# A Generalization of $m$-topology and $U$-topology on rings of measurable
functions

Soumyadip Acharyya  Division of Science, Mathematics, and Engineering,
University of South Carolina Sumter, 200 Miller Road, Sumter, SC 29150, USA
<EMAIL_ADDRESS>,  Rakesh Bharati  Department of Pure Mathematics,
University of Calcutta, 35, Ballygunge Circular Road, Kolkata - 700019, INDIA
<EMAIL_ADDRESS>,  Atasi Deb Ray  Department of Pure Mathematics,
University of Calcutta, 35, Ballygunge Circular Road, Kolkata - 700019, INDIA
<EMAIL_ADDRESS>and  Sudip Kumar Acharyya  Department of Pure
Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata -
700019, INDIA<EMAIL_ADDRESS>
###### Abstract.

For a measurable space ($X,\mathcal{A}$), let $\mathcal{M}(X,\mathcal{A})$ be
the corresponding ring of all real valued measurable functions and let $\mu$
be a measure on ($X,\mathcal{A}$). In this paper, we generalize the so-called
$m_{\mu}$ and $U_{\mu}$ topologies on $\mathcal{M}(X,\mathcal{A})$ via an
ideal $I$ in the ring $\mathcal{M}(X,\mathcal{A})$. The generalized versions
will be referred to as the $m_{\mu_{I}}$ and $U_{\mu_{I}}$ topology,
respectively, throughout the paper. $L_{I}^{\infty}\left(\mu\right)$ stands
for the subring of $\mathcal{M}(X,\mathcal{A})$ consisting of all functions
that are essentially $I$-bounded (over the measure space
($X,\mathcal{A},\mu$)). Also let
$I_{\mu}(X,\mathcal{A})=\big{\\{}f\in\mathcal{M}(X,\mathcal{A}):\,\text{for
every}\,g\in\mathcal{M}(X,\mathcal{A}),fg\,\,\text{is
essentially}\,I$-$\text{bounded}\big{\\}}$. Then $I_{\mu}(X,\mathcal{A})$ is
an ideal in $\mathcal{M}(X,\mathcal{A})$ containing $I$ and contained in
$L_{I}^{\infty}\left(\mu\right)$. It is also shown that
$I_{\mu}(X,\mathcal{A})$ and $L_{I}^{\infty}\left(\mu\right)$ are the
components of $0$ in the spaces $m_{\mu_{I}}$ and $U_{\mu_{I}}$, respectively.
Additionally, we obtain a chain of necessary and sufficient conditions as to
when these two topologies coincide. In particular, it is proved that they
coincide if and only if the three sets $\mathcal{M}(X,\mathcal{A})$,
$L_{I}^{\infty}\left(\mu\right)$, and $I_{\mu}(X,\mathcal{A})$ coincide and
this holds when and only when $\mathcal{M}(X,\mathcal{A})$, with either
topology, is connected, which is further equivalent to each of these two
topological spaces being a topological ring as well as a topological vector
space. Furthermore, it is easy to verify that $L_{I}^{\infty}\left(\mu\right)$
is a pseudonormed linear space regardless of the choice of $I$. Assuming $I$
is a real maximal ideal in $\mathcal{M}(X,\mathcal{A})$, we show that
$L_{I}^{\infty}\left(\mu\right)$ is complete. We construct two examples of
spaces of the form $L_{I}^{\infty}\left(\mu\right)$, each of which is not only
non-separable but is also based on a non-trivial ideal $I$.

###### Key words and phrases:

$m_{\mu_{I}}$-topology, $U_{\mu_{I}}$-topology, Essentially $I$-bounded
measurable functions, $\mu$-stable family, Component, Quasicomponent, Non-
separable pseudonormed linear space

###### 2020 Mathematics Subject Classification:

54C40, 46E30

The first author acknowledges that this research was partially supported by
the University of South Carolina Sumter 2021 Summer Research Stipend Program.

The second author acknowledges financial support from University Grants
Commission, New Delhi, for the award of research fellowship (F.No. 16-9(June
2018)/2019 (NET/CSIR))

##  1\. Introduction

We start with a measure space $(X,\cal{A},\mu)$, where $X$ is a non-empty set,
$\cal{A}$ is a $\sigma$-algebra of subsets of $X$ and
$\mu:\cal{A}\rightarrow[0,\infty]$ is a measure. A function
$f:X\rightarrow\mathbb{R}$ is called $\cal{A}$-measurable or simply measurable
if for any $\alpha\in\mathbb{R}$, $f^{-1}(-\infty,\alpha)$ is a member of
$\cal{A}$. Members of $\cal{A}$ are often called $\cal{A}$-measurable sets or
just measurable sets if there is no chance of any confusion. The collection
$\mathcal{M}(X,\mathcal{A})$ of all measurable functions constitutes a
commutative lattice ordered algebra with unity if the operations are defined
pointwise on $X$. It may be mentioned that a number of properties of
$\mathcal{M}(X,\mathcal{A})$ and a few of its chosen subalgebras have already
been investigated in [2],[4],[5],[10].

Suppose for $f\in\mathcal{M}(X,\mathcal{A})$ and a positive unit $u$ of this
ring and $\epsilon>0$ in $\mathbb{R}$,
$m_{\mu}(f,u)=\\{g\in\mathcal{M}(X,\mathcal{A}):|f(x)-g(x)|<u(x)$ for all $x$,
almost every where on $X$ with respect to the measure $\mu$} and
$U_{\mu}(f,\epsilon)=\\{g\in\mathcal{M}(X,\mathcal{A}):\sup_{x\in X\setminus
A_{g}}|f(x)-g(x)|<\epsilon$ for some $\cal{A}$-measurable set $A_{g}$ with
$\mu(A_{g})=0\\}$. Then {$m_{\mu}(f,u):f\in\mathcal{M}(X,\mathcal{A}),u,$ a
positive unit in $\mathcal{M}(X,\mathcal{A})$} is an open base for a uniquely
determined topology, which is called $m_{\mu}$-topology on
$\mathcal{M}(X,\mathcal{A})$ in [3]. $\mathcal{M}(X,\mathcal{A})$ with the
$m_{\mu}$-topology becomes a topological ring. Incidentally the family
$\\{U_{\mu}(f,\epsilon):f\in\mathcal{M}(X,\mathcal{A}),\epsilon>0\\}$ turns
out to be an open base for a topology called $U_{\mu}$-topology on
$\mathcal{M}(X,\mathcal{A})$ in [2]. The $U_{\mu}$-topology is coarser than
the $m_{\mu}$-topology on $\mathcal{M}(X,\mathcal{A})$ and
$\mathcal{M}(X,\mathcal{A})$ equipped with the $U_{\mu}$-topology is an
additive topological group, which is in general neither a topological ring nor
a topological vector space. However it is established in [2], Theorem 3.7 that
$\mathcal{M}(X,\mathcal{A})$ is a topological ring in the $U_{\mu}$-topology
if and only if it is a topological vector space with respect to the same
topology and this precisely happens when and only when each function in
$\mathcal{M}(X,\mathcal{A})$ is essentially bounded on $X$ with respect to the
measure $\mu$. In the present article our intention is to initiate a further
generalization of the $m_{\mu}$-topology and the $U_{\mu}$-topology both via
an ideal $I$ in the ring $\mathcal{M}(X,\mathcal{A})$ and to place the last
mentioned facts on a wider setting. To be specific with respect to any such
ideal $I$ in $\mathcal{M}(X,\mathcal{A})$, we call a function
$f\in\mathcal{M}(X,\mathcal{A})$ a $\mu_{I}$-unit if there exists $Z\in
Z[I]=\\{Z(g):g\in I\\}$ such that $\mu\\{x\in Z:f(x)=0\\}=0$. Let
$U_{\mu_{I}}$ be the family of all $\mu_{I}$-units in
$\mathcal{M}(X,\mathcal{A})$ and $U_{\mu_{I}}^{+}=\\{u\in U_{\mu_{I}}:$ there
exists $Z\in Z[I]$ and $E\subseteq Z$ with $\mu(E)=0$ such that $u(x)>0$}. Let
for $f\in\mathcal{M}(X,\mathcal{A}),\ u\in U_{\mu_{I}}^{+}$ and $\epsilon>0$,
$U_{\mu_{I}}(f,\epsilon)=\\{g\in\mathcal{M}(X,\mathcal{A}):\sup_{x\in
Z\setminus E}|f(x)-g(x)|<\epsilon$ for some $Z\in Z[I]$ and $E\subseteq Z$
with $\mu(E)=0$} and $m_{\mu_{I}}(f,u)=\\{g\in\mathcal{M}(X,\mathcal{A}):$
there exist $Z\in Z[I]$ such that $|g(x)-f(x)|<u(x)$ for all $x\in Z$ almost
everywhere}. It needs some routine computations to prove that the family
{$m_{\mu_{I}}(f,u):f\in\mathcal{M}(X,\mathcal{A}),u\in U_{\mu_{I}}^{+}$} is an
open base for some topology which we wish to designate as
$m_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$. Furthermore, it is not
at all hard to establish that $\mathcal{M}(X,\mathcal{A})$ becomes a
topological ring in the $m_{\mu_{I}}$-topology. Incidentally the family
$\\{U_{\mu_{I}}(f,\epsilon):f\in\mathcal{M}(X,\mathcal{A}),\epsilon>0\\}$ also
turns out to be an open base for a uniquely determined topology, which we call
as $U_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$.
$\mathcal{M}(X,\mathcal{A})$ equipped with the $U_{\mu_{I}}$-topology is an
additive topological group but it may not be a topological ring. With the
special choice $I=\\{0\\}$, the $m_{\mu_{I}}$-topology and the
$U_{\mu_{I}}$-topology reduce to $m_{\mu}$-topology and $U_{\mu}$-topology
respectively described earlier.

We let $I\mu(X,\cal{A})=\\{f\in\mathcal{M}(X,\mathcal{A}):$ given
$g\in\mathcal{M}(X,\mathcal{A})$ there exists $Z\in Z[I]$ and a measurable
subset $E$ of $Z$ with $\mu(E)$=0 such that $f.g$ is bounded on $Z\setminus
E\\}$ and $L_{I}^{\infty}(X,\cal{A},\mu)\equiv
L_{I}^{\infty}(\mu)=\\{f\in\mathcal{M}(X,\mathcal{A}):$ there exists $Z\in
Z[I]$ and $E\subseteq Z$ with $\mu(E)=0$ such that $f$ is bounded on
$Z\setminus E\\}$. These two sets play a key role in enabling us to establish
a number of main results in the present paper. It turns out that
$I\mu(X,\cal{A})$ is an ideal in $\mathcal{M}(X,\mathcal{A})$ and $I\subseteq
I\mu(X,\cal{A})\subseteq L_{I}^{\infty}(\mu)$. We realize that
$I\mu(X,\cal{A})$ and $L_{I}^{\infty}(\mu)$ are the components of $0$ in
$\mathcal{M}(X,\mathcal{A})$ equipped with the $m_{\mu_{I}}$-topology and the
$U_{\mu_{I}}$-topology respectively. With the special choice $I=\\{0\\}$, the
second fact in the last assertion reads: the space $L^{\infty}(\mu)$ of all
essentially bounded measurable functions over the measure space
$(X,\cal{A},\mu)$ is just the component of the function $0$ in the topoloical
group $\mathcal{M}(X,\mathcal{A})$ equipped with the $U_{\mu}$-topology. The
set $I\mu(X,\cal{A})$, though not in general the component of $0$ in
$\mathcal{M}(X,\mathcal{A})$ in the $U_{\mu_{I}}$-topology, happens to be the
maximal connected ideal in this ring with the same topology. Before narrating
the organization of the technical results in this article, we like to mention
that similar kind of topologies viz. $m_{\mu_{I}}$-topology and
$U_{\mu_{I}}$-topology, as generalizations of the well-known $m$-topology and
the topology of uniform convergence also called the $U$-topology on the ring
$C(X)$ of all real-valued continuous functions on a completely regular
Hausdorff topological space $X$, via an ideal $I$ in $C(X)$ are initiated in
[1] and [6] respectively.

