\mathit{\Lambda}}{}^{B}=\mathit{\Lambda}$ (i.e., $\mathit{\Lambda}$
is complete) if and only if $\widehat{\mathds{k}}=\mathds{k}$. By using
Proposition 2.16 again, we have that
$\widehat{\mathit{\Lambda}}=\mathit{\Lambda}$ if and only if
$\widehat{\mathds{k}}=\mathds{k}$. Then $\widehat{\mathds{k}}=\mathds{k}$ if
and only if
$\widehat{\mathit{\Lambda}}{}^{B}=\mathit{\Lambda}=\widehat{\mathit{\Lambda}}$,
Equivalently,

$\widehat{\mathit{\Lambda}}{}^{B}=\bigg{(}\widehat{\sum_{i=1}^{n}\mathds{k}b_{i}}\bigg{)}^{B}=\sum_{i=1}^{n}\widehat{\mathds{k}}b_{i}=\widehat{\sum_{i=1}^{n}\mathds{k}b_{i}}=\widehat{\mathit{\Lambda}}.$

∎

######  Remark 3.4.

(1) Note that the norms defined on $A$ is not unique. In Section 4, we will
introduce normed $\mathit{\Lambda}$-modules $N$ over any finite-dimensional
normed $\mathds{k}$-algebra $\mathit{\Lambda}$. In this case, we need a
homomorphism $\tau:\mathit{\Lambda}\to\mathit{\Lambda}^{\prime}$ between two
finite-dimensional normed $\mathds{k}$-algebras $\mathit{\Lambda}$ and
$\mathit{\Lambda}^{\prime}$, and the norms $\|\cdot\|$ and
$\|\cdot\|^{\prime}$ respectively defined on $\mathit{\Lambda}$ and
$\mathit{\Lambda}^{\prime}$ may not necessarily be the form of
$\|\cdot\|_{p}$.

(2) If $A=\mathds{k}$ and $\mathfrak{n}(1)=1$, then the norm $\|\cdot\|_{p}$
given in Proposition 3.1 is the norm $|\cdot|$, i.e,
$\|a\|_{p}=(|a|^{p})^{\frac{1}{p}}=|a|$.

###  3.3.  Elementary simple functions

Denote $\mathbb{I}_{\mathit{\Lambda}}$ by the subset

$\bigg{\\{}\sum_{i=1}^{n}k_{i}b_{i}\mid
k_{i}\in\mathbb{I}\bigg{\\}}\mathop{\longleftrightarrow}\limits^{1-1}\prod_{i=1}^{n}(\mathbb{I}\times\\{b_{i}\\})$

of $\mathit{\Lambda}$. A function defined on $\mathbb{I}_{\mathit{\Lambda}}$
is a map $f:\mathbb{I}_{\mathit{\Lambda}}\to\mathds{k}$ from
$\mathbb{I}_{\mathit{\Lambda}}$ to $\mathds{k}$. Since
$(\mathit{\Lambda},\mathfrak{n},\|\cdot\|_{p})$ is a normed space,
$\mathit{\Lambda}$ is also a topological space induced by the norm
$\|\cdot\|_{p}$, and so is $\mathbb{I}_{\mathit{\Lambda}}$. Thus, we can
define the open set for every subset of $\mathit{\Lambda}$, including
$\mathbb{I}_{\mathit{\Lambda}}$. The function $f$ is called continuous if the
preimage of any open subset of $\mathds{k}$ is an open set of
$\mathbb{I}_{\mathit{\Lambda}}$.

An elementary simple function on $\mathbb{I}_{\mathit{\Lambda}}$ is a finite
sum

$\sum_{i=1}^{t}k_{i}\mathbf{1}_{I_{i}},$

where

  * (1)

for any $1\leq i\leq t$, $k_{i}\in\mathds{k}$;

  * (2)

$I_{i}=I_{i1}\times\cdots\times I_{in}$, and, for any $1\leq j\leq n$,
$I_{ij}$ is a subset of $\mathbb{I}$ which is one of the following forms

    * (a)

$(c_{ij},d_{ij})_{\mathds{k}}:=\\{k\in\mathds{k}\mid c_{ij}\prec k\prec
d_{ij}\\}$,

    * (b)

$[c_{ij},d_{ij})_{\mathds{k}}:=\\{k\in\mathds{k}\mid c_{ij}\preceq k\prec
d_{ij}\\}$,

    * (c)

$(c_{ij},d_{ij}]_{\mathds{k}}:=\\{k\in\mathds{k}\mid c_{ij}\prec k\preceq
d_{ij}\\}$,

    * (d)

$[c_{ij},d_{ij}]_{\mathds{k}}:=\\{k\in\mathds{k}\mid c_{ij}\preceq k\preceq
d_{ij}\\}$,

where $a\preceq c_{ij}\prec d_{ij}\preceq b$;

  * (3)

and $\mathbf{1}_{I_{i}}$ is the function $I_{i}\to\\{1\\}$ such that
$I_{i}\cap I_{j}=\varnothing$ holds for all $1\leq i\neq j\leq t$.

We denote $\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$ by the set of all
elementary simple functions. Then $\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$
is a $\mathds{k}$-linear space, and
$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$ induces the direct sum
$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})^{\oplus 2^{n}}$ whose element can
be seen as the sequence

$\bigg{\\{}f_{(\delta_{1},\ldots,\delta_{n})}\bigg{(}\sum_{i=1}^{n}k_{i}b_{i}\bigg{)}\bigg{\\}}_{(\delta_{1},\ldots,\delta_{n})\in\\{a,b\\}\times\cdots\times\\{a,b\\}}=:\boldsymbol{f}(k_{1},\ldots,k_{n}),$

$\sum_{i=1}^{n}k_{i}b_{i}$ is written as $(k_{1},\ldots,k_{n})$ since
$\\{b_{1}\mid 1\leq i\leq n\\}=B_{\mathit{\Lambda}}$ is the $\mathds{k}$-basis
of $\mathit{\Lambda}$. Then we can characterize
$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$ together with two further pieces
of data: the function
$\mathbf{1}_{\mathbb{I}_{\mathit{\Lambda}}}:\mathbb{I}_{\mathit{\Lambda}}\to\\{1\\}$
($1$ is the identity element of $\mathds{k}$), and the map

$\displaystyle\gamma_{\xi}:\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})^{\oplus
2^{n}}\to\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}}),$ (3.2)

say juxtaposition map, sending $\boldsymbol{f}$ to the function

$\gamma_{\xi}(\boldsymbol{f})(k_{1},\ldots,k_{n})=\sum_{(\delta_{1},\ldots,\delta_{n})}\mathbf{1}_{\kappa_{\delta_{1}}(\mathbb{I})\times\cdots\times\kappa_{\delta_{n}}(\mathbb{I})}\cdot
f_{(\delta_{1},\ldots,\delta_{n})}(\kappa_{\delta_{1}}^{-1}(k_{1}),\ldots,\kappa_{\delta_{n}}^{-1}(k_{n})),$

($k_{1}\neq\xi$, $\ldots$, $k_{n}\neq\xi$),

where $\xi$ is an element with $a\prec\xi\prec b$ such that the order
preserving bijections

$\kappa_{a}:\mathbb{I}\to[a,\xi]_{\mathds{k}}$ and
$\kappa_{b}:\mathbb{I}\to[\xi,b]_{\mathds{k}}$

exist.

######  Example 3.5.

(1) Take $\mathit{\Lambda}$ is the $\mathds{k}$-algebra whose dimension is
$2$, and assume that basis of $\mathit{\Lambda}$ is $\\{b_{1},b_{2}\\}$. Then
$\mathbb{I}_{\mathit{\Lambda}}\cong_{\mathds{k}}[a,b]_{\mathds{k}}b_{1}\times[a,b]_{\mathds{k}}b_{2}$.

For any element

$\boldsymbol{f}=(f_{(a,a)},f_{(b,a)},f_{(a,b)},f_{(b,b)})\in\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})^{\oplus
4},$

where $f_{(\delta_{1},\delta_{2})}:\mathbb{I}_{\mathit{\Lambda}}\to\mathds{k}$
is a function in $\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$ sending each
$k_{1}b_{1}+k_{2}b_{2}$ to the element
$f_{(\delta_{1},\delta_{2})}(k_{1},k_{2})$ in $\mathds{k}$,
$(\delta_{1},\delta_{2})\in\\{a,b\\}\times\\{a,b\\}=\\{(a,a),(b,a),(a,b),(b,b)\\}$,
$\gamma_{\xi}$ juxtapose $f_{(a,a)}$, $f_{(b,a)}$, $f_{(a,b)}$ and $f_{(b,b)}$
into a new function

$\gamma_{\xi}(f_{(a,a)},f_{(b,a)},f_{(a,b)},f_{(b,b)})(k_{1},k_{2})=\tilde{f}_{(a,a)}(k_{1},k_{2})+\tilde{f}_{(b,a)}(k_{1},k_{2})+\tilde{f}_{(a,b)}(k_{1},k_{2})+\tilde{f}_{(b,b)}(k_{1},k_{2})$

as shown in Figure 3.1, where

$\displaystyle\tilde{f}_{(a,a)}(k_{1},k_{2})$
$\displaystyle=\mathbf{1}_{[a,\xi)\times[a,\xi)}\cdot
f_{(a,a)}(\kappa_{a}^{-1}(k_{1}),\kappa_{a}^{-1}(k_{2})),$
$\displaystyle\tilde{f}_{(b,a)}(k_{1},k_{2})$
$\displaystyle=\mathbf{1}_{(\xi,b]\times[a,\xi)}\cdot
f_{(b,a)}(\kappa_{b}^{-1}(k_{1}),\kappa_{a}^{-1}(k_{2})),$
$\displaystyle\tilde{f}_{(a,b)}(k_{1},k_{2})$
$\displaystyle=\mathbf{1}_{[a,\xi)\times(\xi,b]}\cdot
f_{(a,b)}(\kappa_{a}^{-1}(k_{1}),\kappa_{b}^{-1}(k_{2})),$
$\displaystyle\tilde{f}_{(b,b)}(k_{1},k_{2})$
$\displaystyle=\mathbf{1}_{(\xi,b]\times(\xi,b]}\cdot
f_{(b,b)}(\kappa_{b}^{-1}(k_{1}),\kappa_{b}^{-1}(k_{2})).$

$\boldsymbol{\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})^{\oplus
4}}$$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})^{\oplus
4}$$\boldsymbol{\mathbf{S}(\mathbb{I}_{A})}$$\mathbf{S}(\mathbb{I}_{A})$$a$$a$$b$$b$$a$$a$$b$$b$$a$$a$$b$$b$$a$$a$$b$$b$$a$$a$$b$$b$$\oplus$$\oplus$$\oplus$$\oplus$$\bullet$$\xi$$\bullet$$\xi$$\kappa_{a}$$\kappa_{a}$$f_{(a,a)}$$\tilde{f}_{(a,a)}$$\kappa_{b}$$\kappa_{a}$$f_{(b,a)}$$\tilde{f}_{(b,a)}$$\kappa_{b}$$\kappa_{b}$$f_{(b,b)}$$\tilde{f}_{(b,b)}$$\kappa_{a}$$\kappa_{b}$$f_{(b,a)}$$\tilde{f}_{(b,a)}$
Figure 3.1. Juxtaposition map

(2) This example is used to establish the relation between Banach space and
Lebesgue intersections in [16]. Take $\mathds{k}=\mathbb{R}$,
$\mathbb{I}=[0,1]$, $\xi=\frac{1}{2}$, $\mathit{\Lambda}=\mathbb{R}$ and the
order preserving bijections
$\kappa_{0}:\mathbb{I}=[0,1]\to\mathds{k}=\mathbb{R}$ and
$\kappa_{1}:\mathbb{I}=[0,1]\to\mathds{k}=\mathbb{R}$ are given by
$x\mapsto\frac{x}{2}$ and $\frac{1+x}{2}$, respectively. Then
$\mathbf{S}(\mathbb{I}_{\mathbb{R}})=\mathbf{S}([0,1])$ is a normed space
together with two further pieces of data: the function
$\mathbf{1}_{[0,1]}:[0,1]\to\\{1\\}$ and the juxtaposition map

$\gamma_{\frac{1}{2}}:\mathbf{S}([0,1])\oplus\mathbf{S}([0,1])\to\mathbf{S}([0,1])$

sending $(f_{1},f_{2})$ to the following function

$\displaystyle\gamma_{\frac{1}{2}}(f_{1},f_{2})(x)$
$\displaystyle=\mathbf{1}_{\kappa_{0}([0,1))}\cdot
f_{1}({\kappa_{0}^{-1}(x)})+\mathbf{1}_{\kappa_{1}((0,1])}\cdot
f_{1}({\kappa_{1}^{-1}(x)})$
$\displaystyle=\begin{cases}f_{1}(2x)&x\in\kappa_{0}([0,1))=[0,\tfrac{1}{2});\\\
f_{2}(2x-1)&x\in\kappa_{1}((0,1])=(\tfrac{1}{2},1].\end{cases}$

######  Lemma 3.6.

The map $\gamma_{\xi}$ is a $\mathds{k}$-linear map.

