# Mathematical Foundations of Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# Measure- and Convexity-Theoretic Foundations of Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# Convexity-Theoretic Foundations of Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# Measure-Theoretic Foundations of
Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# A Theory of
Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# A Measure-Theoretic Treatment
of Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# Theoretical Underpinnings of
Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# On The Measure-Theoretic Underpinnings of
Lorenz Curves and Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# Measure Theory of
Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# On The Measure-Theoretic Underpinnings of Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# A Measure Theory of
Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# A Self-Contained Theory of
Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# Mathematical Underpinnings of
Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# A Mathematical Theory of
Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# Lorenz Curves and Lorenz Hulls: Measure Theory and Mathematical
Unpderpinnings

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# A Distributive and Associative Product for Zonoids Based on Product Measures

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# A Product on Zonoids and Lorenz Hulls

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

# A Product on Lorenz Hulls, Zonoids, and Vector Measure Ranges

John P. Steinberger111Chief Scientist, Auki Labs., Zhe Zhang222IIIS, Tsinghua
University.

###### Abstract

A _Lorenz hull_ is the convex hull of the range of an $n$-dimensional vector
of finite signed measures defined on a common measurable space. We show that
the set of $n$-dimensional Lorenz hulls is endowed with a natural product that
is commutative, associative, and distributive over Minkowski sums. The same
holds with “zonoid” in place of “Lorenz hull” as the two concepts give rise to
the same set of subsets of $\mathbb{R}^{n}$. The product is defined via the
common notion of a product measure.

##  1 Introduction

Let ${\bm{\mu}}=(\mu_{1},\ldots,\mu_{n})$ an $n$-tuple of signed finite (and
hence bounded) measures on a common space $(S,\mathcal{F})$ where $S$ is a
ground set and $\mathcal{F}$ is a $\sigma$-algebra of subsets of $S$. We refer
to ${\bm{\mu}}$ as a _finite signed vector measure_. The _Lorenz hull_ of
${\bm{\mu}}$, denoted $\mathsf{LH}({\bm{\mu}})$, is the convex hull of the
range of ${\bm{\mu}}$; i.e.,

$\mathsf{LH}({\bm{\mu}}):=\mathsf{conv.hull}(\\{{\bm{\mu}}(A):A\in\mathcal{F}\\})$

where $\mathsf{conv.hull}(F)$ is the convex hull of a bounded set
$F\subseteq\mathbb{R}^{n}$. We also refer to the range of ${\bm{\mu}}$ itself
as the _Lorenz skeleton_ of ${\bm{\mu}}$, denoted $\mathsf{LS}({\bm{\mu}})$;
i.e., $\mathsf{LS}({\bm{\mu}}):=\\{{\bm{\mu}}(A):A\in\mathcal{F}\\}$ and
$\mathsf{LH}({\bm{\mu}})=\mathsf{conv.hull}(\mathsf{LS}({\bm{\mu}}))$.

In relation to previous terminology, Harremoës [14] refers to $2$-dimensional
Lorenz hulls as _Lorenz diagrams_. Moreover the Lorenz skeleton of a tuple of
continuous measures is well-known as a _zonoid_ [2, 33]. In the latter case,
by a classical theorem of Lyapunov [21], the Lorenz skeleton is convex, and,
therefore, the Lorenz hull and the Lorenz skeleton coincide. Moreover, while a
Lorenz hull is a nominally broader concept than a zonoid one can show that for
every finite signed vector measure ${\bm{\mu}}$ there is some finite signed
vector measure ${\bm{\mu}}^{\prime}$ with continuous elements such that
$\mathsf{LH}({\bm{\mu}})=\mathsf{LS}({\bm{\mu}}^{\prime})$, i.e., every Lorenz
hull also happens to be a zonoid [2]. For the type of theorem that we present
below, however, the strongest statement is obtained by allowing the largest
possible class of vector measures to be counted as part of the “preimage” of
the hull/zonoid, so we enlarge the scope of the discussion to Lorenz hulls. We
note that the “Lorenz” moniker enters into the scene via the _Lorenz curve_ ,
popular in the social sciences, that can be characterized as the lower
boundary of a two-dimensional Lorenz hull, more specifically the lower
boundary of the Lorenz hull of a pair of nonnegative (i.e., unsigned) measures
on a common space.333The _Gini coefficient_ of a Lorenz curve can also be
given an elegant description in terms of Lorenz hull, being the area of the
hull. Koshevoy [17] seems to be the first author to draw a connection between
zonoids and Lorenz curves.