In the technical section 2 of this article we establish several facts related
to the $m_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$. As in [1], for
any $f\in\mathcal{M}(X,\mathcal{A})$, the map

$\displaystyle\Phi_{f}:\mathbb{R}$
$\displaystyle\rightarrow\mathcal{M}(X,\mathcal{A})$ $\displaystyle r$
$\displaystyle\mapsto r.f$

is introduced. It is proved that for any $f\in I\mu(X,\cal{A})$, $\Phi_{f}$ is
a continuous map if the $m_{\mu_{I}}$-topology is imposed on
$\mathcal{M}(X,\mathcal{A})$ (Theorem 2.4). This fact together with the result
that $L_{I}^{\infty}(\mu)$ is a clopen subset of $\mathcal{M}(X,\mathcal{A})$
in the $m_{\mu_{I}}$-topology (Theorem 2.2) yields that $I\mu(X,\cal{A})$ is
the component of $0$ in the $m_{\mu_{I}}$-topology (Theorem 2.5). We realize
that $L_{I}^{\infty}(\mu)$ becomes a pseudonormed linear space with respect to
a natural definition of generalized essential supremum $||f||_{I}^{\infty}$
for functions $f\in L_{I}^{\infty}(\mu)$. It is easy to verify that the
pseudonorm topology on $L_{I}^{\infty}(\mu)$ is weaker than the subspace
topology on this set relative to the $m_{\mu_{I}}$-topology on
$\mathcal{M}(X,\mathcal{A})$. These two topologies coincide if and only if the
$m_{\mu_{I}}$-topology becomes connected/ locally connected. Indeed several
other necessary and sufficient conditions (each equivalent to the coincidence
of these two topologies) involving the sets $I\mu(X,\cal{A})$ and
$L_{I}^{\infty}(\mu)$ are also found out (Theorem 2.7). This is followed by a
complete description of the connected ideals in $\mathcal{M}(X,\mathcal{A})$
in the $m_{\mu_{I}}$-topology (Theorem 2.8).

In the technical section 3 of this paper, our main focus is on proving a
number of results concerning the $U_{\mu_{I}}$-topology on
$\mathcal{M}(X,\mathcal{A})$. Since the map $\Phi_{f}$ is proved to be
continuous for each $f\in I\mu(X,\cal{A})$ in the $m_{\mu_{I}}$-topology, it
follows that the same map is also continuous for each $f\in I\mu(X,\cal{A})$
if $\mathcal{M}(X,\mathcal{A})$ is given the $U_{\mu_{I}}$-topology as this
topology is weaker than the $m_{\mu_{I}}$-topology. However we make a minor
improvement of the last assertion by proving that the map $\Phi_{f}$ is
continuous for each $L_{I}^{\infty}(\mu)$ in the $U_{\mu_{I}}$-topology
(Theorem 3.1). This has ultimately led to the fact that $L_{I}^{\infty}(\mu)$
is the component of $0$ in the space $\mathcal{M}(X,\mathcal{A})$ with the
$U_{\mu_{I}}$-topology (Theorem 3.3). It is well-known that the component of
$0$ in a topological ring is necessarily an ideal in this ring (Theorem 2.6,
[6]). However if $R$ is a ring with a topology $\tau$ for which $(R,\tau)$ is
just an additive topological group without being a topological ring, then the
component of $0$ in $R$ may not be an ideal in $R$. A counterexample for this
purpose in rings of continuous functions $C(X)$ is already provided in [6].
Here in this article we provide yet another counterexample in rings of
measurable functions by choosing appropriately $X,\cal{A},\mu$ and the ideal
$I$ in $\mathcal{M}(X,\mathcal{A})$ (Example 3.5). It is already mentioned in
this introductory section that $\mathcal{M}(X,\mathcal{A})$ equipped with the
$U_{\mu_{I}}$-topology is in general neither a topological ring nor a
topological vector space. It is realized however that the
$U_{\mu_{I}}$-topology makes $\mathcal{M}(X,\mathcal{A})$ a topological ring
if and only if it renders $\mathcal{M}(X,\mathcal{A})$ a topological vector
space. Such things happen precisely when (amongst other necessary and
sufficient conditions) the $U_{\mu_{I}}$-topology coincides with the
$m_{\mu_{I}}$-topology (Theorem 3.7).

In the technical section 4 of the paper we investigate mainly a few problems
concerning the pseudonorm topology of the space $L_{I}^{\infty}(\mu)$. It is
realized that $L_{I}^{\infty}(\mu)$ is a complete pseudonormed linear space if
the filter $Z[I]$ of $\cal{A}$-measurable sets is closed under countable
intersection (Theorem 4.2), in particular if $I$ is a real maximal ideal in
$\mathcal{M}(X,\mathcal{A})$. It is folklore that the space $L^{\infty}(\mu)$
of all essentially bounded measurable functions over a measure space
$(X,\cal{A},\mu)$ are mostly non-separable (see the comments in page 580 in
[7]). In the fitness of things, by making suitable special choices of
$X,\cal{A},\mu$ and non-zero ideals $I$ in the rings
$\mathcal{M}(X,\mathcal{A})$, we construct two different complete pseudonormed
linear spaces $L_{I}^{\infty}(\mu)$ none of which is separable (Example 4.3,
Example 4.4). We conclude this article after showing that with the special
choice $\mu=$ counting measure on the measurable space $(X,\cal{A})$ that for
two ideals $I$ and $J$ in $\mathcal{M}(X,\mathcal{A})$, $m_{\mu_{I}}$-topology
= $m_{\mu_{J}}$ topology if and only if $U_{\mu_{I}}$-topology =
$U_{\mu_{J}}$-topology if and only if $I=J$. We then check that on choosing
$\mu=$ an appropriately defined Dirac measure on $X$, for any two ideals $I,J$
in $\mathcal{M}(X,\mathcal{A})$, $U_{\mu_{I}}$-topology =
$U_{\mu_{J}}$-topology and $m_{\mu_{I}}$-topology = $m_{\mu_{J}}$-topology. We
like to mention that various other properties enjoyed by the $m$-topology on
the ring $\mathcal{M}(X,\mathcal{A})$ has been established recently in [10].

##  2\. $m_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$

######  Theorem 2.1.

The set $U_{\mu_{I}}$ is open in $\mathcal{M}(X,\mathcal{A})$ in the
$m_{\mu_{I}}$-topology.

###### Proof.

Since $U_{\mu_{I}}=\\{u\in\mathcal{M}(X,\mathcal{A}):$ there exists $Z\in
Z[I]$ and $E\subseteq Z$ with $\mu(E)=0$, such that $|u(x)|>0$ for each $x\in
Z\setminus E\\}$, it suffices to show that $U_{\mu_{I}}^{+}$ is open in the
$m_{\mu_{I}}$-topology. This is indeed the case because for $u\in
U_{\mu_{I}}^{+}$, it is not hard to check that
$m_{\mu_{I}}(u,\frac{u}{2})\subseteq U_{\mu_{I}}^{+}$, from which it follows
that $u$ is an interior point of $U_{\mu_{I}}^{+}$ in the
$m_{\mu_{I}}$-topology. ∎

######  Theorem 2.2.

$L_{I}^{\infty}(\mu)$ is a clopen subset of $\mathcal{M}(X,\mathcal{A})$ in
the $m_{\mu_{I}}$-topology.

###### Proof.

That $L_{I}^{\infty}(\mu)$ is open in $\mathcal{M}(X,\mathcal{A})$ in the
$m_{\mu_{I}}$-topology follows from the easily verifiable result that for
$f\in L_{I}^{\infty}(\mu)$, $m_{\mu_{I}}(f,1)\subseteq L_{I}^{\infty}(\mu)$.
On the other hand if $g\in\mathcal{M}(X,\mathcal{A})\setminus
L_{I}^{\infty}(\mu)$, then it is easy to check that $m_{\mu_{I}}(g,1)\cap
L_{I}^{\infty}(\mu)=\emptyset$. It follows that $L_{I}^{\infty}(\mu)$ is a
closed set in $\mathcal{M}(X,\mathcal{A})$ in the $m_{\mu_{I}}$-topology. ∎

The following result needs only some routine computations for its proof.

######  Theorem 2.3.

$I\mu(X,\cal{A})$ is an ideal in the ring $\mathcal{M}(X,\mathcal{A})$ and
$I\subseteq I\mu(X,\cal{A})\subseteq L_{I}^{\infty}(\mu)$.

In the context of rings of continuous functions $C(X)$, where $X$ is a
completely regular Hausdorff topological space, for $f\in C(X)$ the map

$\displaystyle\Phi_{f}:\mathbb{R}$ $\displaystyle\rightarrow C(X)$
$\displaystyle r$ $\displaystyle\mapsto r.f$

is introduced. In the present paper we exploit a few of the properties of an
analogous map defined by the same notation
$\Phi_{f}:\mathbb{R}\rightarrow\mathcal{M}(X,\mathcal{A})$ defined as
$r\mapsto r.f$ to be useful for our later results.

######  Theorem 2.4.

For each function $f\in I\mu(X,\cal{A})$, the map $\Phi_{f}$ is continuous if
$\mathcal{M}(X,\mathcal{A})$ is given the $m_{\mu_{I}}$-topology. $($A special
case of this result with $I=\\{0\\}$ and $\mu=$ counting measure is already
proved in [10]$)$.

###### Proof.

Let $f\in I\mu(X,\cal{A})$ and $r\in\mathbb{R}$ be arbitrary. Then a typical
basic open neighbourhood of $r.f$ in $\mathcal{M}(X,\mathcal{A})$ in the
$m_{\mu_{I}}$-topology is of the form $m_{\mu_{I}}(rf,u)$, where $u\in
U_{\mu_{I}}^{+}$. Accordingly there exists $Z\in Z[I]$ and $A\in\cal{A}$ with
$A\subseteq Z$ and $\mu(A)=0$ for which $u(x)>0$ for each $x$ in $Z\setminus
A$. Set $g=|u|+\chi_{(X\setminus Z)\cup A}$, here $\chi_{E}$ stands for the
characteristic function of a measurable subset $E$ of $X$. Then $g$ is a
positive multiplicative unit in the ring $\mathcal{M}(X,\mathcal{A})$.
Therefore $\frac{1}{g}\in\mathcal{M}(X,\mathcal{A})$. The hypothesis $f\in
I\mu(X,\cal{A})$ implies that $\frac{f}{g}$ is bounded almost everywhere (with
respect to the measure $\mu$) on some zero-set $Z^{{}^{\prime}}\in Z[I]$. This
means that there exists $B\in\cal{A}$ with $\mu(B)=0$ such that $B\subseteq
Z^{{}^{\prime}}$ for which $|\frac{f}{g}|\leq\lambda$ on
$Z^{{}^{\prime}}\setminus B$ for some $\lambda>0$ in $\mathbb{R}$. Let
$s\in(r-\frac{1}{\lambda},r+\frac{1}{\lambda})$. Then on the set $Z\cap
Z^{{}^{\prime}}\setminus(A\cup B)$,
$|\phi_{f}(r)-\phi_{f}(s)|=|s-r||f|\leq\frac{1}{\lambda}.|\frac{f}{g}|.|g|$$\leq\frac{1}{\lambda}.\lambda.|u|=|u|$.
Since $Z\cap Z^{{}^{\prime}}\in Z[I]$ and $\mu(A\cup B)=0$. It follows that
$\phi_{f}(s)\in m_{\mu_{I}}(\phi_{f}(r),u)$. This ensures the continuity of
the map $\phi_{f}$ at the point $r\in\mathbb{R}$, chosen arbitrarily. We are
now ready to prove the first main technical result in this section. ∎

######  Theorem 2.5.

$I\mu(X,\cal{A})$ is the component of $0$ of the ring
$\mathcal{M}(X,\mathcal{A})$ in the $m_{\mu_{I}}$-topology.

###### Proof.

It is trivial that $I\mu(X,\cal{A})=\bigcup\limits_{f\in
I\mu(X,\cal{A})}\phi_{f}(\mathbb{R})=$ the union of a family of connected
subsets of $\mathcal{M}(X,\mathcal{A})$ in the $m_{\mu_{I}}$-topology each of
which contains the function $0$. Hence $I\mu(X,\cal{A})$ is a connected ideal
in $\mathcal{M}(X,\mathcal{A})$. To show that $I\mu(X,\cal{A})$ is the
component of $0$ in $\mathcal{M}(X,\mathcal{A})$ (equivalently the maximal
connected ideal in $\mathcal{M}(X,\mathcal{A})$), if possible let $J$ be a
connected ideal in $\mathcal{M}(X,\mathcal{A})$ with
$I\mu(X,\cal{A})\subsetneq J$. Choose $f\in J\setminus I\mu(X,\cal{A})$. Then
there exists $g\in\mathcal{M}(X,\mathcal{A})$ such that for each $Z\in Z[I]$
and for each $M>0$ in $\mathbb{R}$, $\mu\\{t\in Z:|f(t)g(t)|>M\\}>0$.
Consequently $f.g\notin L_{I}^{\infty}(\mu)$. But we note that $f.g\in J$,
since $J$ is an ideal in $\mathcal{M}(X,\mathcal{A})$. So we can write
$J=(J\cap L_{I}^{\infty}(\mu))\cup(J\setminus L_{I}^{\infty}(\mu))=$ the union
of two non-empty disjoint open sets in the space $J$. We exploit the
clopenness of the set $L_{I}^{\infty}(\mu)$ in $\mathcal{M}(X,\mathcal{A})$ in
the $m_{\mu_{I}}$-topology to prove the last assertion. This contradicts the
connectedness of $J$ in $\mathcal{M}(X,\mathcal{A})$. ∎