###### Proof.

Take $a,b\in\mathds{k}$, $f,g\in\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$ and
let $(k_{i})_{i}$, $\mathbf{1}$ and $(\delta_{i})_{i}$ be the element
$(k_{1},\ldots,k_{n})$ in $\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})^{\oplus
2^{n}}$, the identity function
$\mathbf{1}_{\kappa_{\delta_{1}}(\mathbb{I})\times\cdots\times\kappa_{\delta_{n}}(\mathbb{I})}$
and the $n$-multiple $(\delta_{1}\times\cdots\times\delta_{n})$, respectively.
Then

$\displaystyle\gamma_{\xi}(af+bg)((k_{i})_{i})$
$\displaystyle=\sum_{(\delta_{i})_{i}}\mathbf{1}\cdot(af+bg)_{(\delta_{i})_{i}}((\kappa_{\delta_{i}}^{-1}(k_{i}))_{i})$
$\displaystyle=\sum_{(\delta_{i})_{i}}\big{(}\mathbf{1}\cdot
af_{(\delta_{i})_{i}}((\kappa_{\delta_{i}}^{-1}(k_{i}))_{i})+\mathbf{1}\cdot
bg_{(\delta_{i})_{i}}((\kappa_{\delta_{i}}^{-1}(k_{i}))_{i})\big{)}$
$\displaystyle=a\sum_{(\delta_{i})_{i}}\mathbf{1}\cdot
f_{(\delta_{i})_{i}}((\kappa_{\delta_{i}}^{-1}(k_{i}))_{i})+b\sum_{(\delta_{i})_{i}}\mathbf{1}\cdot
g_{(\delta_{i})_{i}}((\kappa_{\delta_{i}}^{-1}(k_{i}))_{i})$
$\displaystyle=a\gamma_{\xi}(f)((k_{i})_{i})+b\gamma_{\xi}(g)((k_{i})_{i}).$

Thus, $\gamma_{\xi}$ is a $\mathds{k}$-linear map. ∎

######  Example 3.7.

Take $\mathds{k}=\mathbb{R}$, $\mathbb{I}=[0,1]$, $\xi=\frac{1}{2}$,
$\mathit{\Lambda}=\mathbb{R}$ and the order preserving bijections
$\kappa_{0}:\mathbb{I}=[0,1]\to\mathds{k}=\mathbb{R}$ and
$\kappa_{1}:\mathbb{I}=[0,1]\to\mathds{k}=\mathbb{R}$ are given by
$x\mapsto\frac{x}{2}$ and $\frac{x+1}{2}$, respectively. Then
$\mathbf{S}(\mathbb{I}_{\mathbb{R}})=\mathbf{S}([0,1])$ is a normed space
together with two further pieces of data: the function
$\mathbf{1}_{[0,1]}:[0,1]\to\\{1\\}$ and the juxtaposition map

$\gamma_{\frac{1}{2}}:\mathbf{S}([0,1])\oplus\mathbf{S}([0,1])\to\mathbf{S}([0,1])$

sending $(f_{1},f_{2})$ to the following function

$\displaystyle\gamma_{\frac{1}{2}}(f_{1},f_{2})(x)$
$\displaystyle=\mathbf{1}_{\kappa_{0}([0,1])}\cdot
f_{1}(\kappa_{0}^{-1}(x))+\mathbf{1}_{\kappa_{1}([0,1])}\cdot
f_{1}(\kappa_{1}^{-1}(x))$
$\displaystyle=\begin{cases}f_{1}(2x)&x\in\kappa_{0}([0,1])=[0,\tfrac{1}{2});\\\
f_{2}(2x-1)&x\in\kappa_{1}([0,1])=(\tfrac{1}{2},1].\end{cases}$

##  4\.  Normed modules over $\mathds{k}$-algebras

Let $\mathbb{I}$ be a subset of the field $\mathds{k}=(\mathds{k},\preceq)$
with the totally ordered “$\preceq$”. Then $\mathbb{I}$ is also a total
ordered set. For simplicity, we denote by $[x,y]_{\mathds{k}}$ the set of all
elements $k\in\mathds{k}$ with $x\preceq k\preceq y$ in our paper, that is,

$[x,y]_{\mathds{k}}:=\\{k\in\mathds{k}\mid x\preceq k\preceq y\\}.$

In particular, if $x=y$ then $[x,y]_{\mathds{k}}=\\{x\\}=\\{y\\}$ is a set
containing only one element.

In our paper, assume that $\mathds{k}$ and $[a,b]_{\mathds{k}}$ are infinite
sets and consider only the case for $\mathbb{I}=[a,b]_{\mathds{k}}$ with
$a\prec b$ such that there exists an element $\xi$ with $a\prec\xi\prec b$
such that the order preserving bijections
$\kappa_{a}:\mathbb{I}\to[a,\xi]_{\mathds{k}}$ and
$\kappa_{b}:\mathbb{I}\to[\xi,b]_{\mathds{k}}$ exist (for example, the case of
the cardinal number of $\mathbb{I}$ to be either $\aleph_{0}$ or
$\aleph_{1}$). In this section, we introduce the category
$\mathscr{N}\\!\\!or^{p}$, which is used to explore the categorization of
integration.

###  4.1.  Norms of $\mathit{\Lambda}$-modules

Recall that a left $A$-module (=$A$-module for short) over a
$\mathds{k}$-algebra $A$ is a $\mathds{k}$-linear space $V$ with a
$\mathds{k}$-linear map $h:A\to\mathrm{End}_{\mathds{k}}V$ sending $a$ to
$h_{a}$. Thus, $h$ provides a right action $A\times V\to V$, $(a,v)\mapsto
va:=h_{a}(v)$ which satisfies the following properties:

  * (1)

$a(v+v^{\prime})=av+av^{\prime}$ for any $v,v^{\prime}\in V$ and $a\in A$;

  * (2)

$(a+a^{\prime})v=av+a^{\prime}v$ for any $v\in V$ and $a,a^{\prime}\in A$;

  * (3)

$a^{\prime}(av)=(a^{\prime}a)v$ for any $v\in V$ and $a,a^{\prime}\in A$;

  * (4)

$1v=v$ for any $v\in V$;

  * (5)

$(ka)v=k(av)=a(kv)$ for any $v\in V$, $a\in A$ and $k\in\mathds{k}$.

Take $A=\mathit{\Lambda}$ is the normed space with whose norm
$\|\cdot\|_{p}:\mathit{\Lambda}\to\mathbb{R}^{+}$ given by (3.1), where the
$\mathds{k}$-basis of $\mathit{\Lambda}$ is $B_{\mathit{\Lambda}}=\\{b_{i}\mid
1\leq i\leq n=\dim_{\mathds{k}}\mathit{\Lambda}\\}$.

######  Definition 4.1.

Let $\tau:\mathit{\Lambda}\to\mathds{k}$ be a homomorphism between two normed
$\mathds{k}$-algebras $(\mathit{\Lambda},\|\cdot\|_{p})$ and
$(\mathds{k},|\cdot|)$. A $\tau$-normed $\mathit{\Lambda}$-module is a
$\mathit{\Lambda}$-module $M$ with a norm $\|\cdot\|:M\to\mathbb{R}^{\geq 0}$
such that

$\displaystyle\|am\|=|\tau(a)|\cdot\|m\|\text{ holds for all
}a\in\mathds{k}\text{ and }m\in M.$ (4.1)

Thus, each normed $\mathit{\Lambda}$-module can be seen as a triple
$(M,h,\|\cdot\|)$ of the $\mathds{k}$-linear space $M$, the
$\mathds{k}$-linear map $h:M\to\mathrm{End}_{\mathds{k}}M$ and
$\|\cdot\|:M\to\mathbb{R}^{\geq 0}$ a norm.

The norms of $\mathit{\Lambda}$-modules yield that the following fact.

######  Fact 4.2.

  * (1)

Note that $\|\cdot\|_{p}$ defined by (3.1) is the norm of $\mathit{\Lambda}$
as a $\mathds{k}$-linear space. It is easy to see that $\mathit{\Lambda}$ is
also a left $\mathit{\Lambda}$-module, say regular module, where the scalar
multiplication is given by the multiplication
$\mathit{\Lambda}\times\mathit{\Lambda}\to\mathit{\Lambda},(a,x)\mapsto ax$ of
$\mathit{\Lambda}$ as a finite-dimensional $\mathds{k}$-algebra. Thus, it is
natural to ask whether $\|\cdot\|_{p}$ is a norm of $\mathit{\Lambda}$ as a
$\mathit{\Lambda}$-module. Indeed, the norm of $\mathit{\Lambda}$ as a finite-
dimensional $\mathds{k}$-algebra may not be equal to the norm $\|\cdot\|$ of
$\mathit{\Lambda}$ as a regular module. However, if $\mathit{\Lambda}$ as the
left $\mathit{\Lambda}$-module defined by

$\displaystyle\mathit{\Lambda}\times\mathit{\Lambda}\to\mathit{\Lambda},(a,x)\mapsto
a\star x:=\tau(a)x,$ (4.2)

where $\tau(a)x$ is defined by the scalar multiplication of $\mathit{\Lambda}$
as the $\mathds{k}$-linear space ${{}_{\mathds{k}}}\mathit{\Lambda}$, then,
for any $x=\sum_{i=1}^{n}k_{i}b_{i}\in\mathit{\Lambda}$, we obtain

$\displaystyle\|a\star x\|_{p}$
$\displaystyle=\bigg{\|}\tau(a)\sum_{i=1}^{n}k_{i}b_{i}\bigg{\|}_{p}=\bigg{(}\sum_{i=1}^{n}|\tau(a)k_{i}|^{p}\mathfrak{n}(b_{i})^{p}\bigg{)}^{\frac{1}{p}}$
$\displaystyle=|\tau(a)|\bigg{(}\sum_{i=1}^{n}|k_{i}|^{p}\mathfrak{n}(b_{i})^{p}\bigg{)}^{\frac{1}{p}}=|\tau(a)|\|x\|_{p}.$

To be more precise, $\mathit{\Lambda}$ is a
$(\mathit{\Lambda},\mathit{\Lambda})$-bimodule with two norms, and
$\mathit{\Lambda}$ is a normed module satisfying Definition 4.1 when it is
considered as a module defined in (4.2).

  * (2)

For any $\mathit{\Lambda}$-homomorphism $f:M\to N$ of two
$\mathit{\Lambda}$-modules $M$ and $N$, if $M$ and $N$ are normed
$\mathit{\Lambda}$-modules, that is, $M=(M,h_{M},\|\cdot\|_{M})$ and
$N=(N,h_{N},\|\cdot\|_{N})$, then we have

$\|f(am)\|_{N}=\|af(m)\|_{N}=|\tau(a)|\cdot\|f(m)\|_{N}$

######  Example 4.3.

Let

$\mathit{\Lambda}=\left(\begin{matrix}\mathds{k}&0\\\
\mathds{k}&\mathds{k}\end{matrix}\right).$

Then a $\mathds{k}$-basis of $\mathit{\Lambda}$ is
$B_{\mathit{\Lambda}}=\\{\boldsymbol{E}_{11},\boldsymbol{E}_{21},\boldsymbol{E}_{22}\\}$,
where $\boldsymbol{E}_{11}=\big{(}{}_{0}^{1}\ {}_{0}^{0}\big{)}$,
$\boldsymbol{E}_{21}=\big{(}{}_{1}^{0}\ {}_{0}^{0}\big{)}$,
$\boldsymbol{E}_{22}=\big{(}{}_{0}^{0}\ {}_{1}^{0}\big{)}$. Take
$\mathfrak{n}$ be the map $B_{\mathit{\Lambda}}\to\mathbb{R}^{+}$ defined by
$\mathfrak{n}(\boldsymbol{E}_{11})=\mathfrak{n}(\boldsymbol{E}_{21})=\mathfrak{n}(\boldsymbol{E}_{22})=1$,
then for any element $x=\big{(}{}_{k_{21}}^{k_{11}}\ {}_{k_{22}}^{0}\big{)}$
in $\mathit{\Lambda}$, we have
$\|x\|_{p}=(|k_{11}|^{p}+|k_{21}|^{p}+|k_{22}|^{p})^{\frac{1}{p}}$.

There are three indecomposable $\mathit{\Lambda}$-modules up to
$\mathit{\Lambda}$-isomorphism:

$P(1)=\left(\begin{smallmatrix}\mathds{k}&0\\\
\mathds{k}&0\end{smallmatrix}\right)\cong\left(\begin{smallmatrix}0&\mathds{k}\\\
0&\mathds{k}\end{smallmatrix}\right),P(2)=\left(\begin{smallmatrix}0&0\\\
0&\mathds{k}\end{smallmatrix}\right),\text{ and the cokernel
}\mathrm{coker}\left(P(2)\to P(1)\cdot\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right)\right).$

Then each $\mathit{\Lambda}$-module $M$ is isomorphic to the direct sum
$P(1)^{\oplus t_{1}}\oplus P(2)^{\oplus t_{2}}\oplus(P(1)/P(2))^{\oplus
t_{3}}$ for some $t_{1},t_{2},t_{3}\in\mathbb{N}$. Assume that
$M=(M,h_{M},\|\cdot\|_{M})$ and $N=(N,h_{N},\|\cdot\|_{N})$ are two normed
$\mathit{\Lambda}$-modules. Then, naturally, $M\oplus N$ is also a
$\mathit{\Lambda}$-module, where the left $\mathit{\Lambda}$-action is the map

$h_{M}\oplus h_{N}:=\left(\begin{smallmatrix}h_{M}&0\\\
0&h_{N}\end{smallmatrix}\right):\mathit{\Lambda}\times M\oplus N\to M\oplus N$

which sends $(a,({}_{n}^{m}))$ to

$\left(\begin{smallmatrix}h_{M}&0\\\
0&h_{N}\end{smallmatrix}\right)({}_{n}^{m})=\Big{(}{}_{(h_{N})_{a}(n)}^{(h_{M})_{a}(m)}\Big{)}=({{}_{an}^{am}})$.