Our main result is to show the existence of a natural product, based off of
the common notion of a product measure, on the set of $n$-dimensional Lorenz
hulls (zonoids). The construction is implicit in the following statement:

######  Theorem 1.

Let $H_{1}$, $\dots$, $H_{k}$ be Lorenz hulls. Let ${\bm{\mu}}^{1}$, $\ldots$,
${\bm{\mu}}^{k}$ where ${\bm{\mu}}^{j}=(\mu_{1}^{j},\dots,\mu_{n}^{j})$ be
such that $H_{j}=\mathsf{LH}({\bm{\mu}}^{j})$ for $j=1,\dots,k$. Then
$\mathsf{LH}({\bm{\mu}}^{\times})$ depends only on $H_{1}$, $\dots$, $H_{k}$
and not on the choice of ${\bm{\mu}}^{1}$, $\dots$, ${\bm{\mu}}^{k}$ where
${\bm{\mu}}^{\times}:=(\mu_{1}^{\times},\dots,\mu_{n}^{\times})$, where
$\mu_{i}^{\times}:=\mu_{i}^{1}\times\dots\times\mu_{i}^{k}$ for $i=1,\dots,n$.

To paraphrase, the hull of a tuple of product measures—all of same arity,
named $k$ above—is determined by the $k$ individual “index slice” hulls, i.e.,
the hulls that arise by keeping only the $j$-th term inside each product,
$j=1,\dots,k$.

Whether a similar theorem holds for Lorenz skeletons is left as a main open
problem.

It is easy to check that the case $k=2$ of Theorem 1 implies the general case
and that the binary product defined by the case $k=2$ is associative and
yields the same $k$-ary product via said associativity as when the $k$-ary
product is defined directly as it is in Theorem 1.

We refer to the product defined by Theorem 1 as the _Lorenz product_ , writing
$H_{1}H_{2}$ for the product of hulls $H_{1}$ and $H_{2}$.

One can easily check from the definition that the Lorenz product is
commutative. Moreover, writing $H_{1}+H_{2}$ for the Minkowski sum of Lorenz
hulls $H_{1}$ and $H_{2}$, and noting that the set of Lorenz hulls is closed
under such sums, we also show:

######  Theorem 2.

${H_{1}}{(H_{2}+H_{3})}={H_{1}}{H_{2}}+{H_{1}}{H_{3}}$ for all Lorenz hulls
$H_{1}$, $H_{2}$, $H_{3}\subseteq\mathbb{R}^{n}$.

######  Theorem 3.

$H_{1}\subseteq H_{3}$, $H_{2}\subseteq
H_{4}\implies{H_{1}}{H_{2}}\subseteq{H_{3}}{H_{4}}$ for all Lorenz hulls
$H_{1}$, $H_{2}$, $H_{3}$, $H_{4}\subseteq\mathbb{R}^{n}$.

I.e., the Lorenz product is distributive and “inclusion-preserving”.

One can also easily check that the Lorenz product has a (necessarily unique)
multiplicative identity $\mathsf{conv.hull}(\\{\mathbf{0},\mathbf{1}\\})$
where $\mathbf{0}=(0,\dots,0)$, $\mathbf{1}=(1,\dots,1)\in\mathbb{R}^{n}$ and
that the only Lorenz hulls with a multiplicative inverse are those of the form
$\mathsf{conv.hull}(\\{\mathbf{0},(x_{1},\dots,x_{n})\\})$ where $x_{i}\neq 0$
for $i=1,\dots,n$. (More precisely, one can easily argue that the presence of
non-codirectional points in a hull $H_{1}$ implies the presence of non-
codirectional points in any hull of the form $H_{1}H_{2}$ such that
$H_{1}H_{2}\ni\mathbf{1}$, e.g..)

Interestingly, our results also extend to complex-valued measures, as sketched
in Section 8, but even further generalizations are not pursued here.