An $f\in\mathcal{M}(X,\mathcal{A})$ is called essentially $I$-bounded on $X$
if there exists $Z\in Z[I]$, a measurale subset $E$ of $Z$ with $\mu(E)=0$ and
a real number $\lambda>0$ such that $|f(x)|\leq\lambda$ holds for all $x\in
Z\setminus E$. Any such $\lambda$ is called an essential $I$-upper bound of
$f$ over $X$. With $I=\\{0\\}$, essential $I$-upper bound coincides with
essential upper bound. Thus $L_{I}^{\infty}(\mu)$ is precisely the family of
all essentially $I$-bounded functions (over $X$) in the ring
$\mathcal{M}(X,\mathcal{A})$. For each $f\in L_{I}^{\infty}(\mu)$, let $A_{f}$
be the aggregate of all essential $I$-upper bounds of $f$ over $X$. Suppose
$||f||_{I}^{\infty}=$ infimum of the set $A_{f}$. It is clear that
$||f||_{I}^{\infty}\geq 0$, and if $f=0$ then $||f||_{I}^{\infty}=0$. Also if
$f,g\in L_{I}^{\infty}(\mu)$ and $l\in A_{f}$ and $m\in A_{g}$, then there
exist $Z_{1},Z_{2}\in Z[I]$ and measurable sets $E_{2},E_{2}$ with
$E_{1}\subseteq Z_{1},\ E_{2}\subseteq Z_{2}$ such that
$\mu(E_{1})=0=\mu(E_{2})$ and $|f(x)|\leq l$ for all $x\in Z_{1}\setminus
E_{1}$ and $|g(x)|\leq m$ for all $x\in Z_{2}\setminus E_{2}$. It follows that
for all $x\in(Z_{1}\cap Z_{2})\setminus(E_{1}\cup E_{2})$, $|(f+g)(x)|\leq
l+m$ with $Z_{1}\cap Z_{2}\in Z[J]$ and $\mu(E_{1}\cup E_{2})=0$. Thus $l+m$
is an essential $I$-upper bound of $f+g$ over $X$ and therefore
$||f+g||_{I}^{\infty}\leq l+m$. Since $l\in A_{f}$ and $m\in A_{g}$ are chosen
arbitrarily, this implies that $||f+g||_{I}^{\infty}\leq$ Inf$A_{f}+$
Inf$A_{g}=||f||_{I}^{\infty}+||g||_{I}^{\infty}$. Finally it is routine to
check for any $c\in\mathbb{R}$ and $f\in L_{I}^{\infty}(\mu)$ that
$||cf||_{I}^{\infty}=|c|||f||_{I}^{\infty}$. Thus we have already proved the
following fact.

######  Theorem 2.6.

$(L_{I}^{\infty}(\mu),||.||_{I}^{\infty})$ is a pseudonormed linear space over
$\mathbb{R}$.

Since for $f\in L_{I}^{\infty}(\mu)$ and $\delta>0$ in $\mathbb{R}$,
$m_{\mu_{I}}(f,\delta)\subseteq\\{g\in
L_{I}^{\infty}(\mu):||g-f||_{I}^{\infty}<\delta\\}$, it follows that the
pseudonorm topology on $L_{I}^{\infty}(\mu)$ is weaker than the relative
topology on $L_{I}^{\infty}(\mu)$ induced by the $m_{\mu_{I}}$-topology on
$\mathcal{M}(X,\mathcal{A})$.

The following theorem elaborates in precise terms, when these two topologies
on $L_{I}^{\infty}(\mu)$ coincide.

######  Theorem 2.7.

The following statements are equivalent.

  1. (1)

$L_{I}^{\infty}(\mu)=\mathcal{M}(X,\mathcal{A})$.

  2. (2)

$I\mu(X,\cal{A})=\mathcal{M}(X,\mathcal{A})$.

  3. (3)

The pseudonorm topology on $L_{I}^{\infty}(\mu)$ is identical to the relative
$m_{\mu_{I}}$-topology on it.

  4. (4)

The $m_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$ is connected.

  5. (5)

The $m_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$ is locally
connected.

  6. (6)

$Z[I]$ is a $\mu$-stable set in the sense that given
$f\in\mathcal{M}(X,\mathcal{A})$, there exists $Z\in Z[I]$ and $E\subseteq Z$
with $\mu(E)=0$ such that $f$ is bounded on $Z\setminus E$.

  7. (7)

$\mathcal{M}(X,\mathcal{A})$ with $m_{\mu_{I}}$-topology is a topological
vector space.

###### Proof.

$(1)\Rightarrow(3):$ Let (1) be true. In view of the last observation
preceeding Theorem 2.7, it suffices to show that the relative
$m_{\mu_{I}}$-topology on $L_{I}^{\infty}(\mu)$ is weaker than the pseudonorm
topology on this set. Choose $f\in L_{I}^{\infty}(\mu)$ and $u\in U_{I}^{+}$.
Then there exists $Z\in Z[I]$ and $A\in\cal{A}$, $A\subseteq Z$ with
$\mu(A)=0$ such that $u(x)>0$ for each $x\in Z\setminus A$. Set $v(x)=u(x)$ if
$x\in Z\setminus A$ and $v(x)=1$, otherwise, $x\in X$. Then
$v\in\mathcal{M}(X,\mathcal{A})$ and is a positive unit in the ring. So
$\frac{1}{v}\in\mathcal{M}(X,\mathcal{A})=L_{I}^{\infty}(\mu)$. There exists
therefore $M>0$, $Z^{{}^{\prime}}\in Z[I]$ and $A^{*}\subseteq
Z^{{}^{\prime}}$ with $\mu(A^{*})=0$ for which for each $x\in
Z^{{}^{\prime}}\setminus A^{*},|\frac{1}{v(x)}|\leq M$. This implies that for
each $x\in(Z\cap Z^{{}^{\prime}})\setminus(A\cup A^{*})$,
$u(x)=v(x)\geq\frac{1}{M}=\lambda$, say. Now choose $g\in L_{I}^{\infty}(\mu)$
with $||g-f||_{I}^{\infty}<\lambda$. Then there exists $Z_{1}\in Z[I]$ and
$A_{1}\subseteq Z_{1}$ with $\mu(A_{1})=0$ for which we can write
$|g(x)-f(x)|<\lambda$ for each $x\in Z_{1}\setminus A_{1}$. This implies that
for each $x\in(Z_{1}\cap Z\cap Z^{{}^{\prime}})\setminus(A\cup A_{1}\cup
A^{*})$, $|g(x)-f(x)|<u(x)$ and $Z_{1}\cap Z\cap Z^{{}^{\prime}}\in Z[I]$ and
also $\mu(A\cup A_{1}\cup A^{*})=0$. Thus it follows that $g\in
m_{\mu_{I}}(f,u)$, consequently $\\{g\in
L_{I}^{\infty}(\mu):||g-f||_{I}^{\infty}<\lambda\\}\subseteq
m_{\mu_{I}}(f,u)$.
$(3)\Rightarrow(1):$ Suppose $(1)$ is false. Choose
$f\in\mathcal{M}(X,\mathcal{A})$ such that $f\notin L_{I}^{\infty}(\mu)$ and
$f\geq 1$. Then for each $Z\in Z[I]$ and $n\in\mathbb{N}$, $\mu\\{t\in
Z:f(t)>n\\}>0$ and hence $\mu\\{t\in Z:g(t)<\frac{1}{n}\\}>0$ … $(*)$, on
writing $g(t)=\frac{1}{f(t)},\ t\in X$. Since $g\in L_{I}^{\infty}(\mu)$, we
have then $m_{\mu_{I}}(0,g)\subseteq L_{I}^{\infty}(\mu)$. Thus
$m_{\mu_{I}}(0,g)$ is a neighbourhood of $0$ in the relative
$m_{\mu_{I}}$-topology on $L_{I}^{\infty}(\mu)$. To show that $(3)$ is false,
we shall show that $m_{\mu_{I}}(0,g)$ is not a neighbourhood of $0$ in the
pseudonorm topology on $L_{I}^{\infty}(\mu)$. Indeed for any $\epsilon>0$ and
$Z\in Z[I]$, it follows from the relation $(*)$ above that
$\frac{\epsilon}{2}>g$ on each set $Z\in Z[I]$ with positive $\mu$-measure.
Consequently $\frac{\epsilon}{2}\notin m_{\mu_{I}}(0,g)$. Thus
$\frac{\epsilon}{2}\in\\{h\in
L_{I}^{\infty}(\mu):||h||_{I}^{\infty}<\epsilon\\}\setminus m_{\mu_{I}}(0,g)$
and we are through. Thus the statements $(1)$ and $(3)$ are equivalent.
$(1)\Rightarrow(4)$ is immediate because a pseudonorm topology is path
connected and therefore connected.
$(4)\Rightarrow(1)$ follows from Theorem 2.2.
$(2)\Rightarrow(5):$ Let $(2)$ be true and $\epsilon>0$ be arbitrary. It
suffices to show that $m_{\mu_{I}}(0,\epsilon)$ is connected in the
$m_{\mu_{I}}$-topology. Indeed for each $f\in\mathcal{M}(X,\mathcal{A})$, by
$(2)$, $f\in I\mu(X,\cal{A})$. Consequently by Theorem 2.4, $\phi_{f}$ is a
continuous map. Hence $m_{\mu_{I}}(0,\epsilon)=\bigcup\limits_{f\in
I\mu(X,\cal{A})}\phi_{f}[0,1]=$ the union of a family of connected subsets of
$\mathcal{M}(X,\mathcal{A})$ (in the $m_{\mu_{I}}$-topology), each of which
contains the function $0$. Thus $m_{\mu_{I}}(0,\epsilon)$ is connected in the
$m_{\mu_{I}}$-topology.
$(5)\Rightarrow(2)$. Let $(2)$ be false. Choose
$f\in\mathcal{M}(X,\mathcal{A})\setminus I\mu(X,\cal{A})$. To show that $(5)$
is false, it is sufficient to show that there does not exist any open
connected subset of $\mathcal{M}(X,\mathcal{A})$ containing $0$ (in the
$m_{\mu_{I}}$-topology). If possible let $G$ be an open connected subset of
$\mathcal{M}(X,\mathcal{A})$ in the $m_{\mu_{I}}$-topology containing $0$.
Since from Theorem 2.5, $I\mu(X,\cal{A})$ is the maximal connected subset of
$\mathcal{M}(X,\mathcal{A})$ containing $0$ in this topology, we can write
$0\in m_{\mu_{I}}(0,u)\subseteq G\subseteq I\mu(X,\cal{A})$ for some $u\in
U_{\mu_{I}}^{+}$. Therefore $u(x)>0$ for each $x\in Z\setminus A$ for some
$Z\in Z[I]$ and $A\subseteq Z$ with $\mu(A)=0$.
Define $v:X\rightarrow\mathbb{R}$ as follows $v(x)=u(x)$ if $x\in Z\setminus
A$ and $v(x)=1$ otherwise. Then $v\in\mathcal{M}(X,\mathcal{A})$ and is a
positive unit in $\mathcal{M}(X,\mathcal{A})$. we observe that
$\frac{f}{1+|f|}.v\in m_{\mu_{I}}(0,v)\subseteq m_{\mu_{I}}(0,u)\subseteq
G\subseteq I\mu(X,\cal{A})$. Since $I\mu(X,\cal{A})$ is an ideal in
$\mathcal{M}(X,\mathcal{A})$, it follows that $f\in I\mu(X,\cal{A})$, a
contradiction.
$(2)\Leftrightarrow(6)$ follows by employing routine arguments.
$(4)\Rightarrow(7)$: This follows from the equivalence $(1)\Leftrightarrow(3)$
and $(1)\Leftrightarrow(4)$ and the fact that any pseudonormed linear space is
a topological vector space.
$(7)\Rightarrow(4)$: Let
$L_{I}^{\infty}(\mu)\subsetneq\mathcal{M}(X,\mathcal{A})$. Choose
$f\in\mathcal{M}(X,\mathcal{A})$ such that $f\notin L_{I}^{\infty}(\mu)$.
Without loss of generality we can assume that $f\geq 1$ on $X$. Now for all
$z\in Z[I]$ and $n\in\mathbb{N}$, we should have $\mu\\{x\in
Z:f(x)>n\\}>0\Rightarrow\mu\\{x\in Z:g(x)<\frac{1}{n}\\}>0$ where
$g=\frac{1}{f}\ldots(*)$. We claim that for any two $r,s\in\mathbb{R},\ r\neq
0,\ s\neq 0$, the scalar multiplication function

$\displaystyle\psi:\mathbb{R}\times\mathcal{M}(X,\mathcal{A})$
$\displaystyle\rightarrow\mathcal{M}(X,\mathcal{A})$ $\displaystyle(\alpha,f)$
$\displaystyle\mapsto\alpha.f$

is not continuous at the point $(r,\underline{s})$, here $\underline{s}$ is
the function on $X$ whose value is constantly equal to $s$. If possible let
$\psi$ be continuous at $(r,\underline{s})$. Then
$m_{\mu_{I}}(\underline{rs},g)$ is an open neighbourhood of $\underline{rs}$
in the $m_{\mu_{I}}$-topology in $\mathcal{M}(X,\mathcal{A})$. So $\exists$
$\lambda>0$ and a $v\in U_{\mu_{I}}^{+}$ such that
$(r-\lambda,r+\lambda).m_{\mu_{I}}(\underline{s},v)\subseteq
m_{\mu_{I}}(\bar{rs},g)$. In particular for any $t:r-\lambda<t<r+\lambda$,
$\underline{ts}\in m_{\mu_{I}}(\underline{rs},g)$. Thus $|\underline{ts-
rs}|(x)<g(x)$ for all $x$ almost everywhere with respect to $\mu$ on some
$Z\in Z[I]$, i.e., $g(x)>|(t-r)s|$ for all $x\in Z$ almost everywhere with
respect to $\mu$. Choose $t\neq r$ anywhere in $(r-\lambda,r+\lambda)$ then
$g(x)>(t-r)s$ for all $x\in Z$ almost everywhere with respect to $\mu$. This
implies $g(x)>\frac{1}{n}$ for all $x\in Z$ for some $n\in\mathbb{N}$, almost
everywhere with respect to $\mu\Rightarrow\mu\\{x\in
Z:g(x)\leq\frac{1}{n}\\}=0$, a contradiction to the relation $(*)$ obtained
earlier. ∎

The following theorem determines the exact class of connected ideals in
$\mathcal{M}(X,\mathcal{A})$ in the $m_{\mu_{I}}$-topology.