Furthermore, we can use the $\tau$-norms of $M$ and $N$, that is,
$\|\cdot\|_{M}$ and $\|\cdot\|_{N}$, to define a $\tau$-norm
$\|\cdot\|_{M\oplus N}$ of $M\oplus N$ by

$\|(m,n)\|_{M\oplus
N}:=\left(|k|(\|m\|_{M}^{p}+\|n\|_{N}^{p})\right)^{\frac{1}{p}}$ for given
$k\in\mathds{k}\backslash\\{0\\}$.

Then we have

$\displaystyle\|a(m,n)\|_{M\oplus N}$
$\displaystyle=\left(|k|(\|am\|_{M}^{p}+\|an\|_{N}^{p})\right)^{\frac{1}{p}}=\left(|k|(|\tau(a)|^{p}\|m\|_{M}^{p}+|\tau(a)|^{p}\|n\|_{N}^{p})\right)^{\frac{1}{p}}$
$\displaystyle=|\tau(a)|\left(|k|(\|m\|_{M}^{p}+\|n\|_{N}^{p})\right)^{\frac{1}{p}}=|\tau(a)|\|(m,n)\|_{M\oplus
N}$

for any $a\in\mathit{\Lambda}$.

######  Example 4.4.

The quiver of the $\mathds{k}$-algebra $\mathit{\Lambda}$ given in Example 4.3
is
$\mathcal{Q}=1\mathop{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\longrightarrow}\limits^{\alpha}2$.
By the representation theory all $\mathit{\Lambda}$-modules $M$ can be
represented by
$M_{1}\mathop{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\longrightarrow}\limits^{\varphi_{a}}M_{2}$,
where $M_{1}$ and $M_{2}$ are two $\mathds{k}$-linear spaces and $\varphi_{a}$
is a $\mathds{k}$-linear map. Indeed, the identity element of
$\mathit{\Lambda}$ is
$\boldsymbol{E}=\boldsymbol{E}_{11}+\boldsymbol{E}_{22}$, where
${\boldsymbol{E}_{11},\boldsymbol{E}_{22}}$ are the complete set of primitive
orthogonal idempotents. Thus, $M$, as a $\mathds{k}$-linear space, has a
decomposition $M=\boldsymbol{E}_{11}M\oplus\boldsymbol{E}_{22}M$ (because
$\boldsymbol{E}_{11}\boldsymbol{E}_{22}=0$ yields
$\boldsymbol{E}_{11}M\cap\boldsymbol{E}_{22}M=0$). For any
$a=k_{11}\boldsymbol{E}_{11}+k_{22}\boldsymbol{E}_{22}+k_{21}\boldsymbol{E}_{21}$
and $m\in M$, we have

$\displaystyle am$
$\displaystyle=(k_{11}\boldsymbol{E}_{11}+k_{22}\boldsymbol{E}_{22}+k_{21}\boldsymbol{E}_{21})(\boldsymbol{E}_{11}m+\boldsymbol{E}_{22}m)$
$\displaystyle=k_{11}\boldsymbol{E}_{11}(\boldsymbol{E}_{11}m)+k_{22}\boldsymbol{E}_{22}(\boldsymbol{E}_{22}m)+k_{21}\boldsymbol{E}_{21}(\boldsymbol{E}_{11}m)$
$\displaystyle=k_{11}(h_{M})_{\boldsymbol{E}_{11}}(\boldsymbol{E}_{11}m)+k_{22}(h_{M})_{\boldsymbol{E}_{22}}(\boldsymbol{E}_{22}m)+k_{21}(h_{M})_{\boldsymbol{E}_{21}}(\boldsymbol{E}_{11}m)$
$\displaystyle=(h_{M})_{\boldsymbol{E}_{11}}(k_{11}\boldsymbol{E}_{11}m)+(h_{M})_{k_{22}\boldsymbol{E}_{22}}(\boldsymbol{E}_{22}m)+(h_{M})_{\boldsymbol{E}_{21}}(k_{21}\boldsymbol{E}_{11}m),$
(4.3)

where

  * (a)

$h_{M}:\mathit{\Lambda}\to\mathrm{End}_{\mathds{k}}M$ is a homomorphism of
$\mathds{k}$-algebras sending $a$ to $(h_{M})_{a}$, which satisfies
$\mathbf{1}_{M}=(h_{M})_{\boldsymbol{E}}=(h_{M})_{\boldsymbol{E}_{11}}+(h_{M})_{\boldsymbol{E}_{22}}$;

  * (b)

$(h_{M})_{\boldsymbol{E}_{ii}}=\mathbf{1}_{\boldsymbol{E}_{ii}M}$ ($i=1,2$);

  * (c)

$(h_{M})_{\boldsymbol{E}_{12}}$ is a $\mathds{k}$-linear map from
$\boldsymbol{E}_{11}M$ to $\boldsymbol{E}_{22}M$ (this is equivalent to
(4.3)).

Therefore, we obtain that the representation corresponding to
$M=\boldsymbol{E}_{11}M\oplus\boldsymbol{E}_{22}M$ is

$\boldsymbol{E}_{11}M\mathop{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\longrightarrow}\limits^{\boldsymbol{E}_{21}}\boldsymbol{E}_{22}M.$

Generally,
$M_{1}\mathop{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\longrightarrow}\limits^{\varphi_{a}}M_{2}$
corresponds to the module $M_{1}\oplus M_{2}$, where the
$\mathit{\Lambda}$-action $\mathit{\Lambda}\times M_{1}\oplus M_{2}\to
M_{1}\oplus M_{2}$ is defined by

$\boldsymbol{E}_{11}(m_{1},m_{2})=(m_{1},0)$,
$\boldsymbol{E}_{22}(m_{1},m_{2})=(0,m_{2})$ and
$\boldsymbol{E}_{12}(m_{1},m_{2})=(0,\varphi_{\alpha}(m_{1}))$.

Without loss of generality, for any representation
$M_{1}\mathop{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\longrightarrow}\limits^{\varphi_{a}}M_{2}$
of $\mathcal{Q}$, assume that $M_{1}=\mathds{k}^{\oplus t_{1}}$,
$M_{2}=\mathds{k}^{\oplus t_{2}}$ and $\varphi_{a}\in\mathbf{Mat}_{t_{2}\times
t_{1}}(\mathds{k})$ (up to $\mathit{\Lambda}$-isomorphism), and for any
$i=1,2$, $M_{i}$ is a normed space equipping with the norm
$\|\cdot\|_{M_{i}}:M_{i}=\mathds{k}^{\oplus t_{i}}\to\mathbb{R}^{+}$ sending
$m_{i}=(m_{ij})_{1\leq j\leq t_{i}}$ to
$\left(\sum_{j=1}^{t_{i}}|m_{ij}|^{p}\right)^{\frac{1}{p}}$. Then we can
define a norm $\|\cdot\|_{M_{1}\oplus M_{2}}$ by

$\|(m_{1},m_{2})\|_{M_{1}\oplus
M_{2}}=(|k|(\|m_{1}\|_{M_{1}}^{p}+\|m_{2}\|_{M_{2}}^{p}))^{\frac{1}{p}},$

where $k$ is a given element in $\mathds{k}\backslash\\{0\\}$. The direct sum
“$\oplus$” of $\mathds{k}$-linear spaces is the $p$ powers of the norm
preserving in the case for $k=1$, that is, $\|(m_{1},m_{2})\|_{M_{1}\oplus
M_{2}}^{p}=\|m_{1}\|_{M_{1}}^{p}+\|m_{2}\|_{M_{2}}^{p}$. Furthermore, if
$\|\cdot\|_{M_{1}}$ and $\|\cdot\|_{M_{2}}$ are $\tau$-norms of $M_{1}$ and
$M_{2}$, respectively, then, for any $a\in\mathit{\Lambda}$, we have

$\displaystyle\|a(m_{1},m_{2})\|_{M_{1}\oplus M_{2}}$
$\displaystyle=\big{(}|k|(\|am_{1}\|_{M_{1}}^{p}+\|am_{2}\|_{M_{2}}^{p})\big{)}^{\frac{1}{p}}$
$\displaystyle=\big{(}|k|(|\tau(a)|^{p}\|m_{1}\|_{M_{1}}^{p}+|\tau(a)|^{p}\|m_{2}\|_{M_{2}}^{p})\big{)}^{\frac{1}{p}}$
$\displaystyle=|\tau(a)|\big{(}|k|(\|m_{1}\|_{M_{1}}^{p}+\|m_{2}\|_{M_{2}}^{p})\big{)}^{\frac{1}{p}}$
$\displaystyle=|\tau(a)|\|a(m_{1},m_{2})\|_{M_{1}\oplus M_{2}}.$

###  4.2.  Completions of normed $\mathit{\Lambda}$-modules

Let $N=(N,h,\|\cdot\|)$ be a normed $\mathit{\Lambda}$-module. In this part we
construct its completion. For us, we need the completion of the finite-
dimensional $\mathds{k}$-algebra $\mathit{\Lambda}$. Otherwise, there is at
least one $\mathit{\Lambda}$-module which is not complete, for instance,
$\mathit{\Lambda}$ is a non-complete $\mathit{\Lambda}$-module. Therefore, we
assume that $\mathds{k}$ is complete in this subsection by Propositions 2.16
and 3.3.

Similar to finite-dimensional $\mathds{k}$-algebras, we can define open
neighborhoods $B(0,r)$ of $0$ for any normed $\mathit{\Lambda}$-module
$N=(N,h,\|\cdot\|)$ by

$B(0,r):=\\{x\in N\mid\|x\|<r\\}.$

Let $\mathfrak{U}_{N}^{B}(0)$ be the class of all subsets $U$ of $N$
satisfying the following conditions.

  * (1)

$U$ is the intersection of a finite number of $B(0,r)$;

  * (2)

$U$ is the union of any number of $B(0,r)$.

Then $\mathfrak{U}_{N}^{B}(0)$ is a topology defined on $\mathit{\Lambda}$,
and we can define the Cauchy sequence by the above topology.

######  Lemma 4.5.

Let $\mathfrak{C}^{*}(N)$ be the set of all Cauchy sequences in the normed
$\mathit{\Lambda}$-module $N=(N,h,\|\cdot\|)$. Then $\mathfrak{C}^{*}(N)$ is a
$\mathit{\Lambda}$-module.

###### Proof.

First of all, $\mathfrak{C}^{*}(N)$ is a $\mathds{k}$-linear space whose
addition and $\mathds{k}$-action are given by

$\\{x_{i}\\}_{i\in\mathbb{N}}+\\{y_{i}\\}_{i\in\mathbb{N}}=\\{x_{i}+y_{i}\\}_{i\in\mathbb{N}}\
\
(\forall\\{x_{i}\\}_{i\in\mathbb{N}},\\{y_{i}\\}_{i\in\mathbb{N}}\in\mathfrak{C}^{*}(N))$
$\text{ and }k\\{x_{i}\\}_{i\in\mathbb{N}}=\\{kx_{i}\\}_{i\in\mathbb{N}}\ \
(\forall k\in\mathds{k}),$

respectively. Furthermore, define

$\mathit{\Lambda}\times\mathfrak{C}^{*}(N)\to\mathfrak{C}^{*}(N),\
(a,\\{x_{i}\\}_{i\in\mathbb{N}})\mapsto
a\cdot\\{x_{i}\\}_{i\in\mathbb{N}}:=\\{a\cdot x_{i}\\}_{i\in\mathbb{N}},$

where $a\cdot x_{i}=h_{a}(x_{i})$. Then $\mathfrak{C}^{*}(N)$ is a
$\mathit{\Lambda}$-module. ∎

Two Cauchy sequences $\\{x_{i}\\}_{i\in\mathbb{N}}$ and
$\\{y_{i}\\}_{i\in\mathbb{N}}$ in $N$ are called equivalent, denoted by
$\\{x_{i}\\}_{i\in\mathbb{N}}\sim\\{y_{i}\\}_{i\in\mathbb{N}}$, if for any
$U\in\mathfrak{U}_{N}^{B}(0)$, there is $r\in\mathbb{N}$ such that
$x_{s}-x_{t}\in U$ holds for all $s,t\geq r$. It is easy to see that “$\sim$”
is an equivalence relation. Let $[\\{x_{i}\\}_{i\in\mathbb{N}}]$ be the
equivalent class of Cauchy sequences containing $\\{x_{i}\\}_{i\in\mathbb{N}}$
and let $\mathfrak{C}(N)$ be the set of all equivalent classes. We naturally
obtain a map

$h:\mathfrak{C}^{*}(N)\to\mathfrak{C}(N),\ \
\\{x_{i}\\}_{i\in\mathbb{N}}\mapsto[\\{x_{i}\\}_{i\in\mathbb{N}}].$

We can show that $\mathfrak{C}(N)$ is a $\mathit{\Lambda}$-module by using an
argument similar to that in the proof of Lemma 4.5, and further obtain
$\mathrm{Ker}(h:\mathfrak{C}^{*}(N)\to\mathfrak{C}(N))=[\\{0\\}_{i\in\mathbb{N}}]$.
Thus we have

$\mathfrak{C}(N)\cong\mathfrak{C}^{*}(N)/[\\{0\\}_{i\in\mathbb{N}}].$

Then $\mathfrak{C}(N)$ is complete, and we call it is the completion of $N$.
We use $\widehat{N}$ to denote the completion $\mathfrak{C}(N)$ of $N$. The
$\mathit{\Lambda}$-module $\widehat{N}$ is a normed $\mathit{\Lambda}$-module,
where the norm defined on $\widehat{N}$ is induced by the norm
$\|\cdot\|:N\to\mathbb{R}^{\geq 0}$ defined on $N$.