######  Theorem 2.8.

The following statements are equivalent for an ideal $J$ in the ring
$\mathcal{M}(X,\mathcal{A})$.

  1. (1)

$J$ is a connected ideal in the $m_{\mu_{I}}$-topology.

  2. (2)

$J\subseteq L_{I}^{\infty}(\mu)$.

  3. (3)

$J\subseteq I\mu(X,\cal{A})$.

###### Proof.

$(1)\Rightarrow(2)$. Let $(1)$ be true. If possible let there exist $f\in J$
such that $f\notin L_{I}^{\infty}(\mu)$. Since $0\in J\cap
L_{I}^{\infty}(\mu)$, it follows that $J\cap L_{I}^{\infty}(\mu)$ and
$J\setminus L_{I}^{\infty}(\mu)$ are two non-empty disjoint open sets in the
space $J$. This follows from the clopenness of $L_{I}^{\infty}(\mu)$ in the
space $\mathcal{M}(X,\mathcal{A})$. This contradicts the connectedness of $J$.
$(2)\Rightarrow(3)$. Let $(2)$ be true. If possible let there exist $f\in J$
such that $f\notin I\mu(X,\cal{A})$. Then there exists
$g\in\mathcal{M}(X,\mathcal{A})$ such that $fg\notin L_{I}^{\infty}(\mu)$. But
since $J$ is an ideal in $\mathcal{M}(X,\mathcal{A})$, $f\in J$ implies that
$fg\in J$ and hence by $(2)$, $fg\in L_{I}^{\infty}(\mu)$, a contradiction.
$(3)\Rightarrow(1)$. Let $(3)$ be true. Then for each $f\in J$, it follows
from Theorem 2.4 that $\phi_{f}$ is a continuous map. Therefore we get
$J=\bigcup\limits_{f\in J}\phi_{f}(\mathbb{R})$, a connected subset of
$\mathcal{M}(X,\mathcal{A})$ in the $m_{\mu_{I}}$-topology. ∎

##  3\. $U_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$

######  Theorem 3.1.

If $f\in L_{I}^{\infty}(\mu)$, then the map
$\phi_{f}:\mathbb{R}\rightarrow\mathcal{M}(X,\mathcal{A})$ given by
$\phi_{f}(r)=rf$ is continuous if $\mathcal{M}(X,\mathcal{A})$ is equipped
with the $U_{\mu_{I}}$-topology.

###### Proof.

Consider the basic open set $U_{\mu_{I}}(rf,\epsilon)$, where $r\in\mathbb{R}$
and $\epsilon>0$. Since $f\in L_{I}^{\infty}(\mu)$, there exists $\delta>0$,
$Z\in Z[I],E\in\cal{A},E\subseteq Z$ such that $\mu(E)=0$ and for each $x\in
Z\setminus E$, $|f(x)|\leq\delta$. Let $\lambda=\frac{\epsilon}{2\delta}$.
Then for each $t\in(r-\lambda,r+\lambda)$,
$|\phi_{f}(t)-\phi_{f}(r)|=|t-r||f|\leq\lambda|f|\leq\lambda\delta$ on
$Z\setminus E$. This implies that $\sup\limits_{x\in Z\setminus
E}|\phi_{f}(t)(x)-\phi_{f}(r)(x)|\leq\frac{\epsilon}{2}<\epsilon$. This means
that $\phi_{f}(t)\in U_{\mu_{I}}(\phi_{f}(r),\epsilon)$ where $|t-r|<\lambda$.
Thus the function $\phi_{f}$ becomes continuous at the point $r\in\mathbb{R}$.
∎

Since the $U_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$ is weaker
than the $m_{\mu_{I}}$-topology, the next result is an improvement of Theorem
2.2.

######  Theorem 3.2.

The set $L_{I}^{\infty}(\mu)$ is a clopen subset of
$\mathcal{M}(X,\mathcal{A})$ in the $U_{\mu_{I}}$-topology.

###### Proof.

Let $f\in L_{I}^{\infty}(\mu)$. Then since $Z[I]$ is closed under finite
intersection and the union of two sets of $\mu$-measure zero is again a set of
$\mu$-measure zero, it follows by straight forward arguments that
$U_{\mu_{I}}(f,\frac{1}{3})\subseteq L_{I}^{\infty}(\mu)$ from which we can
say that $f$ is an interior point of $L_{I}^{\infty}(\mu)$ in the
$U_{\mu_{I}}$-topology. Thus $L_{I}^{\infty}(\mu)$ is open in this topology.
To show that $L_{I}^{\infty}(\mu)$ is a closed subset of
$\mathcal{M}(X,\mathcal{A})$ in the same topology, let $f\in$
cl${}_{\mathcal{M}(X,\mathcal{A})}L_{I}^{\infty}(\mu)$. Then for each
$\epsilon>0$ in $\mathbb{R}$, $U_{\mu_{I}}(f,\epsilon)\cap
L_{I}^{\infty}(\mu)\neq\emptyset$. Select a function $g\in
U_{\mu_{I}}(f,\frac{1}{2})\cap L_{I}^{\infty}(\mu)$. Then there exists $Z\in
Z[I],E\subseteq Z$ with $\mu(E)=0$ and $\delta>0$ such that for each $x\in
Z\setminus E$, $|g(x)|\leq\delta$ and $|g(x)-f(x)|\leq\frac{1}{2}$. This
implies that for each $x\in Z\setminus E$, $|f(x)|\leq\delta+\frac{1}{2}$,
from which it follows that $f\in L_{I}^{\infty}(\mu)$. Thus
$L_{I}^{\infty}(\mu)$ is closed in $\mathcal{M}(X,\mathcal{A})$ in the
$U_{\mu_{I}}$-topology. ∎

######  Theorem 3.3.

$L_{I}^{\infty}(\mu)$ is the component of $0$ in $\mathcal{M}(X,\mathcal{A})$
in the $U_{\mu_{I}}$-topology.

###### Proof.

It follows from Theorem 3.1 that for each $f\in L_{I}^{\infty}(\mu)$,
$\phi_{f}(\mathbb{R})$ is a connected subset of $\mathcal{M}(X,\mathcal{A})$
in the $U_{\mu_{I}}$-topology. As $L_{I}^{\infty}(\mu)$ is a vector space over
$\mathbb{R}$, we can write
$L_{I}^{\infty}(\mu)=\displaystyle\bigcup\limits_{f\in
L_{I}^{\infty}(\mu)}\phi_{f}(\mathbb{R})=$ the union of connected sets each of
which contains the function $0$. Thus $L_{I}^{\infty}(\mu)$ is a connected
subset of $\mathcal{M}(X,\mathcal{A})$ in the $U_{\mu_{I}}$-topology. It
follws from Theorem 3.2 therefore that $L_{I}^{\infty}(\mu)$ is a maximal
connected subset of $\mathcal{M}(X,\mathcal{A})$ containing $0$ in the
$U_{\mu_{I}}$-topology. In other wards $L_{I}^{\infty}(\mu)$ is the component
of $0$ in $\mathcal{M}(X,\mathcal{A})$ in this topology. ∎

It is a standard result in general topology that the component of a point in a
topological space is contained in a quasicomponent of the same point. But we
have the following result in our situation.

######  Theorem 3.4.

The component and quasicomponent of $0$ in $\mathcal{M}(X,\mathcal{A})$ in the
$U_{\mu_{I}}$-topology are identical.

###### Proof.

It is sufficient to show in view of the preceeding observation that the
quasicomponent of $0$ in $\mathcal{M}(X,\mathcal{A})$ is contained in the
component of $0$. This is immediate because $L_{I}^{\infty}(\mu)$ is a clopen
subset of $\mathcal{M}(X,\mathcal{A})$ in the $U_{\mu_{I}}$-topology
containing $0$ and the quasicomponent of $0$ is the intersection of all clopen
sets which contain $0$. ∎

It is known that the component of $0$ in a topological ring is an ideal in
this ring (Theorem 2.6, [6]). The following counterexample shows that in a
ring with a topology which is compatible only with the underlying additive
group structure, the component of $0$ may not be an ideal.

######  Example 3.5.

Let $X=[0,1]$, $\mathcal{A}=$ the $\sigma$-algebra of all Lebesgue measurable
subsets of $X$, $\mu$ is the Lebesgue measure. Thus
$\mathcal{M}(X,\mathcal{A})$ is the ring of all real-valued Lebesgue
measurable functions over $[0,1]$. Suppose
$I=\\{f\in\mathcal{M}(X,\mathcal{A}):f[0,\frac{1}{3}]=\\{0\\}\\}$. Then $I$ is
an ideal in $\mathcal{M}(X,\mathcal{A})$. We assert that
$L_{I}^{\infty}(\mu)\subsetneq\mathcal{M}(X,\mathcal{A})$ and hence
$L_{I}^{\infty}(\mu)$ is not an ideal in $\mathcal{M}(X,\mathcal{A})$ because
the function $1\in L_{I}^{\infty}(\mu)$.

###### Proof of the assertion:.

Define the function $f:[0,1]\rightarrow\mathbb{R}$ as follows: corresponding
to a strictly increasing sequence $t_{1}<t_{2}<\ldots<t_{n}<\ldots$ in the
interval $(0,\frac{1}{3})$ with
$\lim\limits_{n\rightarrow\infty}t_{n}=\frac{1}{3}$, set
$f([0,t_{1}))=\\{1\\}$, $f([t_{n-1},t_{n}))=\\{n\\}$ for $n\geq
2,n\in\mathbb{N}$ and $f=0$ on $[\frac{1}{3},1]$. We see that
$f\in\mathcal{M}(X,\mathcal{A})$ (by the well-known pasting lemma for
measurable functions) but $f\notin L_{I}^{\infty}(\mu)$. Thus
$f\in\mathcal{M}(X,\mathcal{A})\setminus L_{I}^{\infty}(\mu)$. ∎

We recall $U_{\mu_{I}}^{+}=\\{f\in\mathcal{M}(X,\mathcal{A}):$ there exists
$Z\in Z[I]$ and $E\subseteq Z$ with $\mu(E)=0$ such that for each $x\in
Z\setminus E,\ f(x)>0\\}$. We now set
$U_{\mu_{I}}^{++}=\\{f\in\mathcal{M}(X,\mathcal{A}):$ there exists $Z\in
Z[I],E\subseteq Z$ with $\mu(E)=0$ and $\lambda>0$ in $\mathbb{R}$ such that
for each $x\in Z\setminus E,\ f(x)\geq\lambda\\}$. It is clear that
$U_{\mu_{I}}^{++}\subseteq U_{\mu_{I}}^{+}$. The following theorem decides
when $U_{\mu_{I}}^{+}$ and $U_{\mu_{I}}^{++}$ are equal.

######  Theorem 3.6.

For an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$,
$U_{\mu_{I}}^{+}=U_{\mu_{I}}^{++}$ if and only if the $m_{\mu_{I}}$-topology
on $\mathcal{M}(X,\mathcal{A})$ coincides with the $U_{\mu_{I}}$-topology.