######  Definition 4.6.

Assume that $\mathit{\Lambda}$ is complete. A normed $\mathit{\Lambda}$-module
$N$ is called a Banach $\mathit{\Lambda}$-module if $\widehat{N}=N$ (i.e. $N$
is complete).

###  4.3.  $\sigma$-algebras and the elementary simple function set
$\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$

Take $\tau$ to be a homomorphism of $\mathds{k}$-algebras
$\tau:\mathit{\Lambda}\to\mathds{k}$. Then the elementary simple function set
$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$ with the above homomorphism
$\tau$, denoted by $\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$, is a
$\mathit{\Lambda}$-module, where the $\mathit{\Lambda}$-action
$\mathit{\Lambda}\times\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})\to\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$
is given by

$(a,f=\sum_{i=1}^{t}k_{i}\mathbf{1}_{I_{i}})\mapsto
af:=\sum_{i=1}^{t}\tau(a)k_{i}\mathbf{1}_{I_{i}}$

because, for all $a\in\mathit{\Lambda}$, $a^{\prime}\in\mathit{\Lambda}$,
$k\in\mathds{k}$,
$f=\sum_{i}k_{i}\mathbf{1}_{I_{i}}\in\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$
and
$f^{\prime}=\sum_{j}k_{j}^{\prime}\mathbf{1}_{I_{j}^{\prime}}\in\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$,
the following conditions are satisfied:

  * (1)

$a(f+f^{\prime})=af+af^{\prime}$ (trivial);

  * (2)

$(a+a^{\prime})f=af+a^{\prime}f$ (trivial);

  * (3)

$(aa^{\prime})f=a(a^{\prime}f)$ because

$\displaystyle(aa^{\prime})f$
$\displaystyle=(aa^{\prime})\sum\nolimits_{i}k_{i}\mathbf{1}_{I_{i}}=\sum\nolimits_{i}\tau(aa^{\prime})k_{i}\mathbf{1}_{I_{i}}=\sum\nolimits_{i}\tau(a)\tau(a^{\prime})k_{i}\mathbf{1}_{I_{i}}$
$\displaystyle=a\sum\nolimits_{i}\tau(a^{\prime})k_{i}\mathbf{1}_{I_{i}}=a(a^{\prime}\sum\nolimits_{i}k_{i}\mathbf{1}_{I_{i}})=a(a^{\prime}f)$

  * (4)

$1f=f$ (trivial);

  * (5)

We have

    * –

$(ka)f=(ka)\sum_{i}k_{i}\mathbf{1}_{I_{i}}=\sum_{i}\tau(ka)(k_{i}\mathbf{1}_{I_{i}})$,

    * –

$k(af)=k(a\sum_{i}k_{i}\mathbf{1}_{I_{i}})=k\sum_{i}\tau(a)k_{i}\mathbf{1}_{I_{i}}=\sum_{i}k(\tau(a)(k_{i}\mathbf{1}_{I_{i}}))$,

    * –

and
$a(kf)=a\sum_{i}k(k_{i}\mathbf{1}_{I_{i}})=\sum_{i}\tau(a)(k(k_{i}\mathbf{1}_{I_{i}}))$.

Since $\tau$ is a homomorphism of $\mathds{k}$-algebras, we have

$\tau(ka)(k_{i}\mathbf{1}_{I_{i}})=k(\tau(a)(k_{i}\mathbf{1}_{I_{i}}))=\sum\nolimits_{i}\tau(a)(k(k_{i}\mathbf{1}_{I_{i}}))=\sum\nolimits_{i}kk_{i}\tau(a)\mathbf{1}_{I_{i}},$

for all $i$. Then $(ka)f=k(af)=a(kf)$.

Now, we introduce a norm for
$\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$ such that it is a normed
$\mathit{\Lambda}$-module. To do this, we firstly recall the definition of
$\sigma$-algebras.

######  Definition 4.7.

Let $S$ be a set and $P(S)$ the power set of $S$, that is, $P(S)$ is the set
of all subsets of $S$. A $\sigma$-algebra is a subset $\mathcal{A}$ of $P(S)$
satisfying the following conditions:

  * (1)

$\varnothing$ and $S$ lie in $\mathcal{A}$;

  * (2)

for any $X\in\mathcal{A}$, the complement set $X^{c}:=S\backslash X$ of $X$
lies in $\mathcal{A}$;

  * (3)

for any $X_{1},\ldots,X_{n}\ldots\in\mathcal{A}$, the union
$\bigcup_{i=1}^{\infty}X_{i}$ is an element in $\mathcal{A}$.

For a class $\mathcal{C}$ of some sets lying in $P(S)$, we call $\mathcal{A}$
is a $\sigma$-algebra generated by $\mathcal{C}$ if $\mathcal{A}$ is the
minimal $\sigma$-algebra containing $\mathcal{C}$.

Let $\Sigma_{\mathds{k}}$ be the $\sigma$-algebra generated by
$\\{(a,b)_{\mathds{k}},[a,b)_{\mathds{k}},(a,b]_{\mathds{k}},[a,b]_{\mathds{k}}\mid
a\preceq b\\}$, and let $\mu:\Sigma_{\mathds{k}}\to\mathbb{R}^{\geq 0}$ be a
measure such that $\mu(\\{k\\})=0$ holds for any $k\in\mathds{k}$, that is,
$\mu$ is a function satisfying the following conditions:

  * (1)

$\mu(\varnothing)=0$;

  * (2)

$\mu(\bigcup_{i\in\mathbb{N}}X_{i})=\sum_{i\in\mathbb{N}}\mu(X_{i})$ holds for
all sets $X_{1},X_{2},\ldots$ satisfying $X_{i}\cap X_{j}=\varnothing$ ($i\neq
j$).

Any two functions $f$ and $g$ in $\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$
are called equivalent if

$\mu(\\{\boldsymbol{k}=(k_{1},\ldots,k_{n})\in\mathds{k}^{\oplus n}\mid
f(\boldsymbol{k})\neq g(\boldsymbol{k})\\})=0$.

The equivalent class containing $f$ is written as $[f]$. Then we obtain an
epimorphism

$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})\to\overline{\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})}:=\\{[f]\mid
f\in\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})\\}$

sending each function to its equivalent classes. It is easy to see that the
kernel of the above epimorphism is $[0]$, then we have

$\overline{\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})}\cong\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})/[0].$

For simplification, we do not differentiate between two equivalent functions
under the above isomorphism. Therefore, we treat
$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$ and the quotient
$\overline{\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})}$ equivalently.

######  Lemma 4.8.

Let $\tau:\mathit{\Lambda}\to\mathds{k}$ be a homomorphism between two
$\mathds{k}$-algebras. Then the $\mathit{\Lambda}$-module
$\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$ with the map

$\|\cdot\|_{p}:\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})\to\mathbb{R}^{\geq
0},\
f=\sum_{i=1}^{t}k_{i}\mathbf{1}_{I_{i}}\mapsto\bigg{(}\sum_{i=1}^{t}(|k_{i}|\mu(I_{i}))^{p}\bigg{)}^{\frac{1}{p}}$

is normed.

###### Proof.

Let $f$ be an arbitrary function lying in
$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$. It is trivial that $\|f\|_{p}$ is
non-negative. Let $a$ be an arbitrary element in $\mathit{\Lambda}$ and assume
$f=\sum_{i=1}^{t}k_{i}\mathbf{1}_{I_{i}}$. We have

$\displaystyle\|af\|_{p}$
$\displaystyle=\Big{\|}\sum_{i=1}^{t}\tau(a)k_{i}\mathbf{1}_{I_{i}}\Big{\|}_{p}=\Big{(}\sum_{i=1}^{t}|\tau(a)k_{i}|^{p}\mu(\mathbf{1}_{I_{i}})^{p}\Big{)}^{\frac{1}{p}}$
$\displaystyle=|\tau(a)|\cdot\Big{(}\sum_{i=1}^{t}|k_{i}|^{p}\mu(\mathbf{1}_{I_{i}})^{p}\Big{)}^{\frac{1}{p}}=|\tau(a)|\cdot\|f\|_{p}$

which satisfies the formula (4.1) for the case
$(\mathit{\Lambda}^{\prime},\|\cdot\|_{2})=(\mathds{k},|\cdot|)$. In
particular, if $\|f\|_{p}=0$, then so is $(|k_{i}|\mu(I_{i}))^{p}=0$ for all
$i$, and we have $|k_{i}|=0$ in the case for $\mu(I_{i})\neq 0$. If
$\mu(I_{j})=0$ holds for some $j\in J$ $(\subseteq\\{1,2,\ldots,t\\})$, then
we have $f=\sum_{j\in J}k_{j}\mathbf{1}_{I_{j}}$. Clearly,

$\mu(\\{x\in\mathbb{I}_{\mathit{\Lambda}}\mid f(x)\neq 0\\})=\sum_{j\in
J}\mu(I_{j})=0,$

that is, $f=0$ in treating $\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$ and the
quotient $\overline{\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})}$ equivalently.
Thus, $\|f\|_{p}=0$ if and only if $f=0$.

Next, we prove the the triangle inequality. For two arbitrary functions
$f=\sum_{i}k_{i}\mathbf{1}_{I_{i}}$ and
$g=\sum_{j}l_{j}\mathbf{1}_{I_{j}^{\prime}}$, we have

$\displaystyle
f+g=\sum\nolimits_{i}k_{i}\mathbf{1}_{I_{i}\backslash\bigcup\nolimits_{j}I_{j}^{\prime}}+\sum\nolimits_{j}l_{j}\mathbf{1}_{I_{j}^{\prime}\backslash\bigcup\nolimits_{i}I_{i}}+\sum_{I_{i}\cap
I_{j}^{\prime}\neq\varnothing}(k_{i}\mathbf{1}_{I_{i}\cap
I_{j}^{\prime}}+l_{j}\mathbf{1}_{I_{i}\cap I_{j}^{\prime}})$ (4.4)

by $I_{i}\cap I_{\imath}=\varnothing$ ($\forall i\neq\imath$) and
$I_{j}^{\prime}\cap I_{\jmath}^{\prime}=\varnothing$ ($\forall j\neq\jmath$).
Then we can compute the norm of $f+g$ by (4.4) as the following formula:

$\|f+g\|_{p}=({R+G+B)^{\frac{1}{p}}},$

where

$\displaystyle R$
$\displaystyle=\sum\nolimits_{i}|k_{i}|^{p}\mu\big{(}I_{i}\big{\backslash}\bigcup\nolimits_{j}I_{j}^{\prime}\big{)}^{p};$
$\displaystyle G$
$\displaystyle=\sum\nolimits_{j}|l_{j}|^{p}\mu\big{(}I_{j}^{\prime}\big{\backslash}\bigcup\nolimits_{i}I_{i}\big{)}^{p};$
$\displaystyle B$ $\displaystyle=\sum_{I_{i}\cap
I_{j}^{\prime}\neq\varnothing}(|k_{i}|^{p}+|l_{j}|^{p})\mu(I_{i}\cap
I_{j}^{\prime})^{p}.$

On the other hand, we have the following inequality by the discrete Minkowski
inequality:

$\displaystyle\|f\|_{p}+\|g\|_{p}=$
$\displaystyle\bigg{(}\sum\nolimits_{i}|k_{i}|^{p}\mu(I_{i})^{p}\bigg{)}^{\frac{1}{p}}+\bigg{(}\sum\nolimits_{j}|l_{i}|^{p}\mu(I_{i}^{\prime})^{p}\bigg{)}^{\frac{1}{p}}$
$\displaystyle\geq$
$\displaystyle\bigg{(}\sum\nolimits_{i}|k_{i}|^{p}\mu(I_{i})^{p}+\sum\nolimits_{j}|l_{i}|^{p}\mu(I_{i}^{\prime})^{p}\bigg{)}^{\frac{1}{p}}=:\mathfrak{S}.$
(4.5)

Since, by the definition of measure, $\mu(X\cup Y)=\mu(X)+\mu(Y)$ holds for
any $X,Y$ with $X\cap Y=\varnothing$, we obtain

$\displaystyle\mu(X\cup Y)^{p}\geq\mu(X)^{p}+\mu(Y)^{p},$ (4.6)

then

$\mu(I_{i})^{p}\geq\mu\big{(}I_{i}\backslash\bigcup\nolimits_{j}I_{j}^{\prime}\big{)}^{p}+\mu\big{(}I_{i}\cap\bigcup\nolimits_{j}I_{j}^{\prime}\big{)}^{p}.$

Thus,

$\displaystyle\sum\nolimits_{i}|k_{i}|^{p}\mu(I_{i})^{p}\geq\ $
$\displaystyle\sum\nolimits_{i}|k_{i}|^{p}\mu\big{(}I_{i}\backslash\bigcup\nolimits_{j}I_{j}^{\prime}\big{)}^{p}+\sum\nolimits_{i}|k_{i}|^{p}\mu\big{(}I_{i}\cap\bigcup\nolimits_{j}I_{j}^{\prime}\big{)}^{p}$
$\displaystyle=\ $ $\displaystyle
R+\sum\nolimits_{i}|k_{i}|^{p}\bigg{(}\mathop{\sum\nolimits_{j}}\limits_{I_{i}\cap
I_{j}^{\prime}\neq\varnothing}\mu(I_{i}\cap I_{j}^{\prime})\bigg{)}^{p}$
$\displaystyle\mathop{\geq}\limits^{(\ref{formula:mu-addi})}$ $\displaystyle
R+\sum_{I_{i}\cap I_{j}^{\prime}\neq\varnothing}|k_{i}|^{p}\mu(I_{i}\cap
I_{j}^{\prime})^{p}.$ (4.7)

Similarly,

$\displaystyle\sum\nolimits_{j}|l_{j}|^{p}\mu(I_{j}^{\prime})^{p}\geq
G+\sum_{I_{j}^{\prime}\cap
I_{i}\neq\varnothing}|l_{j}|^{p}\mu(I_{j}^{\prime}\cap I_{i})^{p}.$ (4.8)

Notice that

$\sum_{I_{i}\cap I_{j}^{\prime}\neq\varnothing}|k_{i}|^{p}\mu(I_{i}\cap
I_{j}^{\prime})^{p}+\sum_{I_{j}^{\prime}\cap
I_{i}\neq\varnothing}|l_{j}|^{p}\mu(I_{j}^{\prime}\cap
I_{i})^{p}=\sum_{I_{i}\cap
I_{j}^{\prime}\neq\varnothing}(|k_{i}|^{p}+|l_{j}|^{p})\mu(I_{i}\cap
I_{j}^{\prime})^{p}=B,$

then (4.3)$+$(4.8) induces $\mathfrak{S}^{p}\geq{R+G+B}$. Thus, the triangle
inequality $\|f\|_{p}+\|g\|_{p}\geq\|f+g\|_{p}$ holds. ∎

##  5\.  Two categories

Let $\dim_{\mathds{k}}\mathit{\Lambda}=n$, and let $N$ be a normed
$\mathit{\Lambda}$-module equipped with two additional pieces of data: an
element $v\in N$ such that $\|v\|\leq\mu(\mathbb{I}_{\mathit{\Lambda}})$, and
a continuous $\mathit{\Lambda}$-homomorphism $\delta:N^{\oplus_{p}2^{n}}\to
N$. Here, $\oplus_{p}$ denotes the direct sum of $2^{n}$
$\mathit{\Lambda}$-modules $X_{1},\ldots,X_{2^{n}}$ with the norm defined as
follows:

$\|\cdot\|_{p}:{\bigoplus_{i=1}^{2^{n}}{}_{p}}\ X_{i}\to\mathbb{R}^{\geq
0},(x_{1},x_{2},\ldots,x_{2^{n}})\mapsto\bigg{(}\bigg{(}\frac{\mu(\mathbb{I})}{\mu(\mathbb{I}_{\mathit{\Lambda}})}\bigg{)}^{n}\sum_{i=1}^{2^{n}}\|x_{i}\|^{p}\bigg{)}^{\tfrac{1}{p}}.$

###  5.1.  The categories $\mathscr{N}\\!\\!or^{p}$ and $\mathscr{A}^{p}$

Let $\mathscr{N}\\!\\!or^{p}$ be a class of triples which are of the form
$(N,v,\delta)$, where $N$ is a normed $\mathit{\Lambda}$-module, $v\in N$ is
an element with $\|v\|_{p}\leq\mu(\mathbb{I}_{\mathit{\Lambda}})$ and
$\delta:N^{\oplus_{p}2^{n}}\to N$ is a $\mathit{\Lambda}$-homomorphism
satisfying $\delta(v,v,\ldots,v)=v$ such that for any Cauchy sequence
$\\{x_{i}\\}_{i\in\mathbb{N}}\in\widehat{N^{\oplus_{p}2^{n}}}\cong\widehat{N}^{\oplus_{p}2^{n}}$,
the commutativity

$\displaystyle\underleftarrow{\lim}\delta(x_{i})=\delta(\underleftarrow{\lim}x_{i})$
(5.1)

of the inverse limit and the $\mathit{\Lambda}$-homomorphism holds. For any
two triples $(N,v,\delta)$ and $(N^{\prime},v^{\prime},\delta^{\prime})$ in
$\mathscr{N}\\!\\!or^{p}$, we define the morphism
$(N,v,\delta)\to(N^{\prime},v^{\prime},\delta^{\prime})$ to be the
$\mathit{\Lambda}$-homomorphism $\theta:N\to N^{\prime}$ with
$\theta(v)=v^{\prime}$ such that the following diagram

$\textstyle{N^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\theta^{\oplus
2^{n}}=\left(\begin{smallmatrix}\theta&&\\\ &\ddots&\\\
&&\theta\end{smallmatrix}\right)_{2^{n}\times
2^{n}}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta\
}$$\textstyle{N^{\prime\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\prime}}$$\textstyle{N^{\prime}}$

commutes, that is, for any $(v_{1},\ldots,v_{2^{n}})\in N^{\oplus_{p}2^{n}}$,
$\theta(\delta(v_{1},\ldots,v_{2^{n}}))=\delta^{\prime}(\theta(v_{1}),\ldots,\theta(v_{2^{n}}))$.
Then it is easy to check that $\mathscr{N}\\!\\!or^{p}$ is a category.

######  Lemma 5.1.

Let

  * (1)

$\xi$ be an element in $\mathbb{I}=[a,b]_{\mathds{k}}$ with $a\prec\xi\prec b$
such that there exists an element $\xi$ with $a\prec\xi\prec b$ such that the
order preserving linear bijections
$\kappa_{a}:\mathbb{I}\to[a,\xi]_{\mathds{k}}$ and
$\kappa_{b}:\mathbb{I}\to[\xi,b]_{\mathds{k}}$ exist,

  * (2)

$\mathbf{1}$ be the identity function
$\mathbf{1}_{\mathbb{I}_{\mathit{\Lambda}}}:\mathbb{I}_{\mathit{\Lambda}}\to\\{1\\}$,

  * (3)

$\gamma_{\xi}$ be the map given in (3.2),

  * (4)

$\tau:\mathit{\Lambda}\to\mathds{k}$ be the homomorphism of
$\mathds{k}$-algebras given in Lemma 4.8.

Then the following statements hold.

  * (a)

$\gamma_{\xi}(\mathbf{1},\mathbf{1},\cdots,\mathbf{1})=\mathbf{1}$;

  * (b)

$\gamma_{\xi}$ is a $\mathit{\Lambda}$-homomorphism.

First, we provide a remark for the above lemma.

######  Remark 5.2.

Indeed,
$(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})$
is an object in the category $\mathscr{N}\\!\\!or^{p}$. However, Lemma 5.1
points out that
$\gamma_{\xi}(\mathbf{1},\mathbf{1},\cdots,\mathbf{1})=\mathbf{1}$ and
$\gamma_{\xi}$ is a $\mathit{\Lambda}$-homomorphism. Thus, we need to show
that the commutativity of the inverse limit and $\gamma_{\xi}$ holds. We will
prove this result in the following content, as shown in Lemma 5.5.

Next, we prove Lemma 5.1.

###### Proof.

(a) We have that $\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$ is a
normed $\mathit{\Lambda}$-module by Lemma 4.8, and $\gamma_{\xi}$ is a
$\mathds{k}$-linear map by Lemma 3.6. The formula
$\gamma_{\xi}(\mathbf{1},\ldots,\mathbf{1})=\mathbf{1}$ can be directly
induced by the definition of $\gamma_{\xi}$.

(b) Take $\lambda\in\mathit{\Lambda}$,
$f\in\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$ and let $(k_{i})_{i}$,
$\mathbf{1}$ and $(\delta_{i})_{i}$ be an arbitrary element
$(k_{1},\ldots,k_{n})$ in $\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})^{\oplus
2^{n}}$, the identity function
$\mathbf{1}_{\kappa_{\delta_{1}}(\mathbb{I})\times\cdots\times\kappa_{\delta_{n}}(\mathbb{I})}$
and the $n$-multiple $(\delta_{1}\times\cdots\times\delta_{n})$, respectively.
Then we have

$\displaystyle\gamma_{\xi}(\lambda\cdot f)((k_{i})_{i})$ $\displaystyle=\ $
$\displaystyle\sum_{(\delta_{i})_{i}}\mathbf{1}\cdot(\tau(\lambda)f)_{(\delta_{i})_{i}}((\kappa_{\delta_{i}}^{-1}(k_{i}))_{i})$
$\displaystyle=\ $ $\displaystyle\tau(\lambda)\gamma_{\xi}(f)((k_{i})_{i})\ \
(\text{similar to Lemma \ref{lemm:gamma linear}})$ $\displaystyle=\ $
$\displaystyle\lambda\cdot\gamma_{\xi}(f)((k_{i})_{i}).$

Thus $\gamma_{\xi}$ is a $\mathit{\Lambda}$-homomorphism. ∎

Let $\mathscr{A}^{p}$ denote a class of triples which are of the form
$(\widehat{N},v,\widehat{\delta})$, where $\widehat{N}$ is a Banach
$\mathit{\Lambda}$-module (see Definition 4.6), $v\in\widehat{N}$ is an
element with $\|v\|_{p}\leq\mu(\mathbb{I}_{\mathit{\Lambda}})$ and
$\delta:\widehat{N}^{\oplus_{p}2^{n}}\to\widehat{N}$ is a
$\mathit{\Lambda}$-homomorphism satisfying $\widehat{\delta}(v,v,\ldots,v)=v$.
Obviously, $\mathscr{A}^{p}$ is a full subcategory of
$\mathscr{N}\\!\\!or^{p}$.

###  5.2.  The triple
$(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})$

Let $(N,v,\delta)$ be an object in $\mathscr{N}\\!\\!or^{p}$ and $\widehat{N}$
the completion of the $\mathit{\Lambda}$-module $N$. Then $\widehat{N}$, as a
$\mathds{k}$-linear space, is a Banach space which is a Banach
$\mathit{\Lambda}$-module. And, naturally, we obtain the
$\mathit{\Lambda}$-homomorphism

$\widehat{\delta}:\widehat{N}^{\oplus_{p}2^{n}}\to\widehat{N}$

induced by the $\mathit{\Lambda}$-homomorphism $\delta$. Furthermore, we have
that $(\widehat{N},v,\widehat{\delta})$ is also an object in
$\mathscr{N}\\!\\!or^{p}$, and there is a naturally embedding morphism

$\mathrm{emb}:(N,v,\delta)\hookrightarrow(\widehat{N},v,\widehat{\delta})$

which is induced by $N\subseteq\widehat{N}$.

######  Notation 5.3.

Keep the notations $\xi=:\xi_{11}$, $\kappa_{a}$, $\kappa_{b}$, $\mathbf{1}$,
$\gamma_{\xi}$ and $\tau$ as in Lemma 5.1. Then $\xi_{11}$ divides
$\mathbb{I}=:\mathbb{I}^{(01)}$ to two subsets
$[a,\xi_{11}]_{\mathds{k}}=:\mathbb{I}^{(11)}$ and
$[\xi_{11},b]_{\mathds{k}}=:\mathbb{I}^{(12)}$. Next, let $\xi_{22}=\xi_{11}$
$(=\xi)$, and denote by $\xi_{21}$ and $\xi_{23}$ the two elements in
$\mathbb{I}_{\mathit{\Lambda}}$ such that

  * •

$a\prec\xi_{21}=\kappa_{a}\kappa_{a}(b)=\kappa_{a}\kappa_{b}(a)=\kappa_{b}\kappa_{a}(a)=\kappa_{a}(\xi_{11})\prec\xi_{22}$;

  * •

$\xi_{22}\prec\xi_{23}=\kappa_{b}\kappa_{b}(a)=\kappa_{b}\kappa_{a}(b)=\kappa_{b}\kappa_{a}(b)=\kappa_{b}{\xi_{11}}\prec
b$.

Then $\mathbb{I}_{\mathit{\Lambda}}$ is divided to four subsets which are of
the form $\mathbb{I}^{(2t)}=[\xi_{2t},\xi_{2\ t+1}]_{\mathds{k}}$ ($0\leq
t\leq 3$) by
$a=\xi_{20}\prec\xi_{21}\prec\xi_{22}\prec\xi_{23}\prec\xi_{24}=b$. Repeating
the above step $t$ times, we obtain a sequence of $2^{t}-1$ elements lying in
$\mathbb{I}_{\mathit{\Lambda}}$

$a=\xi_{t0}\prec\xi_{t1}\prec\xi_{t2}\prec\cdots\prec\xi_{t2^{t}}=b,$

all $2^{t}$ subsets which are of the form $\mathbb{I}^{(t\
s+1)}=[\xi_{ts},\xi_{t\ s+1}]_{\mathds{k}}$, and $2^{t}$ order preserving
bijections $\kappa_{\xi_{ts}}$ from $\mathbb{I}^{(t\ s+1)}$ to
$\mathbb{I}^{(01)}$.