###### Proof.

First assume that $m_{\mu_{I}}$-topology = $U_{\mu_{I}}$-topology. If possible
let $U_{\mu_{I}}^{++}\subsetneq U_{\mu_{I}}^{+}$. So we can select a $\omega$
from $U_{\mu_{I}}^{+}$ such that $\omega\notin U_{\mu_{I}}^{++}$. Now for any
$\epsilon>0$, there does not exist any $Z\in Z[I]$ and an $E\subseteq Z$ with
$\mu(E)=0$ for which we can write $|\omega(x)|\geq\frac{\epsilon}{3}$ for all
$x\in Z\setminus E$. This implies that for any
$f\in\mathcal{M}(X,\mathcal{A})$, $f+\frac{\epsilon}{3}\notin
m_{\mu_{I}}(f,\epsilon)$. On the other hand $f+\frac{\epsilon}{3}\in
U_{\mu_{I}}(f,\epsilon)$. Thus $U_{\mu_{I}}(f,\epsilon)\nsubseteq
m_{\mu_{I}}(f,\epsilon)$. This shows that the set $m_{\mu_{I}}(f,\omega)$,
which is open in the $m_{\mu_{I}}$-topology is not open in the
$U_{\mu_{I}}$-topology, a contradiction. Therefore
$U_{\mu_{I}}^{+}=U_{\mu_{I}}^{++}$.

To prove the converse, let $U_{\mu_{I}}^{++}=U_{\mu_{I}}^{+}$. Then for $v\in
U_{\mu_{I}}^{+}$, there exists $M>0,\ Z\in Z[I]$ and $E\subseteq Z$ with
$\mu(E)=0$ such that for each $x\in Z\setminus E$, $v(x)>M$. This implies that
for any $f\in\mathcal{M}(X,\mathcal{A})$, $f\in U_{\mu_{I}}(f,M)\subseteq
m_{\mu_{I}}(f,v)$. Thus each open set in the $m_{\mu_{I}}$-topology is open in
the $U_{\mu_{I}}$-topology and hence these two topologies on
$\mathcal{M}(X,\mathcal{A})$ coincide. ∎

The follwing comprehensive theorem offers several conditions each necessary
and sufficient for the coincidence of the $m_{\mu_{I}}$-topology and the
$U_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$.

######  Theorem 3.7.

The following statements are equivalent.

  1. (1)

The $U_{\mu_{I}}$-topology = the $m_{\mu_{I}}$-topology on
$\mathcal{M}(X,\mathcal{A})$.

  2. (2)

$\mathcal{M}(X,\mathcal{A})$ with the $U_{\mu_{I}}$-topology is a topological
ring.

  3. (3)

$\mathcal{M}(X,\mathcal{A})$ with the $U_{\mu_{I}}$-topology is a topological
vector space.

  4. (4)

$L_{I}^{\infty}(\mu)=\mathcal{M}(X,\mathcal{A})$.

  5. (5)

$I\mu(X,\cal{A})=L_{I}^{\infty}(\mu)$.

  6. (6)

$I\mu(X,\cal{A})=\mathcal{M}(X,\mathcal{A})$.

  7. (7)

$\mathcal{M}(X,\mathcal{A})$ with the $U_{\mu_{I}}$-topology is connected.

  8. (8)

$Z[I]$ is a $\mu$-stable family.

###### Proof.

$(1)\Rightarrow(2)$ is immediate because $\mathcal{M}(X,\mathcal{A})$ with the
$m_{\mu_{I}}$-topology is a topological ring. $(2)\Rightarrow(3)$ is also
immediate because each (real-valued) constant function on $(X,\cal{A})$ is
measurable.
$(2)\Rightarrow(4):$ Let $(2)$ be true. Then since the component of $0$ in a
topological ring is an ideal in this ring (Theorem 2.6, [6]), it follows from
Theorem 3.3, that $L_{I}^{\infty}(\mu)$ is an ideal in
$\mathcal{M}(X,\mathcal{A})$. Since $1\in L_{I}^{\infty}(\mu)$, this implies
that $L_{I}^{\infty}(\mu)=\mathcal{M}(X,\mathcal{A})$.
$(4)\Rightarrow(5):$ Let $(4)$ be true. Choose $f\in
L_{I}^{\infty}(\mu)=\mathcal{M}(X,\mathcal{A})$. Then for each
$g\in\mathcal{M}(X,\mathcal{A})$, $g\in L_{I}^{\infty}(\mu)$ implies that
$fg\in L_{I}^{\infty}(\mu)$. Thus $f\in I\mu(X,\cal{A})$. Hence
$L_{I}^{\infty}(\mu)\subseteq I\mu(X,\cal{A})\subseteq L_{I}^{\infty}(\mu)$.
Therefore $L_{I}^{\infty}(\mu)=I\mu(X,\cal{A})$.
$(5)\Rightarrow(6)$ is immediate because $I\mu(X,\cal{A})$ is an ideal in the
ring $\mathcal{M}(X,\mathcal{A})$ and $1\in L_{I}^{\infty}(\mu)$.
$(6)\Rightarrow(7)$ follows from Theorem 2.5, in conjunction with the fact
that $m_{\mu_{I}}$-topology on $\mathcal{M}(X,\mathcal{A})$ is finer than the
$U_{\mu_{I}}$-topology.
$(5)\Leftrightarrow(8)$ is the same as $(2)\Leftrightarrow(6)$ in Theorem 2.7.
$(7)\Rightarrow(4)$ follows directly from Theorem 3.2 as the set
$L_{I}^{\infty}(\mu)$ is non-void.
$(4)\Rightarrow(1)$. Let $(4)$ be true. Then
$\mathcal{M}(X,\mathcal{A})=\bigcup\limits_{\epsilon>0}U_{\mu_{I}}(0,\epsilon)$.
Let $u\in U_{\mu_{I}}^{+}$. To prove the validity of $(1)$, it suffices to
show in view of Theorem 3.6 that $u\in U_{\mu_{I}}^{++}$. Now $u\in
U_{\mu_{I}}^{+}$ implies that $u(x)>0$ for each $x\in Z(f)\setminus E$ for a
suitable $f\in I$ and an appropriate $E\subseteq Z(f)$ with $\mu(E)=0$.
Consequently then $Z(u)\cap(Z(f)\setminus E)=\emptyset$. Define the function
$u^{*}:X\rightarrow\mathbb{R}$ as follows:

$u^{*}(x)=\begin{cases}u(x)&if\ x\in Z(u)\setminus E\\\
1&otherwise.\end{cases}$

Then $u^{*}\in\mathcal{M}(X,\mathcal{A})$ and $Z(u^{*})\cap Z(f)=\emptyset$.
This implies that $Z(u^{*^{2}}+f^{2})=\emptyset$. Thus
$\frac{1}{u^{*^{2}}+f^{2}}\in\mathcal{M}(X,\mathcal{A})=L_{I}^{\infty}(\mu)$.
Hence there exists $\epsilon>0$ such that $\frac{1}{u^{*^{2}}+f^{2}}\in
U_{\mu_{I}}(0,\epsilon)$. Therefore there exists $Z\in Z[I]$ and
$E^{*}\subseteq Z$ with $\mu(E^{*})=0$ such that $\sup\limits_{x\in Z\setminus
E^{*}}\frac{1}{(u^{*^{2}}+f^{2})(x)}<\epsilon$. This implies that for each
$x\in Z\setminus E^{*}$, $(u^{*^{2}}+f^{2})(x)>\frac{1}{\epsilon}$ and hence
for each $x\in(Z\setminus E^{*})\cap Z(f)$, $u(x)>\frac{1}{\sqrt{\epsilon}}$.
This implies that $u\in U_{\mu_{I}}^{++}$. Thus
$U_{\mu_{I}}^{+}=U_{\mu_{I}}^{++}$. Thus far we have established that the
seven statements $(1),(2),(4),(5),(6),(7),(8)$ are equivalent. Since
$(2)\Rightarrow(3)$ is already established, to complete this theorem it
suffices to prove that $(3)\Rightarrow(4)$. So assume that $(4)$ is false.
Then there exists an $f\in\mathcal{M}(X,\mathcal{A})$, $f\geq 1$ on $X$ with
the following property: for each $Z\in Z[I]$ and each $E\subseteq Z$ with
$\mu(E)=0$ and for each $n\in\mathbb{N},\ \mu\\{x\in Z\setminus
E:f(x)>n\\}>0$. We assert that the scalar multiplication:

$\displaystyle\mathbb{R}\times\mathcal{M}(X,\mathcal{A})$
$\displaystyle\rightarrow\mathcal{M}(X,\mathcal{A})$ $\displaystyle(r,g)$
$\displaystyle\mapsto r.g$

is not continuous at the point $(0,f)$ in the $U_{\mu_{I}}$-topology and
therefore $(3)$ is false.

∎

##  4\. Question of completeness and separability of the pseudonormed linear
space $L_{I}^{\infty}(\mu)$

At the very beginning of this section, we make two additional assumptions:

  1. (1)

For the ideal $I$ in $\mathcal{M}(X,\mathcal{A})$, the associated filter
$Z[I]=\\{Z(f):f\in I\\}$ of measurable sets is closed under countable
intersection. We note that this condition is fulfilled if $I$ is a real
maximal ideal in $\mathcal{M}(X,\mathcal{A})$ and is trivially fulfilled with
$I=\\{0\\}$.

  2. (2)

For each $E\in Z[I],\mu(E)>0$. Observe that this condition is also obeyed by
$I=\\{0\\}$ if $\mu(X)>0$.

The following lemma will be helpful towards the completeness of
$L_{I}^{\infty}(\mu)$.

######  Lemma 4.1.

For each $f\in L_{I}^{\infty}(\mu)$, $||f||_{I}^{\infty}$ is itself an
essential $I$-upper bound of $f$ over $X$ $($thus the set of all essential
$I$-upper bounds of $f$ over $X$ has a smallest member$)$.

###### Proof.

By a known property of infimum, there exists a decreasing sequence
$t_{1}>t_{2}>\ldots$ of positive real numbers with the following property:
each $t_{n}$ is an essential $I$-upper bound of $f$ and
$\lim\limits_{n\rightarrow\infty}t_{n}=||f||_{I}^{\infty}$. Consequently for
each $n\in\mathbb{N}$, there exists $Z_{n}\in Z[I]$ and a measurable subset
$E_{n}$ of $Z_{n}$ such that $\mu(E_{n})=0$ and for each $x\in Z_{n}\setminus
E_{n}$, $|f(x)|\leq t_{n}$. Take $Z=\displaystyle\bigcap_{n=1}^{\infty}Z_{n}$.
Then by the additional hypothesis $Z\in Z[I]$ and $\\{x\in
Z:|f(x)|>||f||_{I}^{\infty}\\}\subseteq\displaystyle\bigcup_{n=1}^{\infty}\\{x\in
Z_{n}:|f(x)|>t_{n}\\}$. This implies that

$\mu\\{x\in Z:|f(x)|>||f||_{I}^{\infty}\\}\leq\sum_{n=1}^{\infty}\mu\\{x\in
Z_{n}:|f(x)|>||f||_{I}^{\infty}\\}\ \ldots(*)$

We recall that $|f(x)|\leq t_{n}$ for each $x\in Z_{n}\setminus E_{n}$ and
$\mu(E_{n})=0$. This implies that $\mu\\{x\in Z_{n}:|f(x)|>t_{n}\\}=0$ for
each $n\in\mathbb{N}$. It follows from the above relation $(*)$ that
$\mu\\{x\in Z:|f(x)|>||f||_{I}^{\infty}\\}=0$. Hence $||f||_{I}^{\infty}$ is
an essential $I$-upper bound of $f$ over $X$. ∎

######  Theorem 4.2.

The pseudonormed linear space $(L_{I}^{\infty}(\mu),||.||_{I}^{\infty})$ is
complete.