For any family of subsets $(\mathbb{I}^{(u_{i}v_{i})})_{1\leq i\leq n}$
($1\leq v_{i}\leq 2^{u_{i}}$), we denote by $\mathbf{1}_{(u_{i}v_{i})_{i}}$
the function

$\mathbf{1}_{(u_{i}v_{i})_{i}}:=\mathbf{1}_{\mathbb{I}_{\mathit{\Lambda}}}\Big{|}{}_{\prod_{i=1}^{n}\mathbb{I}^{(u_{i}v_{i})}}:\
\mathbb{I}_{\mathit{\Lambda}}\to\\{0,1\\},\
x\mapsto\begin{cases}1,&x\in\prod_{i=1}^{n}\mathbb{I}^{(u_{i}v_{i})};\\\
0,&\text{otherwise},\end{cases}$

where
$\mathbb{I}^{(u_{i}v_{i})}\cong\mathbb{I}^{(u_{i}v_{i})}\times\\{b_{i}\\}\subseteq\mathbb{I}_{\mathit{\Lambda}}$
holds for all $i$ and $B_{\mathit{\Lambda}}=\\{b_{i}\mid 1\leq i\leq n\\}$ is
the $\mathds{k}$-basis of $\mathit{\Lambda}$.

Let $E_{u}$ be the set of all step functions constant on each of
$\prod_{i=1}^{n}\mathbb{I}^{(u_{i}v_{i})}$ ($1\leq v_{i}\leq 2^{u_{i}}$ for
all $i$), that is, every step function in $E_{u}$ is of the form

$\sum_{(u_{i}v_{i})_{i}}k_{(u_{i}v_{i})_{i}}\mathbf{1}_{(u_{i}v_{i})_{i}},$

where each $k_{(u_{i}v_{i})_{i}}$ lies in $\mathds{k}$, the number of summands
is $(2^{u})^{n}=2^{un}$, and each $(u_{i}v_{i})_{i}$ corresponds to the
Cartesian product $\prod_{i=1}^{n}\mathbb{I}^{(u_{i}v_{i})}$. Then it is easy
to check that each $E_{n}$ is a normed submodule of
$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$, and $E_{u}\subseteq E_{u+1}$
because each step function constant on each of $\mathbb{I}^{(uv)}$ is
equivalent to a step function constant on each of $\mathbb{I}^{(u+1\ v)}$.
Thus,

$\mathds{k}\cong E_{0}\subseteq E_{1}\subseteq\ldots\subseteq
E_{t}\subseteq\ldots\subseteq\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})\subseteq\widehat{\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})}.$

Moreover, for any $\mathbb{I}^{(uv)}=[\xi_{u\ v-1},\xi_{uv}]_{\mathds{k}}$, we
have two cases (i) $\xi_{uv}\preceq\xi$ and (ii) $\xi\preceq\xi_{u\ v-1}$ by
the definition of $E_{u}$. Therefore, we obtain a map

$\mathfrak{p}:\\{\mathbb{I}^{(uv)}\mid u\in\mathbb{N}\\}\to\\{a,b\\},\
\mathbb{I}^{(uv)}\mapsto\begin{cases}a,&\mathbb{I}^{(uv)}\text{ lies in case
(i); }\\\ b,&\mathbb{I}^{(uv)}\text{ lies in case (ii).}\end{cases}$

Now we use the above map to prove the following lemma.

######  Lemma 5.4.

The map
$\gamma_{\xi}:\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})^{\oplus_{p}2^{n}}\to\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$
induces the following $\mathds{k}$-linear map

$\gamma_{\xi}:E_{u}^{\oplus_{p}2^{n}}\mathop{\longrightarrow}\limits^{\cong}E_{u+1}$

which is an isomorphism of $\mathit{\Lambda}$-modules.

###### Proof.

The $\mathds{k}$-linear space $E_{u}$ is a $\mathit{\Lambda}$-module, where
$\mathit{\Lambda}\times E_{u}\to E_{u}$ is defined by

$(a,f=\sum_{i}1\cdot\mathbf{1}_{I_{i}})\mapsto a\cdot
f=\sum_{i}\tau(a)\cdot\mathbf{1}_{I_{i}}$.

Then it is easy to see that $\gamma_{\xi}$ is a
$\mathit{\Lambda}$-homomorphism. Since $\mathrm{Ker}(\gamma_{\xi})=0$, we have
$\gamma_{\xi}$ is injective. Next, we prove that it is surjective.

Any step function $f:\mathds{k}^{\oplus n}\to\mathds{k}$ lying in $E_{u+1}$
can be written as

$\displaystyle
f(k_{1},\ldots,k_{n})=\sum_{(u_{i}v_{i})_{i}}f_{i}=\sum_{(\omega_{1},\ldots,\omega_{n})\in\\{a,b\\}\times\cdots\times\\{a,b\\}}f_{(\omega_{1},\ldots,\omega_{n})}$

where

  * •

$f_{i}=k_{(u_{i}v_{i})_{i}}\mathbf{1}_{(u_{i}v_{i})_{i}}$;

  * •

$f_{(\omega_{1},\ldots,\omega_{n})}(k_{1},\ldots,k_{n})=\sum_{\Pi_{i=1}^{n}\mathfrak{p}(\mathbb{I}^{(u_{i}v_{i})})=(\omega_{1},\ldots,\omega_{n})}f_{i},$

thus, the number of all summands of it is $(2^{u})^{n}=2^{un}$;

  * •

the number of all summands of
$\sum_{(\omega_{1},\ldots,\omega_{n})\in\\{a,b\\}\times\cdots\times\\{a,b\\}}$
is $2^{n}$ (thus the number of all summands of $\sum_{(u_{i}v_{i})_{i}}f_{i}$
is $2^{un}\cdot 2^{n}=2^{(u+1)n}$).

Then

$\tilde{f}_{(\omega_{1},\ldots,\omega_{n})}(k_{1},\ldots,k_{n})=f_{(\omega_{1},\ldots,\omega_{n})}(\kappa_{\omega_{1}}^{-1}(k_{1}),\ldots,\kappa_{\omega_{n}}^{-1}(k_{n}))\in
E_{u},$

and $\gamma_{\xi}$ sends
$\\{f_{(\omega_{1},\ldots,\omega_{n})}\\}_{(\omega_{1},\ldots,\omega_{n})\in\\{a,b\\}\times\cdots\times\\{a,b\\}}$
to $f$ by the definition of $\gamma_{\xi}$, see (3.2). We obtain that
$\gamma_{\xi}$ is surjective. Therefore, $\gamma_{\xi}$ is a
$\mathit{\Lambda}$-isomorphism. ∎

By Lemma 5.4, the following result holds.

######  Lemma 5.5.

The commutativity of the inverse limit and the map
$\widehat{\gamma}_{\xi}:\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}}\to\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$
induced by the completion of
$\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})$ holds, that is, for any
sequence $\\{\boldsymbol{f}_{i}\\}_{i\in\mathbb{N}^{+}}$ in
$\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}}$,
if its inverse limit exists, then we have

$\widehat{\gamma}_{\xi}(\underleftarrow{\lim}\boldsymbol{f}_{i})=\underleftarrow{\lim}\widehat{\gamma}_{\xi}(\boldsymbol{f}_{i}).$

Furthermore,
$(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})$
is an object in $\mathscr{N}\\!\\!or^{p}$.

###### Proof.

Since $\gamma_{\xi}$ is a $\mathit{\Lambda}$-isomorphism, it is clear that
$\widehat{\gamma}_{\xi}$ is also a $\mathit{\Lambda}$-isomorphism. Then, the
commutativity of the inverse limit and the map $\widehat{\gamma}_{\xi}$ holds.
Thus, for any sequence $\\{\boldsymbol{f}_{i}\\}_{i\in\mathbb{N}^{+}}$ in
$\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}){}^{\oplus_{p}2^{n}}$, if its
inverse limit exists, then this inverse limit is also an element in
$\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}}$,
and so

$\gamma_{\xi}(\underleftarrow{\lim}\boldsymbol{f}_{i})=\widehat{\gamma}_{\xi}(\underleftarrow{\lim}\boldsymbol{f}_{i})\mathop{=}^{\spadesuit}\underleftarrow{\lim}\widehat{\gamma}_{\xi}(\boldsymbol{f}_{i})=\underleftarrow{\lim}\gamma_{\xi}(\boldsymbol{f}_{i}),$

where $\spadesuit$ holds since $\widehat{\gamma}_{\xi}$ is a
$\mathit{\Lambda}$-isomorphism (see Lemma 5.4). Therefore, by Lemma 5.1,
$(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})$
is an object in $\mathscr{N}\\!\\!or^{p}$. ∎

###  5.3.  $\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$ is a
direct limit

Let $\mathsf{nor}\mathit{\Lambda}$ be the category of normed
$\mathit{\Lambda}$-modules and $\mathit{\Lambda}$-homomorphisms between them.
Then it is easy to check that all $E_{u}$ are objects in
$\mathsf{nor}\mathit{\Lambda}$. Furthermore, for any $u\leq v$, we have a
$\mathit{\Lambda}$-homomorphism $\varphi_{uv}:E_{u}\to E_{v}$ which is induced
by $E_{u}\subseteq E_{v}$. Thus we obtain a direct system
$((E_{i})_{i\in\mathbb{N}},(\varphi_{uv})_{u\leq v})$ in
$\mathsf{nor}\mathit{\Lambda}$ over $\mathbb{N}$. Let
$\mathsf{Ban}\mathit{\Lambda}$ be the category of Banach
$\mathit{\Lambda}$-modules and continuous $\mathit{\Lambda}$-homomorphisms
between them. Then $\mathsf{Ban}\mathit{\Lambda}$ is a full subcategory of
$\mathsf{nor}\mathit{\Lambda}$, and so, naturally, we obtain a direct system
$((E_{i})_{i\in\mathbb{N}},(\varphi_{uv})_{u\leq v})$ in
$\mathsf{Ban}\mathit{\Lambda}$ if $\mathit{\Lambda}$ is a complete
$\mathds{k}$-algebra.

The following lemma establishes the relation between $E_{n}$ and
$\mathbf{S}(\mathbb{I}_{\mathit{\Lambda}})$.

######  Lemma 5.6.

Let $\mathit{\Lambda}$ be a complete $\mathds{k}$-algebra. Consider the
category $\mathsf{Ban}\mathit{\Lambda}$ and take
$(\alpha_{i}:E_{i}\to\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})})_{i\in\mathbb{N}}$,
where every $\alpha_{i}$ is the embedding given by
$E_{i}\subseteq\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$.
Then
$\underrightarrow{\lim}E_{i}\cong\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$.

(Note that we assume that all morphisms in $\mathsf{Ban}(\mathit{\Lambda})$
are continuous, which ensures the commutativity
$\underleftarrow{\lim}\vartheta(x_{i})=\vartheta(\underleftarrow{\lim}x_{i})$
between the inverse limit and any morphism starting from
$\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$.)

###### Proof.

Let $X$ be an arbitrary object in $\mathsf{nor}\mathit{\Lambda}$ such that
there is $(f_{i}:E_{i}\to X)_{i\in\mathbb{N}}$ satisfying
$f_{i}\varphi_{ij}=f_{j}$ for all $i\leq j$. Then we can find the
$\mathit{\Lambda}$-homomorphism
$\theta:\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}\to X$ in
the following way.

For any $x\in\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$,
there exists a sequence $\\{x_{t}\\}_{t\in\mathbb{N}}$ in $\bigcup_{i}E_{i}$
such that $\\{\|x_{t}-x\|_{p}\\}_{t}$ is a monotonically decreasing sequence
of positive real numbers. Then we have
$\underleftarrow{\lim}\\{\|x_{t}-x\|_{p}\\}_{t}=0$ by Example 2.4 which
induces $\underleftarrow{\lim}x_{t}=x.$ Since $\alpha_{i}$, $\alpha_{j}$ and
$\varphi_{ij}$ ($\forall i\leq j$) are $\mathit{\Lambda}$-homomorphisms
induced by “$\subseteq$” (thus they are $\mathds{k}$-linear maps induced by
“$\subseteq$”) and every $x_{t}$ has a preimage in some $E_{u(t)}$, then
$\mathit{\Lambda}$-homomorphisms $(f_{i})_{i\in\mathbb{N}}$ send
$\\{x_{t}\\}_{t\in\mathbb{N}}$ to a sequence
$\\{f_{u(t)}(x_{t})\\}_{t\in\mathbb{N}}$ in $X$. By the completeness of $X$,
$\underleftarrow{\lim}f_{u(t)}(x_{t})\in X$ holds. Define

$\theta(x)=\underleftarrow{\lim}f_{u(t)}(x_{t})=\underleftarrow{\lim}f|_{E_{u(t)}}(x_{t})=\underleftarrow{\lim}f(x_{t}),$

where $f$ is the map $\underleftarrow{\lim}E_{u}\to X$ induced by the direct
limit of $((E_{i})_{i\in\mathbb{N}},(\varphi_{uv})_{u\leq v})$. Then one can
check that $\theta$ is well-defined and is a $\mathit{\Lambda}$-homomorphism
making the following diagram commute.