###### Proof.

Let $\\{f_{k}\\}$ be a cauchy sequence in $L_{I}^{\infty}(\mu)$. Since by
Lemma 4.1 for each pair of indices $k,j\in\mathbb{N}$,
$||f_{k}-f_{j}||_{I}^{\infty}$ is an essential $I$-upper bound of the function
$f_{k}-f_{j}$, it follows that there exists a set $Z_{k,j}\in Z[I]$ and a
measurable set $E_{k,j}\subseteq Z_{k,j}$ with $\mu(E_{k,j})=0$ for which we
can write: for each $x\in Z_{k,j}\setminus E_{k,j}$,
$|f_{k}(x)-f_{j}(x)|\leq||f_{k}-f_{j}||_{I}^{\infty}$. Let
$Z=\bigcap\limits_{k=1}^{\infty}\bigcap\limits_{j=1}^{\infty}Z_{k,j}$ and
$E=\bigcup\limits_{k=1}^{\infty}\bigcup\limits_{j=1}^{\infty}E_{k,j}$. Then by
hypothesis $Z\in Z[I]$ and $\mu(E)=0$. Thus we can write for each $x\in
Z\setminus E$, for each $k,j\in\mathbb{N}$,

$\ \ \ \ \ \ \ \ \ \ \ |f_{k}(x)-f_{j}(x)|\leq||f_{k}-f_{j}||_{I}^{\infty}\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ldots(1)$

Since $\\{f_{k}\\}$ is initially chosen to be a cauchy sequence in
$L_{I}^{\infty}(\mu)$, it follows from the relation $(1)$ above that, for each
$x\in Z\setminus E$, $\\{f_{k}(x)\\}_{k=1}^{\infty}$ is a cauchy sequence in
$\mathbb{R}$ and hence $\lim\limits_{k\rightarrow\infty}f_{k}(x)$ exists in
$\mathbb{R}$. Define the function $g:X\rightarrow\mathbb{R}$ as follows:
$g(x)=$ $\lim\limits_{k\rightarrow\infty}f_{k}(x)$ if $x\in Z\setminus E$ and
$g(x)=0$ otherwise. Then $g\in\mathcal{M}(X,\mathcal{A})$. To complete this
theorem, it remains to check that $g\in L_{I}^{\infty}(\mu)$ and
$\lim\limits_{k\rightarrow\infty}f_{k}=g$ in this space. Choose $\epsilon>0$.
Then there exists $k_{0}\in\mathbb{N}$ such that for all $j\geq k_{0}$, for
all $k\geq k_{0}$, $||f_{k}-f_{j}||_{I}^{\infty}<\epsilon$. This inequality in
conjunction with the inequality $(1)$ yields: for all $j\geq k_{0}$, for all
$k\geq k_{0}$, for all $x\in Z\setminus E$, $|f_{k}(x)-f_{j}(x)|<\epsilon$ on
taking limit as $k\rightarrow\infty$, this further yields that:

$\displaystyle\forall\ x\in Z\setminus E,\ \forall\ j\geq$ $\displaystyle
k_{0},\ |\lim\limits_{k\rightarrow\infty}f_{k}(x)-f_{j}(x)|\leq\epsilon$
$\displaystyle\text{i.e.,}\ |g(x)-f_{j}(x)|\leq\epsilon\ \ \ \ \ \ \ \ \ \
\ldots(2)$

It follows from $(2)$ that $g-f_{k_{0}}$ is essentially $I$-bounded on $X$,
i.e., $g-f_{k_{0}}\in L_{I}^{\infty}(\mu)$. Since $f_{k_{0}}\in
L_{I}^{\infty}(\mu)$, it follows therefore that $g\in L_{I}^{\infty}(\mu)$.
The relation $(2)$ further implies that for all $j\geq k_{0}$, the positive
number $\epsilon$ is an essential $I$-upper bound of the function $f_{j}-g\in
L_{I}^{\infty}(\mu)$. It follows that $||f_{j}-g||_{I}^{\infty}\leq\epsilon$
for each $j\geq k_{0}$. Hence $\lim\limits_{j\rightarrow\infty}f_{j}=g$ in
$L_{I}^{\infty}(\mu)$. The proof is complete. ∎

Normally the space $L^{\infty}(\mu)$ of all essentially bounded measurable
functions over a measure space $(X,\cal{A},\mu)$ is not separable (see the
comments in page 580 in [7]). It is well-known that the space of all
essentially bounded Lebesgue measurable functions over the closed interval
$[0,1]$ is not separable [9]. In our terminology this reads on choosing
$X=[0,1]$, $\cal{A}=$ the $\sigma$-algebra of all Lebesgue measurable subsets
of $[0,1]$, $\mu=$ the Lebesgue measure and $I=\\{0\\}$, that
$L_{I}^{\infty}(\mu)$ is not separable. It is also known that the space
$l^{\infty}$ of all bounded sequence of real numbers (or complex numbers) with
familier supremum norm is non separable. We can realize $l^{\infty}$ as the
space $L_{I}^{\infty}(\mu)$ on specialising $X=\mathbb{N},\
\cal{A}=P(\mathbb{N}),\ \mu=$ the counting measure on $\mathbb{N}$ and
$I=\\{0\\}$. We shall exhibit a non-trivial ideal $I\neq\\{0\\}$ in the ring
$\mathcal{M}(X,\mathcal{A})$ arising out in each of the above two examples to
give two examples of spaces of the form $L_{I}^{\infty}(\mu)$, none of which
is separable.

######  Example 4.3.

Let $X=[0,1]$, $\cal{A}=$ the $\sigma$-algebra of all Lebesgue measurable
subsets of $[0,1]$, $\mu=$ the Lebesgue measure and

$I=\\{f\in\mathcal{M}(X,\mathcal{A}):f(x)=0\ \text{for each}\
x\in[0,\frac{1}{2}]\\}.$

Let $t_{1}<t_{2}<\ldots$ be a strictly increasing sequence in the open
interval $(0,\frac{1}{2})$ which converges to $\frac{1}{2}$. Let
$A_{n}=(t_{n-1},t_{n})$, $n\geq 1$, writing $t_{0}=0$. Suppose $J=\\{f\in
L_{I}^{\infty}(\mu):f(t_{i})=0$ for all $i\in\mathbb{N}\cup\\{0\\},\ f=0\
\text{on}\ [\frac{1}{2},1]$ and for each $n\in\mathbb{N}$, either
$f(A_{n})=\\{0\\}$ or $f(A_{n})=\\{1\\}$}. We assert that $J$ is an
uncountable set. To prove this assertion, if possible let $J$ be a countable
set: $J=\\{f_{1},f_{2},\ldots f_{n},\ldots\\}$. Define a function
$g:X\rightarrow\mathbb{R}$ as follows
$g(\\{t_{0},t_{1},\ldots,t_{n},\ldots\\}\cup[\frac{1}{2},1])=0$ and given
$n\in\mathbb{N}$, $g(A_{n})=1$ if $f_{n}(A_{n})=\\{0\\}$ and $g(A_{n})=0$ if
$f_{n}(A_{n})=1$. Then it is easy to check that $g\in J$, but $g$ is different
from each $f_{n}$ (on a set of positive Lebesgue measure), a contradiction. We
further note that whenever $f,g\in J$ and $f\neq g$, then there exists
$n\in\mathbb{N}$ for which $f(A_{n})=\\{0\\}$ and $g(A_{n})=\\{1\\}$, say.
This implies that $g-f=1$ on $A_{n}$ and $\mu(A_{n})>0$. It follows that no
number $<1$ can be an essential $I$-upper bound of $g-f$ over $[0,1]=X$. In
other words $||f-g||_{I}^{\infty}\geq 1$. Indeed, it is easy to check that
$||f-g||_{I}^{\infty}=1$ if $f,g\in J$ with $f\neq g$. If for $h\in
L_{I}^{\infty}(\mu)$ and $\epsilon>0$ in $\mathbb{R}$, $B(h,\epsilon)=\\{g\in
L_{I}^{\infty}(\mu):||g-h||_{I}^{\infty}\\}=$ the open ball centered at ‘$h$’
with radius $\epsilon$ in the pseudonormed linear space $L_{I}^{\infty}(\mu)$
then the last observation implies that $\\{B(f,\frac{1}{3}):f\in J\\}$ is an
uncountable family of pairwise disjoint open balls in this space. Thus
$L_{I}^{\infty}(\mu)$ turns out to be a non-separable pseudonormed linear
space.

######  Example 4.4.

Take $X=\mathbb{N}$, $\cal{A}=P(\mathbb{N})=$ the power set of $\mathbb{N}$,
$\mu=$ the counting measure on $(X,\cal{A})$ and $S=\\{n_{k}\\}$ be any
subsequence of $\mathbb{N}$, i.e., $n_{1}<n_{2}<\ldots$ is an increasing
sequence of natural numbers without any upper bound.

Suppose $I=\\{f\in\mathcal{M}(X,\mathcal{A}):f(n_{k})=0$ for each
$k\in\mathbb{N}\\}$. Then $I$ is an ideal in the ring
$\mathcal{M}(X,\mathcal{A})$. In this setting $L_{I}^{\infty}(\mu)$ consists
precisely all those sequences $\\{x_{n}\\}$ of real numbers for which the
subsequences $\\{x_{n_{k}}\\}$ are bounded.

Let $J=\\{f\in L_{I}^{\infty}(\mathbb{N},P(\mathbb{N}),\mu):f(n)=0$ if
$n\in\mathbb{N}\setminus S$ and for each $n_{k}\in S,\ f(n_{k})\in\\{0,1$}.
Then $J$ is an uncountable set and for any two distinct $f,g\in J$, it is
clear that there exists some $n_{k}$ for which $f(n_{k})=1$ and $g(n_{k})=0$
(say). Then $1$ is an essential $I$-upper bound of $f-g$ on $\mathbb{N}$
because $|f-g|\leq 1$ on the whole of $S\in Z[I]$. On the other hand no number
$\lambda\ (0<\lambda<1)$ can be an essential $I$-upper bound of $g-f$ on
$\mathbb{N}$ as
$\\{n\in\mathbb{N}:|(g-f)(x)|>\lambda\\}=\\{n\in\mathbb{N}:|g(x)-f(x)|=1\\}\supseteq\\{n_{k}\\}$.
This implies that
$\mu\\{n\in\mathbb{N}:|(g-f)(x)|>\lambda\\}\geq\mu\\{n_{k}\\}=1$ because $\mu$
is the counting measure on $\mathbb{N}$. Thus $||g-f||_{I}^{\infty}=1$. Now
arguing as in the previous example, taking into consideration the
uncountability of the set $J$, we can conclude that
$L_{I}^{\infty}(\mathbb{N},P(\mathbb{N}),\mu)$ is a non-separable Banach
space.

We shall conclude this article after making a few remarks in connection with
the following classification of ideals in the ring
$\mathcal{M}(X,\mathcal{A})$, effected by the $U_{\mu_{I}}$ and
$m_{\mu_{I}}$-topologies, with two rather special choices of the measure
$\mu$, viz the counting measure and Dirac measure. With $\mu=$ counting
measure, let us call $U_{\mu_{I}}$-topology to be simply $U_{I}$-topology and
the $m_{\mu_{I}}$-topology as $m_{I}$-topology. Also for
$f\in\mathcal{M}(X,\mathcal{A})$, $u\in U_{\mu_{I}}^{+}$ and $\epsilon>0$ in
$\mathbb{R}$, we prefer to write $U_{I}(f,\epsilon)$ and $m_{I}(f,u)$ instead
of $U_{\mu_{I}}(f,\epsilon)$ and $m_{\mu_{I}}(f,u)$ respectively. To address
this problem and to make this paper self-contained, we write the following
background material, which we reproduce from the article [4].

Suppose $\widehat{X}$ is an enlargement of the set $X$ (in any convenient
manner) with the purpose that it will serve as an index set for the aggregate
of all maximal ideals in the ring $\mathcal{M}(X,\mathcal{A})$. For each
measurable set $A\in\cal{A}$,
$\bar{A}=\\{p\in\widehat{X}:A\in\mathcal{U}^{p}\\}$, here $M^{p}$ is the
maximal ideal in $\mathcal{M}(X,\mathcal{A})$ that corresponds to the point
$p\in\widehat{X}$ and $\cal{U}=Z[M^{p}]$ is the associated ultrafilter of
measurable sets on $X$. If $p\in X$, then
$\cal{U}^{p}=\cal{U}_{p}=\\{E\in\cal{A}:p\in E\\}=$ the fixed ultrafilter of
measurable sets on $X$ corresponding to the point $p$. Also $\bar{A}\cap X=A$
for each $A\in\cal{A}$. The family $\\{\bar{A}:A\in\cal{A}\\}$ constitutes a
base for the closed sets of some topology, familiarly known as the Stone-
topology on $\widehat{X}$. Using this topology on $\widehat{X}$, a complete
description of the maximal ideals in $\mathcal{M}(X,\mathcal{A})$ is provided
by the following list $\\{M^{p}:p\in\widehat{X}\\}$, where for
$p\in\widehat{X}$, $M^{p}=\\{f\in\mathcal{M}(X,\mathcal{A}):p\in$
cl${}_{\widehat{X}}Z(f)\\}$. This may be called the measure theoretic analogue
of Gelfand-Kolmogoroff theorem in the theory of rings of continuous functions.
Since every ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ happens to be an
intersection of maximal ideals [4], we can write $I=\bigcap\limits_{p\in
Y}M^{p}$ for some subset $Y$ of $\widehat{X}$.

We now define two binary relations ‘$\sim$’ and ‘$\approx$’ on the set
$\cal{I}$ of all ideals in $\mathcal{M}(X,\mathcal{A})$ as follows: for
$I,J\in\cal{I}$, $I\sim J$ if and only if $m_{I}$-topology = $m_{J}$-topology,
and $I\approx J$ if and only if $U_{I}$-topology = $U_{J}$-topology. Clearly
‘$\sim$’ and ‘$\approx$’ are equivalence relations on $\cal{I}$.

The following theorem essentially says that, these two classifications of
ideals in $\mathcal{M}(X,\mathcal{A})$ via ‘$\sim$’ and ‘$\approx$’, albeit
outwardly different are essentially the same. Indeed, each equivalence class
in both the quotient sets $\cal{I}/_{\sim}$ and $\cal{I}/_{\approx}$ is a
singleton.

######  Theorem 4.5.

For two ideals $I$ and $J$ in $\cal{I}$, $I\sim J$ if and only if $I\approx J$
if and only if $I=J$.

###### Proof.

If $I=J$, then it is trivial that $I\sim J$ and $I\approx J$. To prove the
non-trivial part of this theorem assume that $I\sim J$. we shall prove that
$J\subseteq I$, from symmetry it will follow that $I\subseteq J$ and hence
$I=J$. If possible let $J\nsubseteq I$. We can choose $f\in J$ such that
$f\notin I$. Now $I=\bigcap\limits_{p\in Y}M^{p}$ for some subset $Y$ of
$\widehat{X}$. Therefore $f\notin M^{q}$ for some point $q\in\widehat{X}$. It
follows on using the above Gelfand-Kolmogoroff like theorem that $q\notin$
cl${}_{\widehat{X}}Z(f)$. But as $\\{\bar{A}:A\in\cal{A}\\}$ constitutes a
closed base for the Stone-topology on $\widehat{X}$, this further implies that
there exists $A\in\cal{A}$ such that $q\notin\bar{A}$ yet
cl${}_{\widehat{X}}Z(f)\subseteq\bar{A}$. It follows that
cl${}_{\widehat{X}}Z(f)\cap X\subseteq\bar{A}\cap X$, this means that
$Z(f)\subseteq A$.

Let $h\in\mathcal{M}(X,\mathcal{A})$ be the characteristic function of
$X\setminus A$, i.e., $h=\chi_{X\setminus A}$. Now $m_{I}(0,\frac{1}{2})$ is
open in $\mathcal{M}(X,\mathcal{A})$ in the $m_{I}$-topology. To get the
desired contradiction we shall check that $m_{I}(0,\frac{1}{2})$ is not open
in the $m_{J}$-topology and hence $I$ is not related to $J$ by the binary
relation ‘$\sim$’ on $\cal{I}$. If possible let $m_{I}(0,\frac{1}{2})$ be open
in the $m_{J}$-topology. Then there exists $v\in U_{\mu_{J}}^{+}=U_{J}^{+}$
such that $m_{J}(0,v)\subseteq m_{I}(0,\frac{1}{2})$. We note that $v>0$ on
some $Z_{0}\in Z[J]$. Now since $f\in J$, if follows that $Z(f)\cap Z_{0}\in
Z[J]$. We recall that $Z(f)\subseteq A$, from which it follows that $h=0$ on
$Z(f)$ and hence $h=0$ on $Z(f)\cap Z_{0}\in Z[J]$. Since $v$ is already ‘$0$’
on this last measurable set, this implies that $h\in m_{J}(0,v)\subseteq
m_{I}(0,\frac{1}{2})$. Therefore there exists $g\in I$ such that
$|h(x)|<\frac{1}{2}$ for each $x\in Z(g)$ on the other hand
$I=\bigcap\limits_{p\in Y}M^{p}$ implies that $g\in M^{q}$ and hence $q\in$
cl${}_{\widehat{X}}Z(g)$. But as $q\notin\bar{A}$, we get that
$\widehat{X}\setminus\bar{A}$ is an open neighbourhood of $q$ in the space
$\widehat{X}$. This surely implies that $(\widehat{X}\setminus\bar{A})\cap
Z(g)\neq\emptyset$. Since $Z(g)\subseteq X$, this further implies that
$(X\setminus A)\cap Z(g)\neq\emptyset$. Choose a point $x\in(X\setminus A)\cap
Z(g)$. Then $x\in X\setminus A$ implies that $h(x)=1$. On the other hand $x\in
Z(g)$ implies that $|h(x)|<\frac{1}{2}$, this is a contradiction. Thus far we
have proved that $I=J$ if and only if $m_{J}$-topology = $m_{I}$-topology on
$\mathcal{M}(X,\mathcal{A})$.

Now if we make a close scrutiny in the above arguments, then it is revealed
that the assumption $J\nsubseteq I$, made at the beginning of the proof of the
present theorem, would have led to the conclusion that the open set
$U_{I}(0,\frac{1}{2})$ in $\mathcal{M}(X,\mathcal{A})$ in the $U_{I}$-topology
(we write $U_{\mu_{I}}(0,\frac{1}{2})=U_{I}(0,\frac{1}{2})$) is not open in
the $m_{I}$-topology and consequently not open in $U_{I}$-topology (because
$U_{I}$-topology $\subseteq$ $m_{I}$-topology). Thus it is proved that
$I\approx J$ if and only if $I=J$. The theorem is completely proved. ∎

Our next classification of ideals in $\mathcal{M}(X,\mathcal{A})$ is with
respect to the choice $\mu=$ an appropriately defined Dirac measure. So let us
prepare our set up once again. Here $X$ is an infinite set, $\cal{A}=P(X)$,
the family of all subsets of $X$ and for any chosen $x_{0}\in X$, the measure
$\delta_{x_{0}}:P(X)\rightarrow\\{0,1\\}$ is defined as follows:
$\delta_{x_{0}}(E)=\\{1\\}$ if $x_{0}\in E$, $\delta_{x_{0}}(E)=\\{0\\}$ if
$x_{0}\notin E$. $\delta_{x_{0}}$ is called a Dirac measure on $X$. As in the
preceeding theorem $\cal{I}$ stands for the set of all ideals in the measure
space $(X,P(X),\delta_{x_{0}})$. In view of the additional assumption (2) made
at the beginning of this section, for any $I\in\cal{I}$ and for each $E\in
Z[I]$, $\delta_{x_{0}}(E)>0$, this means that $x_{0}\in E$. Thus
$x_{0}\in\bigcap Z[I]$. Now as in the preceeding theorem define the binary
relation ‘$\sim$’ and ‘$\approx$’ on $\cal{I}$ as follows: for
$I,J\in\cal{I}$, $I\sim J$ if and only if $m_{{\delta_{x_{0}}}_{I}}$-topology
= $m_{{\delta_{x_{0}}}_{J}}$-topology on $\cal{M}(X,P(X),\delta_{x_{0}})$.
Here we write $m_{{\delta_{x_{0}}}_{I}}$ in palce of $m_{\mu_{I}}$ on choosing
$\mu=\delta_{x_{0}}$. Also writing $U_{{\delta_{x_{0}}}_{I}}$-topology =
$U_{\mu_{I}}$-topology with this choice of $\mu$, we define our second binary
relation ‘$\approx$’ on $\cal{I}$ as follows: for $I,J\in\cal{I}$, $I\approx
J$ if and only if $U_{{\delta_{x_{0}}}_{I}}$-topology =
$U_{{\delta_{x_{0}}}_{J}}$-topology on $\cal{M}(X,P(X),\delta_{x_{0}})$. It is
easy to check that for any $f\in\cal{M}(X,P(X))$, $u\in
U_{{\delta_{x_{0}}}_{I}}^{+}$ and $\epsilon>0$ in $\mathbb{R}$,
$m_{{\delta_{x_{0}}}_{I}}(f,u)=m_{{\delta_{x_{0}}}_{J}}(f,u)$ and
$U_{{\delta_{x_{0}}}_{I}}(f,\epsilon)=U_{{\delta_{x_{0}}}_{J}}(f,\epsilon)$
for any two ideals $I$ and $J$ in the family $\cal{I}$. This leads to the
following proposition which as something far opposite to what happens in
Theorem 4.5 that each of the quotient sets $\cal{I}/_{\sim}$ and
$\cal{I}/_{\approx}$, degenerates into a one-membered set.

######  Theorem 4.6.

For any two ideals $I$ and $J$ in $\cal{I}$, $I\sim J$ and also $I\approx J$.

######  Question 4.7.

The following problems are still open.

  1. (1)

Suppose an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ is such that the
pseudonormed linear sapce $L_{I}^{\infty}(\mu)$ is complete. Is the filter
$Z[I]$ closed under countable intersection?

  2. (2)

Does there exist a measure space $(X,\cal{A},\mu)$ and a non-zero ideal $I$ in
$\cal{M}(X,\cal{A},\mu)$ for which the pseudonormed linear space
$L_{I}^{\infty}(\mu)$ is separable?

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# A Security and Trust Framework for
Decentralized 5G Marketplaces

José María Jorquera Valero, Manuel Gil Pérez, and Gregorio Martínez Pérez
Faculty of Computer Science, University of Murcia, 30100 Murcia, Spain
Email: {josemaria.jorquera, mgilperez<EMAIL_ADDRESS>
###### Abstract

5G networks intend to cover user demands through multi-party collaborations in
a secure and trustworthy manner. To this end, marketplaces play a pivotal role
as enablers for network service consumers and infrastructure providers to
offer, negotiate, and purchase 5G resources and services. Nevertheless,
marketplaces often do not ensure trustworthy networking by analyzing the
security and trust of their members and offers. This paper presents a security
and trust framework to enable the selection of reliable third-party providers
based on their history and reputation. In addition, it also introduces a
reward and punishment mechanism to continuously update trust scores according
to security events. Finally, we showcase a real use case in which the security
and trust framework is being applied.

###### Index Terms:

Security, Trust Management, 5G, Trustworthy Relationships

Tipo de contribución: Investigación en desarrollo

##  I Introduction

The fifth generation of mobile networks (5G) entails a continuously increasing
figure of interconnected end-users as well as relationships among entities.
Thereby, 5G resource, service, and infrastructure providers are constantly
working on innovative solutions to deal with the huge demand from tenants and
users for data capacity, bandwidth, network coverage, and latency. One of the
most promising approaches is decentralized marketplaces. A marketplace is
normally described as a centralized or decentralized repository in which
thousand of offers are advertised by resource and infrastructure providers.
The trading of such resources and services is carried out by consumers who
need to cover dynamic requirements and high Quality-of-Service (QoS); for
instance, high speed, low energy consumption, and automatically satisfy user
demands. In this context, marketplaces boost the generation of service chains
across operators with security and trustworthiness [1].

Conventionally, trust has been employed as a mechanism capable of determining
the trustworthiness level that a trustor has in a trustee. In this regard,
there are multiple models for determining a trust level such as PKI-based,
role-based, or reputation-based, among others. The latter is one of the most
considered models, since it not only allows the trust to be calculated based
on entity’s behavior in previous interactions but also allows gathering
feedback from reliable recommenders who also interacted with the entity. Yet,
reputation-based trust models also open the door to key challenges to be
addressed whether they want to be one of the technologies that await to
support next-generation solutions (5G and beyond) [2].

Especially, trust models should guarantee a minimum set of features in order
to be integrated with promising 5G solutions. First and foremost, trust models
need to provide highly dynamic and context-dependence solutions as 5G
ecosystems tend to support scenarios where main actors scale and migrate
flexibly. In this sense, trust models should continuously collect information
from the principal actors involved in the relationship through automatic
mechanisms triggered by events, rules, time, etc. By means of these
mechanisms, the trust score can be rapidly adapted in real-time, and in
consequence, adjust the participating entities, if necessary. Besides, trust
models should ensure reliable end-to-end establishments, therefore any
intermediate entities must be analyzed. Another pivotal aspect to be covered
is the automated management following a zero-touch approach. On the one hand,
trust models should leverage tools that enable the automatization of network
and service management via high-level policies, and artificial intelligence
algorithms. On the other hand, they should empower a flexible integration with
other 5G essential services and their workflows. Lastly, trust models should
fulfill the zero trust principle [3]. It attempts not to attribute implicit
trust to an entity regardless of whether the trustor and trustee had
previously a relationship or whether both entities belong to the same domain.
Therefore, trust models would avoid utilizing an outdated trust value without
being updated at a given time as well as assign trust-by-default to entities
under our reliable zone.

Figure 1: Modular architecture of the proposed security and trust framework

Despite the progress of literature, more efforts are needed to enhance the
trust and reputation models in 5G scenarios as prior trust models did not
contemplate all requirements set out above. Furthermore, the trust concept
also includes security aspects, therefore considering security-related
features for profiling actor behavior would help to detect feasible threats as
well as to have a broader view when determining trust. In this regard, this
paper presents a security and trust framework capable of ensuring a
trustworthy network inside of a decentralized marketplace. In particular, such
a framework enables consumers to establish trustworthy end-to-end connections
across domains, at the same time it covers the aforementioned challenges. To
this end, the security and trust framework analyzes a set of product offers,
published by resource and infrastructure providers in a decentralized
marketplace, in order to predict trust scores from previous interactions as
well as recommendations from trustworthy third parties. It should be pointed
out that such a framework is being developed in the 5GZORRO H2020 European
project [4], which enables a secure, flexible, and distributed multi-
stakeholder combination and composition of resources and services in 5G
networks. Additionally, the security and trust framework also introduces a
reward and punishment mechanism to continuously update trust levels from
multiple security events obtained in real-time. Finally, the paper introduces
a real use case to demonstrate how the framework is integrated with other 5G
services and helps to perform a smart resource and service discovery.

The remainder of this paper is structured as follows. Section II presents an
overview of our security and trust framework, detailing the principal modules
and the main actions to be performed. Section III introduces the use case in
which our framework is being applied. Finally, Section IV expounds some
conclusions an open perspectives for future work.