$\textstyle{\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{X}$$\textstyle{E_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{f_{i}}$$\scriptstyle{\varphi_{ij}}$$\scriptstyle{(i\preceq
j)}$$\textstyle{E_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{j}}$$\scriptstyle{f_{j}}$

Next, we show that the existence of $\theta$ is unique. Assume that
$\theta^{\prime}$ is also a $\mathit{\Lambda}$-homomorphism with
$\theta^{\prime}\alpha_{i}=f_{i}$ for all $i$. Then for any
$x\in\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$, taking the
sequence $\\{x_{t}\\}_{t\in\mathbb{N}}$ in $\bigcup_{i}E_{i}$ satisfying
$\underleftarrow{\lim}x_{t}=x$, we have

$\theta^{\prime}(x)=\theta^{\prime}\big{(}\underleftarrow{\lim}\alpha_{i}(x_{t})\big{)}=\underleftarrow{\lim}\theta^{\prime}(\alpha_{i}(x_{t}))=\underleftarrow{\lim}f_{i}(x_{t})=\underleftarrow{\lim}\theta(\alpha_{i}(x_{t}))=\theta\big{(}\underleftarrow{\lim}\alpha_{i}(x_{t})\big{)}=\theta(x),$

that is, $\theta=\theta^{\prime}$. Therefore, by the definition of direct
limits, we have
$\underrightarrow{\lim}E_{i}\cong\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$.
∎

##  6\.  The $\mathscr{A}^{p}$-initial object in $\mathscr{N}\\!\\!or^{p}$

Let $\mathcal{C}$ be a category. Recall that an object $O$ in $\mathcal{C}$ is
an initial object if for any object $Y$ we have
$\mathrm{Hom}_{\mathcal{C}}(O,Y)$ contains only one morphism. Obviously, if
$\mathcal{C}$ has initial objects, then the initial object is unique up to
isomorphism. Let $\mathcal{D}$ be a full subcategory of $\mathcal{C}$. An
object $C\in\mathcal{C}$ is called a $\mathcal{D}$-initial object if it is a
subobject of the initial object in $\mathcal{D}$, that is, there is an object
${C^{\prime}}$ in $\mathcal{D}$ such that the following conditions hold:

  * •

there is a monomorphism from $C$ to ${C^{\prime}}$;

  * •

for any $D\in\mathcal{D}$, $\mathrm{Hom}_{\mathcal{D}}({C^{\prime}},D)$
contains a unique morphism.

It is trivial that an initial object in $\mathcal{C}$ is a
$\mathcal{C}$-initial object.

Let $\mathit{\Lambda}$ is a complete $\mathds{k}$-algebra. In this section, we
show that
$(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})$
is an $\mathscr{A}^{p}$-initial object in $\mathscr{N}\\!\\!or^{p}$. The proof
is divided to two parts: (1) there is at least one morphism from
$(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})$
to any object in $\mathscr{A}^{p}$; (2) the above morphism is unique.

###  6.1. The existence of morphism from
$(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})$

In this subsection we show that
$\mathrm{Hom}_{\mathscr{A}^{p}}((\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi}),(V,v,\delta))$
is not empty for every object $(V,v,\delta)$ in $\mathscr{A}^{p}$.

######  Lemma 6.1.

For any object $(V,v,\delta)\in\mathscr{A}^{p}$, we have

$\mathrm{Hom}_{\mathscr{A}^{p}}((\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi}),(V,v,\delta))\neq\varnothing.$

###### Proof.

For each $u\in\mathbb{N}$, consider the map $\theta_{n}:E_{n}\to V$ as
follows:

  * (i)

$\theta_{0}:E_{0}\to V$ is a map induced by the $\mathds{k}$-linear map
$\mathds{k}\to V$ sending $1$ to $v$ (note that $E_{0}\cong\mathds{k}$). Then
one can check that $\theta$ is a $\mathit{\Lambda}$-homomorphism.

  * (ii)

$\theta_{u+1}$ is induced by $\theta_{u}$ through the composition

$\theta_{u+1}:=\bigg{(}E_{u+1}\mathop{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\longrightarrow}\limits^{\gamma_{\xi}^{-1}}E_{u}^{\oplus_{p}2^{n}}\mathop{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\longrightarrow}\limits^{\theta_{u}^{\oplus
2^{n}}}V^{\oplus_{p}2^{n}}\mathop{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\longrightarrow}\limits^{\delta}V\bigg{)},$

where the inverse $\gamma_{\xi}^{-1}$ of the map $\gamma_{\xi}$ is given in
Lemma 5.4.

Notice that $\gamma_{\xi}^{-1}(f)\in E_{u-1}$ for any $f\in E_{u}\subseteq
E_{u+1}$, then for the case $u=0$, we have that $f=k\mathbf{1}_{E_{0}}$ is a
constant defined on $E_{0}$, and

$\theta_{1}(f)=\delta(\theta_{0}^{\oplus
2^{n}}(\gamma_{\xi}^{-1}(f)))=\delta(\theta_{0}(k\mathbf{1}_{E_{0}}),\theta_{0}(k\mathbf{1}_{E_{0}}),\ldots,\theta_{0}(k\mathbf{1}_{E_{0}}))=kv,$

that is, $\theta_{1}$ is an extension of $\theta_{0}$. It yields
$\theta_{1}(\mathbf{1}_{E_{1}})=v$ by $\theta_{0}(\mathbf{1}_{E_{0}})=v$ (see
(i)). Furthermore, we can check that $\theta_{u+1}$ is an extension of
$\theta_{u}$ and

$\displaystyle\theta_{u}(\mathbf{1}_{E_{u}})=v\ (\forall u\in\mathbb{N})$
(6.1)

by induction, that is, the following diagram

$\textstyle{\underrightarrow{\lim}E_{i}}$$\textstyle{V}$$\textstyle{E_{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{u}}$$\scriptstyle{\theta_{u}}$$\scriptstyle{\alpha_{u\
u+1}}$$\textstyle{E_{u+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{u+1}}$$\scriptstyle{\theta_{u+1}}$

commutes, where $\alpha_{i}:E_{i}\to\underrightarrow{\lim}E_{i}$ and
$\alpha_{ij}:E_{i}\to E_{j}$ ($i\leq j$) are the embeddings induced by
$E_{i}\subseteq\underrightarrow{\lim}E_{i}$ and $E_{i}\subseteq E_{j}$,
respectively. Then, for any $i\leq j$, there is a unique
$\mathit{\Lambda}$-homomorphism $\theta$ such that the following diagram

$\textstyle{\underrightarrow{\lim}E_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{V}$$\textstyle{E_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{\theta_{i}}$$\scriptstyle{\alpha_{ij}}$$\textstyle{E_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{j}}$$\scriptstyle{\theta_{j}}$

commutes. By Lemma 5.6, we have that
$\theta:\underrightarrow{\lim}E_{i}\cong\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}\to
V$ is a $\mathit{\Lambda}$-homomorphism in
$\mathrm{Hom}_{\mathit{\Lambda}}(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},V)$.

Next, we prove that $\theta$ is a morphism in $\mathscr{N}\\!\\!or^{p}$. First
of all, we have

$\theta(\mathbf{1})=\underleftarrow{\lim}\theta|_{E_{i}}(\mathbf{1}_{E_{i}})=\underleftarrow{\lim}\theta(\alpha_{i}(\mathbf{1}_{E_{i}}))=\underleftarrow{\lim}\theta_{i}(\mathbf{1}_{E_{i}})\mathop{=\\!=\\!=}\limits^{(\ref{formula:theta(1)=v})}\underleftarrow{\lim}v=v.$

In the following, we show that the following diagram

(6.6)

is commutative. Notice that each
$\boldsymbol{f}=(f_{1},\ldots,f_{2^{n}})\in\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}}$
can be seen as the inverse limit $\underleftarrow{\lim}\boldsymbol{f}_{i}$ of
some sequence
$\\{\boldsymbol{f}_{i}=(f_{1i},\ldots,f_{2^{n}i})_{i\in\mathbb{N}}\\}$ in
$\bigcup_{u\in\mathbb{N}}E_{u}^{\oplus_{p}2^{n}}$, where $f_{ji}\in E_{u_{i}}$
($1\leq j\leq 2^{n}$), $u_{i}\in\mathbb{N}$, such that for any $i\leq j$, we
have $u_{i}\leq u_{j}$. Thus, naturally, we need consider the following
diagram
($e_{u_{i}}:E_{u_{i}}\to\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$
is the embedding induced by
$E_{u_{i}}\subseteq\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$):

$\textstyle{E_{u_{i}}^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{\xi}\big{|}_{E_{u_{i}}^{\oplus_{p}2^{n}}}}$$\scriptstyle{\cong}$$\scriptstyle{e_{u_{i}}^{\oplus
2^{n}}}$$\scriptstyle{\theta_{u_{i}}^{\oplus
2^{n}}}$$\textstyle{E_{u_{i}+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{u_{i}+1}}$$\scriptstyle{\theta_{u_{i}}}$$\textstyle{\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{\xi}}$$\scriptstyle{\theta^{\oplus
2^{n}}}$$\textstyle{\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{V^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{V.}$

Since

$\displaystyle\theta(\gamma_{\xi}(\boldsymbol{f}))$
$\displaystyle=\underleftarrow{\lim}\ \theta(\gamma_{\xi}(e^{\oplus
2^{n}}_{u_{i}}(\boldsymbol{f}_{i})))$ $\displaystyle=\underleftarrow{\lim}\
\theta(e_{u_{i}+1}(\gamma_{\xi}\big{|}_{E^{\oplus_{p}2^{n}}}(\boldsymbol{f}_{i})))$
$\displaystyle(\gamma_{\xi}e^{\oplus
2^{n}}_{u_{i}}=e_{u_{i}+1}\gamma_{\xi}\big{|}_{E^{\oplus_{p}2^{n}}})$
$\displaystyle=\underleftarrow{\lim}\
\theta_{u_{i}}(\gamma_{\xi}\big{|}_{E^{\oplus_{p}2^{n}}}(\boldsymbol{f}_{i}))$
$\displaystyle(\theta e=\theta_{u_{i}})$ $\displaystyle=\underleftarrow{\lim}\
\delta(\theta_{u_{i}}^{\oplus 2^{n}}(\boldsymbol{f}_{i}))$
$\displaystyle(\theta_{u_{i}}\gamma_{\xi}\big{|}_{E^{\oplus_{p}2^{n}}}=\delta\theta_{u_{i}}^{\oplus
2^{n}})$ $\displaystyle=\underleftarrow{\lim}\ \delta(\theta^{\oplus
2^{n}}(e^{\oplus 2^{n}}_{u_{i}}(\boldsymbol{f}_{i})))$
$\displaystyle(\theta_{u}^{\oplus 2^{n}}=\theta^{\oplus 2^{n}}e^{\oplus
2^{n}}_{u_{i}})$ $\displaystyle=\delta(\theta^{\oplus
2^{n}}(\underleftarrow{\lim}\ e^{\oplus
2^{n}}_{u_{i}+1}(\boldsymbol{f}_{i})))=\delta(\theta^{\oplus
2^{n}}(\boldsymbol{f})),$ $\displaystyle(\text{by }(\ref{formula:comm}))$

the assertion follows. ∎

###  6.2. The uniqueness of morphism from
$(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})$

Now, we show that, for any object $(V,v,\delta)$ in $\mathscr{A}^{p}$, if the
morphism in the category $\mathscr{A}^{p}$ from
$(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})$
exists, then it is unique.

######  Lemma 6.2.

Let $(V,v,\delta)\in\mathscr{A}^{p}$ be an object in $\mathscr{A}^{p}$. If

$\mathrm{Hom}_{\mathscr{A}^{p}}((\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi}),(V,v,\delta))\neq\varnothing,$

then
$\mathrm{Hom}_{\mathscr{A}^{p}}((\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi}),(V,v,\delta))$
contains a unique morphism.

###### Proof.

Let $\theta$ and $\theta^{\prime}$ be two $\mathit{\Lambda}$-homomorphisms
from
$(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})$
to $(V,v,\delta)$ in $\mathscr{A}^{p}$. Then
$\theta(\mathbf{1})=v=\theta^{\prime}(\mathbf{1})$. Since $\theta$ and
$\theta^{\prime}$ are maps in $\mathscr{A}^{p}$, the square

$\textstyle{E_{u}^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{\xi}|_{E_{u}^{\oplus
2^{n}}}}$$\scriptstyle{\cong}$$\scriptstyle{(\theta|_{E_{u}}-\theta^{\prime}|_{E_{u}})^{\oplus
2^{n}}}$$\textstyle{E_{u+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta|_{E_{u+1}}-\theta^{\prime}|_{E_{u+1}}}$$\textstyle{V^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{V}$

commutes. Then for any $f\in E_{u+1}$, we have

$(\theta|_{E_{u+1}}-\theta^{\prime}|_{E_{u+1}})(f)=(\delta\circ(\theta|_{E_{u}}-\theta^{\prime}|_{E_{u}})^{\oplus
2^{n}}\circ(\gamma_{\xi}|_{E_{u}^{\oplus 2^{n}}})^{-1})(f),$

that is, $\theta|_{E_{u+1}}-\theta^{\prime}|_{E_{u+1}}$ is determined by
$\theta|_{E_{u}}-\theta^{\prime}|_{E_{u}}$. Consider the case for $u=0$, since
$\theta|_{E_{0}}$ and $\theta^{\prime}|_{E_{0}}:E_{0}\to V$ are defined by
$\theta_{0}(\mathbf{1}_{E_{0}})=v$, we have

$(\theta|_{E_{0}}-\theta^{\prime}|_{E_{0}})(k\mathbf{1}_{E_{0}})=k(\theta|_{E_{0}}(\mathbf{1}_{E_{0}})-\theta^{\prime}|_{E_{0}}(\mathbf{1}_{E_{0}}))=k(v-v)=0.$

Therefore $\theta|_{E_{u}}-\theta^{\prime}|_{E_{u}}=0$ for all
$u\in\mathbb{N}$ by induction.