##  II The Security and Trust Framework

This section describes a modular architecture of the proposed security and
trust framework which enables to cater to a trustworthy ecosystem among
multiple domains and operators. As Fig. 1 sketches, the security and trust
framework is made up of four principal modules: the information gathering and
sharing, trust computation, trust storage, and continuous updates modules. In
this vein, this section is going to thoroughly explain them as well as the
capital actions carries out.

###  II-A Information gathering and sharing module

One of the uppermost important steps in a security and trust framework is the
collection of meaningful features to afterward evaluate entity behavior. In
this sense, it is pivotal to find out a set of information sources from which
the trustor can retrieve information to ascertain the credibility and
effectiveness of the assigned trust scores. Due to the fact that this
framework is designed bearing in mind the 5GZORRO ecosystem, we have selected
four principal software modules from which we will infer features (see
interfaces in Fig. 1). On one side, we consider the Catalog module [5] which
stores the Product Offerings (POs) published by the stakeholders registered in
the 5GZORRO system. Note that there are seven types of product offers defined
in the Catalog: radio access network (RAN), spectrum, virtual/container
network function (VNF, CNF), slice, network service, cloud, and edge. By means
of this dimension, the security and trust framework is capable of deriving
statistic features such as the number of POs available per provider, per
location, per type of resource, or what POs were available but are not
currently, among other features. On another side, we contemplate the Data Lake
platform [5] which stores all the interactions performed by stakeholders in
the past. Thereby, a consumer would be able to know what stakeholders
previously interacted with a specific resource and service provider and, if
necessary, requests recommendations (indirect trust) about the provider. This
information source plays an important pillar for the security and trust
framework as this is based on a reputation model. Thence, recommendations from
third parties are considered one of the two types of trust (direct and
indirect trust) to compute a final trust score. Concerning direct trust, it is
formulated from the previous interactions with the target stakeholder as well
as new features inferred from Catalog.

In addition, the security and trust framework also gathers information from
the Security Analysis Service (SAS) [5] which is instantiated once an end-to-
end relationship is set up. Hence, it is not leveraged when the framework is
analyzing a set of POs but when a final PO is selected. Through the SAS
component, the framework can detect security threats or misbehavior in the
network traffic as the SAS monitors, through the Zeek tool [6], network
packets sent by consumers and providers. Moreover, the SAS component has
activated several security policy rules that allow triggering alerts when
unusual behaviors happen in real-time. Last but not least, the security and
trust framework also recaps information from the Intelligent Service Level
Agreement Breach Predictor (ISBP) module [5]. In this case, the framework
leverages events published by the ISBP so as to predict the provider’s
reputation and how it can be affected based on current events such as Service
Level Agreement (SLA) violations.

###  II-B Trust computation module

Once the framework collects raw data from the aforementioned information
sources, it is time to determine a trust score per each PO to be analyzed.
Note that this step is performed regardless of whether stakeholders have a
previous trust relationship for a while or whether they belong to the same
domain (zero trust). Since the 5GZORRO ecosystem boosts a decentralized
philosophy where peer-to-peer communications are carried out, we have selected
the PeerTrust model [7] as a statistic algorithm to prognosticate a trust
score. The PeerTrust model is an algorithm based on the reputation that has
been utilized in decentralized scenarios for ages. Concretely, the PeerTrust
model determines trust scores from four primary pillars: Satisfaction (S),
Credibility (Cr), Transaction Factor (TF), and Community Factor (CF), as
depicted in Eq. (1). Note that the PeerTrust model only introduces the above-
mentioned generic pillars but each researcher should find out how each one
will be formulated. Thus, our security and trust framework has carried out an
adapted PeerTrust model.

$T(u)=\alpha\cdot\displaystyle\sum_{i=1}^{I(u)}S(u,i)\cdot Cr(p(u,i))\cdot
TF(u,i)+\beta\cdot CF(u),$ (1)

Since each pillar is composed of multiple equations, this paper does not
display the 19 equations formulated to cover all pillars. Instead, we broadly
explain below how each pillar is computed as well as presenting some of the
utmost important equations. In the case of Satisfaction (S), the framework
predicts the provider’s satisfaction and the offer’s satisfaction. The main
difference between them is the former considers all assets linked to a
provider whereas the latter only envisages a given type of PO. Thence, the
provider’s and the offer’s satisfactions are formulated from the reputation, a
set of recommendations about the target, and the last trust score we have for
each recommender in the previous set. When it comes to Credibility (Cr), the
adapted PeerTrust model leverages a Personalized Similarity Metric (PSM)
through which we can discover how similar two stakeholders are when they are
evaluating the same targets. Thus, the PSM allows us to know the distance of
credibility of a set of evaluated stakeholders as well as contrast their
opinions.

Additionally, the adapted PeerTrust model also takes into account two context
factors: the Transaction Factor (TF) and the Community Factor (CF). Concerning
the TF, it intends to predict a trust value linked to the current interaction,
with a particular stakeholder or product offer, from the number of feedbacks
published in the Data Lake from different time windows. As a result, a higher
number of feedbacks published in the Data Lake, a higher number of
recommenders to be contemplated for finally computing a stable reputation. The
TF then rewards stakeholders who publish their interactions with others in the
Data Lake since it spurs future stakeholders to look into the Data Lake,
request recommendations to other stakeholders, and grow the community.
Instead, the CF intends to assess the stakeholder participation within the
community to give greater or lesser weight to their recommendations over time.
To this end, the CF collects multiple feedback from $j$-th trustworthy
opinions on a target stakeholder $u$ for computation, see Eq. (2), and notices
dishonest recommendations as rated by the confidence from recommenders (CR) on
$u$. By CF, we can thus detect conventional attacks in trust models such as
bad-mouthing attacks.

$CF(u)=\frac{\frac{R(u)}{I(u)}+\displaystyle\bigoplus_{j=1}^{n}CR(v,j,u)\cdot
Inf(v,j)}{2},$ (2)

where $R(u)$ denotes the number of published recommendations and $I(u)$ the
total number of interactions of $u$; and $Inf(v,j)$ the recommender’s
influence on all contemplated recommendations $Rec(j,u)$. Furthermore, Eq. (3)
shows how to measure the confidence of a $j$-th recommender on $u$, in
accordance to the trust on that recommender $T(v,j)$ and the recommendation
trust $RT(v,j)$.

$CR(v,j,u)=\alpha\cdot T(v,j)+(1-\alpha)\cdot\big{(}RT(v,j)\cdot
Rec(j,u)\big{)},$ (3)

Moreover, the CF also measures the number of interactions that a specific
stakeholder had in the community through the contribution of services or
resources with other stakeholders. In the end, multiple recommendations
together with the credibility of the recommender are contemplated through an
aggregation function to achieve the general reputation of the community about
a target stakeholder.

###  II-C Trust storage module

Another capital step of the security and trust framework is the storage of
non-public data, raw data, and new interactions between stakeholders. In this
vein, the framework employs two kinds of storage sources. On the one hand, the
security and trust framework manages both raw data and all features inferred
from them which are stored in a private database per domain. In this case, we
are handling information which must not be public to other stakeholders. On
the other side of the coin, the 5GZORRO ecosystem takes advantage of a Data
Lake platform that allows users to push any type of information. By means of
the Data Lake, the framework informs other stakeholders about the
interactions. Concretely, after establishing a trust relationship the trustor
pushes a new object into the Data Lake with information on the parties
involved, the start date, the number of interactions between them, etc. In
this way, newcomers can look at the Data Lake and identify what trustworthy
recommenders had interaction with a particular target, and consequently,
request their feedback about their relationships. In addition to that, the
Data Lake introduces pivotal characteristics such as decentralization, a long-
term reputation reflection, and security.

###  II-D Continuous update module

Parallel to the trust storage module, the next step is to continuously monitor
an ongoing trust relationship in order to adapt a previous trust score to the
events occurring in real-time. Hence, this module plays an essential role
since it not only enables early threat identification but also enhances the
security capabilities of the framework through dynamicity, context-dependence,
and end-to-end automatization. This module includes a reward and punishment
mechanism which allows collecting security-based network monitoring events so
as to increase or dwindle a trust score. This mechanism follows a time-driven
approach so we need to declare pre-established time windows in which the
previous trust score will be updated using the network monitoring events. To
update the previous trust scores, the security and trust framework makes use
of four main log files gathered by Zeek: conn.log, that gathers the tracking
of general information regarding TCP, UDP, and ICMP traffics; notice.log, that
collects likely monitoring events which are potentially odd; weird.log, that
takes action when unusual or exceptional activity appears; and stat.log, that
obtains memory, packet, and log statistics. These four weighted logs are shown
in Eq. (4) to compute reward and punishment (RP) values.

$\begin{split}RP(v,u)=\alpha\cdot Conn(v,u)+\beta\cdot Notice(v,u)+\\\ \cdot
Weird(v,u)+\phi\cdot Stat(v,u)\end{split}$ (4)

Finally, depending on the RP value obtained for the current time window, the
framework will update the prior trust value ($O_{ts}$) to a new trust score
($N_{ts}$) accordingly, as given in Eq. (5). Note that a value close to the
extremes will result in a larger increase or decrease than if the RP value is
around 0.5.

$N_{ts}(v,u)=\begin{cases}$$O_{ts}(v,u)+(RP(v,u)-0.5)\cdot\frac{\Big{(}1-O_{ts}(v,u)\Big{)}}{10}$$,&\text{if
}RP(v,u)\geq 0.5\\\
$$O_{ts}(v,u)-(0.5-RP(v,u))\cdot\frac{\Big{(}1-O_{ts}(v,u)\Big{)}}{10}$$,&\text{if
}RP(v,u)<0.5\end{cases}$

(5)

##  III 5GZORRO Use Case

This section describes how the security and trust framework can assist in the
process of selecting trustworthy resources and services available in a
distributed marketplace (see Fig. 2).

Figure 2: Trust-based resource discovery

Since the generation and registering of offers are not the main focus of our
use case, we have simplified the sub-tasks of these steps in order to fully
focus on the consumer side. In this case, the consumer detects a lack of
resource capabilities to cover the agreed QoS and the dynamic requirements. To
overcome the above problems, the consumer decides to extend its current
capabilities to satisfy a signed contract and its SLA. At this point, the
5GZORRO marketplace [5] aims at facilitating multi-party collaboration in
dynamic 5G environments where consumers and service providers often need to
employ third-party resources. As a result, resource providers usually offer
their resources available by promoting them through the 5GZORRO marketplace.
In general terms, the distributed marketplace allows creating and acquiring
product offers that cover a variety of telco digital assets.

To cope with this, the resource consumer leverages the Intelligent Slice and
Service Manager (ISSM) [5] to request new resource capabilities as well as
looking at the available offers in the . In this sense, two different
discovery processes take place via the Smart Resource and Service Discovery
(SRSD) [5]. On the one hand, the SRSD conducts a pre-classification of new POs
based on imperative constraints such as category, geographic location, price,
etc. On the other hand, the ISSM launches an intent-based discovery request
employing priorities such as performance, proximity, cost, etc. Afterward, the
SRSD determines trust scores and sends back to the ISSM a ranked list of
product offers based on the reputation of both the provider and its type of
offer. Therefore, the security and trust framework contributes directly to the
intelligent discovery process classifying the most trustworthy POs which
previously matched the intent-based criteria.

Afterward, the ISSM carries out additional optimization steps to find out the
best offer from the list ranked by trust scores. As a next step, the ISSM
informs the security and trust framework, called 5G-enabled Trust and
Reputation Management Framework (5G-TRMF) in the 5GZORRO ecosystem [5], about
the final offer selected by the consumer. To close the lifecycle of trust-
based resource discovery, the security and trust framework enables, as a
background process, ongoing monitoring of network traffic between the resource
consumer and the new third-party via the Security Analysis Service. As shown
in Fig 2, our framework can perfectly work with other pivotal modules such as
the ISSM, Catalog, and SRSD to help consumers to address day-to-day problems
such as resource shortages, while respecting automation and ensuring security
and trustworthiness throughout the process.

##  IV Conclusion

This paper presents a security and trust framework capable of ensuring a
reliable ecosystem where stakeholders can set up trustworthy end-to-end
connections across domains. In this vein, a modular architecture of the
proposed framework has been explained, as well as the capital steps under each
module. At the same time, the framework also introduces a reward and
punishment mechanism to continuously update an ongoing trust relationship via
security-based monitoring events generated in real-time.

As future work, we will carry out the validation of the current security and
trust framework in real testbeds such as 5GBarcelona and 5TONIC. Besides, the
continuous update module will be enhanced by adding a novel event-driven
mechanism to adjust trust scores from SLA prediction notifications.
Additionally, further computation models will be contemplated to analyze their
performance and accuracy, as well as enlarging the resilience of the framework
to other conventional trust attacks such as on-off or Sybil attacks.

## Acknowledgment

This work has been supported by the European Commission through 5GZORRO
project (grant no. 871533) part of the 5G PPP in Horizon 2020.

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