On the other hand, consider the embeddings
$e_{u}:E_{u}\to\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}$ and
$e_{uv}:E_{u}\to E_{v}$ ($u\leq v$) induced by “$\subseteq$” and the direct
system

$\big{(}E_{u}^{\oplus_{p}2^{n}},(e_{u}^{\oplus
2^{n}}:E_{u}^{\oplus_{p}2^{n}}\to\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}})_{u\in\mathbb{N}}\big{)},$

we have the following commutative diagram

$\textstyle{\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}}}$$\textstyle{V}$$\textstyle{E_{i}^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{i}^{\oplus
2^{n}}}$

$\begin{smallmatrix}\theta|_{E_{i}}-\theta^{\prime}|_{E_{i}}\\\ \ \\\ (=0)\\\
\ \end{smallmatrix}$

$\scriptstyle{e_{ij}^{\oplus
2^{n}}}$$\textstyle{E_{ij}^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{j}^{\oplus
2^{n}}}$$\scriptstyle{\theta|_{E_{j}}-\theta^{\prime}|_{E_{j}}=0.}$

Since

$\underrightarrow{\lim}E_{i}^{\oplus_{p}2^{n}}\cong(\underrightarrow{\lim}E_{i})^{\oplus_{p}2^{n}}\cong\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}},$

there is a unique $\mathit{\Lambda}$-homomorphism
$\phi:\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}}\to
V$ such that the following diagram

$\textstyle{\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}{}^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{V}$$\textstyle{E_{i}^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{i}^{\oplus
2^{n}}}$

$\begin{smallmatrix}\theta|_{E_{i}}-\theta^{\prime}|_{E_{i}}\\\ \ \\\ (=0)\\\
\ \end{smallmatrix}$

$\scriptstyle{e_{ij}^{\oplus
2^{n}}}$$\textstyle{E_{ij}^{\oplus_{p}2^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{j}^{\oplus
2^{n}}}$$\scriptstyle{\theta|_{E_{j}}-\theta^{\prime}|_{E_{j}}=0}$

commutes. Since $(\theta-\theta^{\prime})e_{u}^{\oplus
2^{n}}=\theta|_{E_{i}}-\theta^{\prime}|_{E_{j}}$, we know that the case for
$\phi=\theta-\theta^{\prime}$ making the above diagram commute. On the other
hand, the case for $\phi=0$ making the above diagram commute. Thus
$\theta-\theta^{\prime}=0$ and $\theta=\theta^{\prime}$. ∎

###  6.3. The $\mathscr{A}^{p}$-initial object in $\mathscr{N}\\!\\!or^{p}$

Now, we can prove the following result of this paper.

######  Theorem 6.3.

The triple
$(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})$
is an $\mathscr{A}^{p}$-initial object in $\mathscr{N}\\!\\!or^{p}$.

###### Proof.

For any object $(V,v,\delta)$ in $\mathscr{A}^{p}$, the existence of morphisms
in
$\mathrm{Hom}_{\mathscr{A}^{p}}((\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi}),$
$(V,v,\delta))$ is proved in Lemma 6.1, and its uniqueness is proved in Lemma
6.2. Thus, the triple
$(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})$,
as an object in $\mathscr{A}^{p}$, is an initial object in $\mathscr{A}^{p}$.
It follows that
$(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\widehat{\gamma}_{\xi})$
is an $\mathscr{A}^{p}$-initial object in $\mathscr{N}\\!\\!or^{p}$. ∎

We give a remark for Theorem 6.3.

######  Remark 6.4.

For any object $(V,v,\delta)$ in $\mathscr{A}^{p}$, there is a unique morphism

$h:(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})\to(V,v,\delta)$

in $\mathscr{N}\\!\\!or^{p}$, which can be extended to
$\widehat{h}:(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})\to(V,v,\delta)$,
In other words, if there exists a morphism $h$ making the diagram

$\textstyle{(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\subseteq}$$\textstyle{(V,v,\delta)}$$\textstyle{(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{h}}$

commute, then the existence of $h$ is guaranteed to be unique.

##  7\.  The categorization of Lebesgue integration

Take $\mathds{k}=(\mathds{k},|\cdot|,\preceq)$ to be an extension of
$\mathbb{R}$. Recall the symbols given in Notation 5.3, any step function $f$
in $E_{u}$ can be written as

$f=\sum_{(u_{i}v_{i})_{i}}k_{(u_{i}v_{i})_{i}}\mathbf{1}_{(u_{i}v_{i})_{i}}.$

We define the map $T_{u}:E_{u}\to\mathds{k}$ by

$\displaystyle
T_{u}(f)=\sum_{(u_{i}v_{i})_{i}}k_{(u_{i}v_{i})}\mu\bigg{(}\prod\nolimits_{i}\mathbb{I}^{(u_{i}v_{i})}\bigg{)}$
(7.1)

(note that if all coefficients $k_{(u_{i}v_{i})}$ equal to $1$, then
$T_{u}(f)=\mu(E_{u})$).

Then the $\mathit{\Lambda}$-isomorphism $\gamma_{\xi}$ shown in Lemma 5.4
points out the following fact: there is a map
$m_{u}:\mathds{k}^{\oplus_{p}2^{n}}\to\mathds{k}$ such that the following
diagram

(7.6)

commutes. Indeed, for the function
$f_{k}=\frac{k}{\mu(\mathbb{I}_{\mathit{\Lambda}})}\mathbf{1}_{\mathbb{I}_{\mathit{\Lambda}}}$
with $k\in\mathds{k}$, we have

$T_{u}(f_{k})=T_{u}(\frac{k}{\mu(\mathbb{I}_{\mathit{\Lambda}})}\mathbf{1}_{\mathbb{I}_{\mathit{\Lambda}}})=\frac{k}{\mu(\mathbb{I}_{\mathit{\Lambda}})}T_{u}(\mathbf{1}_{\mathbb{I}_{\mathit{\Lambda}}})=k$

by (7.1). Then for any
$\boldsymbol{k}=(k_{1},\ldots,k_{2^{n}})\in\mathds{k}^{\oplus_{p}2^{n}}$,
$\boldsymbol{f}_{\boldsymbol{k}}=(f_{k_{1}},\ldots,f_{k_{2^{n}}})\in
E_{u}^{\oplus_{p}2^{n}}$ is a preimage of $\boldsymbol{k}$ under the
$\mathds{k}$-linear map $T_{u}^{\oplus 2^{n}}$. We define

$m_{u}(\boldsymbol{k})=T_{u+1}(\gamma_{\xi}(\boldsymbol{f}_{\boldsymbol{k}})).$

It is easy to see that $m_{u}$ is a $\mathds{k}$-linear map. In particular,
for the constant function $\mathbf{1}_{\mathbb{I}_{\mathit{\Lambda}}}$ given
by the measure $\mu(\mathbb{I}_{\mathit{\Lambda}})$ of
$\mathbb{I}_{\mathit{\Lambda}}$, we obtain that $f_{\mu(E_{u})}$ is a preimage
of $\mu(\mathbb{I}_{\mathit{\Lambda}})\in\mathds{k}$, and then

$m_{u}(\mu(\mathbb{I}_{\mathit{\Lambda}}),\ldots,\mu(\mathbb{I}_{\mathit{\Lambda}}))=T_{u+1}\gamma_{\xi}(\mathbf{1}_{\mathbb{I}_{\mathit{\Lambda}}},\ldots,\mathbf{1}_{\mathbb{I}_{\mathit{\Lambda}}})=\sum_{(u_{i}v_{i})_{i}}1\cdot\mu\bigg{(}\prod\nolimits_{i}\mathbb{I}^{(u_{i}v_{i})}\bigg{)}=\mu(\mathbb{I}_{\mathit{\Lambda}}).$

######  Lemma 7.1.

Let $\mathds{k}=(\mathds{k},|\cdot|,\preceq)$ be an extension of $\mathbb{R}$.
Then $T_{u}:E_{u}\to\mathds{k}$ is a $\mathit{\Lambda}$-homomorphism.

###### Proof.

Note that $\mathds{k}$ is a $\mathit{\Lambda}$-module given by

$\mathit{\Lambda}\times\mathds{k}\to\mathds{k},(\lambda,k)\mapsto\lambda\cdot
k:=\tau(\lambda)k$

For arbitrary two elements $\lambda_{1},\lambda_{2}\in\mathit{\Lambda}$ and
arbitrary two functions $f=\sum_{i}k_{i}\mathbf{1}_{I_{i}}$,
$g=\sum_{j}k_{j}^{\prime}\mathbf{1}_{I_{j}^{\prime}}\in E_{u}$, we have

$\displaystyle T_{u}(\lambda_{1}\cdot f+\lambda_{2}\cdot g)$
$\displaystyle=T_{u}\bigg{(}\sum\nolimits_{i}\tau(\lambda_{1})k_{i}\mathbf{1}_{I_{i}}+\sum\nolimits_{j}\tau(\lambda_{2})k_{j}^{\prime}\mathbf{1}_{I_{j}^{\prime}}\bigg{)}$
$\displaystyle=\tau(\lambda_{1})T_{u}\bigg{(}\sum\nolimits_{i}k_{i}\mathbf{1}_{I_{i}}\bigg{)}+\tau(\lambda_{2})T_{u}\bigg{(}\sum\nolimits_{j}k_{j}^{\prime}\mathbf{1}_{I_{j}^{\prime}}\bigg{)}$
$\displaystyle=\tau(\lambda_{1})T_{u}(f)+\tau(\lambda_{2})T_{u}(g)$
$\displaystyle=\lambda_{1}\cdot T_{u}(f)+\lambda_{2}\cdot T_{u}(g).$

∎

######  Lemma 7.2.

Let $\mathds{k}=(\mathds{k},|\cdot|,\preceq)$ be an extension of $\mathbb{R}$
and let $m_{u}$ be the $\mathds{k}$-linear map given in the diagram (7.6).
Then $m_{u}$ is a $\mathit{\Lambda}$-homomorphism.

###### Proof.

We can prove that $m_{u}$ is a $\mathit{\Lambda}$-homomorphism by using an
argument similar to proving that $m_{u}$ is a $\mathds{k}$-linear mapping
(note that this proof needs to use Lemma 7.1). ∎

######  Remark 7.3.

Since $E_{0}\subseteq E_{1}\subseteq\cdots\subseteq
E_{u}\subseteq\cdots\subseteq\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})\subseteq\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})}=\underrightarrow{\lim}E_{i}$,
we have that $\mu$ is independent on $u$. Thus, we can use $m$ to present all
maps $m_{i}$ ($i\in\mathbb{N}$) because $m_{0}=m_{1}=m_{2}=\ldots$.

######  Proposition 7.4.

Let $\mathds{k}=(\mathds{k},|\cdot|,\preceq)$ be an extension of $\mathbb{R}$.
Then the triple $(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m)$ is an
object in $\mathscr{N}\\!\\!or^{p}$. Furthermore, since $\mathit{\Lambda}$ is
complete, so is $\mathds{k}$. Then $\mathds{k}^{\oplus_{p}2n}$ is a Banach
$\mathit{\Lambda}$-module, and so
$(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m)$ is an object in
$\mathscr{A}^{p}$.

###### Proof.

It follows from Lemmas 7.1 and 7.2 and Remark 7.3. ∎

The following proposition shows that $T_{u}$ satisfies the triangle
inequality.

######  Proposition 7.5.

If $\mathds{k}=(\mathds{k},|\cdot|,\preceq)$ is an extension of $\mathbb{R}$,
then for any $f\in E_{u}$, the following inequality holds for all
$u\in\mathbb{N}$.

$\displaystyle|T_{u}(f)|\leq T_{u}(|f|).$ (7.7)

###### Proof.

Assume that $f=\sum_{i}k_{i}\mathbf{1}_{I_{i}}\in E_{u}$, where $I_{i}\cap
I_{j}=\varnothing$ for all $i\neq j$. Then
$|f|=|\sum_{i}k_{i}\mathbf{1}_{I_{i}}|$ is also a step function in $E_{u}$,
and we have

$\displaystyle T_{u}(|f|)$
$\displaystyle=T_{u}\Big{(}\Big{|}\sum\nolimits_{i}k_{i}\mathbf{1}_{I_{i}}\Big{|}\Big{)}\mathop{=\\!=}\limits^{(\star)}T_{u}\Big{(}\sum\nolimits_{i}|k_{i}|\mathbf{1}_{I_{i}}\Big{)}$
$\displaystyle=\sum\nolimits_{i}|k_{i}|\mu\Big{(}\prod\nolimits_{i}\mathbb{I}^{(u_{i}v_{i})_{i}}\Big{)}$
$\displaystyle\geq\Big{|}\sum\nolimits_{i}k_{i}\mu\Big{(}\prod\nolimits_{i}\mathbb{I}^{(u_{i}v_{i})_{i}}\Big{)}\Big{|}=|T_{u}(f)|,$

where $(\star)$ is given by $I_{i}\cap I_{j}=\varnothing$. ∎

######  Theorem 7.6.

If $\mathds{k}=(\mathds{k},|\cdot|,\preceq)$ is an extension of $\mathbb{R}$,
then there exists a unique morphism

$T:(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})\to(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m)$

in
$\mathrm{Hom}_{\mathscr{N}\\!\\!or^{p}}((\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi}),(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m))$
such that the diagram

$\textstyle{(\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}}),\mathbf{1},\gamma_{\xi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{\subseteq}$$\textstyle{(\mathds{k},\mu(\mathbb{I}_{\mathit{\Lambda}}),m)}$$\textstyle{(\widehat{\mathbf{S}_{\tau}(\mathbb{I}_{\mathit{\Lambda}})},\mathbf{1},\widehat{\gamma}_{\xi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{T}}$

commutes, where $\widehat{T}$ is the morphism in