Historical work on zonoids. In a famous theorem, Lyapunov [21] (1940) proved
that Lorenz skeletons are closed and that Lorenz hulls of tuples of continuous
(a.k.a., “non-atomic”) vector measures are convex. A simplified proof of
Lyapunov’s theorem was presented in English by Halmos [11], published in 1948.
Lindenstrauss [22] also gave an elegant short proof of the closedness and
convexity of Lorenz skeletons of continuous vector measures in 1966 from a
functional analysis perspective, cited by Rudin [30]. Rickert [28] showed a
bijection between zonoids and measures on certain “standard” spaces,
specifically the projective space. In 1969, Bolker [2] wrote a first survey of
results on zonoids (also introducing the term), where in particular it is
shown that every Lorenz hull444There is no dedicated term to denote the
concept of a Lorenz hull, however, until the afore-mentioned paper by
Harremoës [14]. is a zonoid. A survey including work up to the early 1980s is
given by Schneider and Weil [33].

After the late 1970s one research theme concerned the approximability of
zonoids by zonotopes, their discrete counterparts, culminating in a result by
Talagrand [8, 32, 3, 34].

Two-dimensional zonoids and zonotopes, and in particular inclusion
relationships between these, can also be related to the important topic of
_majorization_ [12, 13, 24]. See for example works Foster [9], Harremoës and
Harremoës and van Erden [14, 7], and by Koshevoy and Koshevoy and Mosler [17,
19, 18].

More recent work. Work on zonoids has continued apace in recent decades. See
[1, 15, 23, 20] for some recent references. We note that one recurring
question has been the issue of proving that a given set is not a zonoid [10,
27, 23].

Proof Summary. We establish Theorem 1 by way of the following more general
inclusion-preservation result, that also establishes Theorem 3:

######  Theorem 4.

Let ${\bm{\alpha}}$, ${\bm{\beta}}$, ${\bm{\alpha}}^{\prime}$,
${\bm{\beta}}^{\prime}$ be four $n$-dimensional signed vector measures such
that $\mathsf{LH}({\bm{\alpha}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime})$,
$\mathsf{LH}({\bm{\beta}})\subseteq\mathsf{LH}({\bm{\beta}}^{\prime})$. Then
$\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$
where ${\bm{\alpha}}\times{\bm{\beta}}$ is the $n$-dimensional signed vector
measure whose $i$-th coordinate is the direct product of the $i$-th
coordinates of ${\bm{\alpha}}$ and ${\bm{\beta}}$, and likewise for
${\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime}$.

Theorem 1 is easily seen to be a corollary of Theorem 4 by application of the
principle that sets $S$ and $T$ are equal if and only if $S\subseteq T$ and
$T\subseteq S$.

To establish the conclusion of Theorem 4 we use a separating hyperplane
argument. Because the Lorenz hulls
$\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})$,
$\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$ are closed
convex sets, specifically, it suffices to show that

$\sup_{{\bm{z}}\in\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})}{{\bm{x}}^{*}}{}^{T}{{\bm{z}}}\leq\sup_{{\bm{z}}\in\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})}{{\bm{x}}^{*}}{}^{T}{{\bm{z}}}$
(1)

for every ${\bm{x}}^{*}\in\mathbb{R}^{n}$ in order to show
$\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$.
In turn, (1) is established by characterizing, for each
${\bm{x}}^{*}\in\mathbb{R}^{n}$, the optimal ${\bm{z}}$ for which the left-
hand side achieves its supremum. Knowing this characterisation allows us to
write the left-hand side as an integral that can be rewritten as an iterated
integral, given the product structure of the measure space. The assumption
that $\mathsf{LH}({\bm{\alpha}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime})$
gives an inequality (in fact, the direct analog of (1)) that can be applied to
the inner integral, effectively replacing ${\bm{\alpha}}$ by
${\bm{\alpha}}^{\prime}$ at that stage in the computation, while introducing
an inequality; proceeding symmetrically, one can rewind, reverse the order of
integration, and replace ${\bm{\beta}}$ by ${\bm{\beta}}^{\prime}$. One can
also rely on symmetry to reduce the theorem to the case
${\bm{\beta}}={\bm{\beta}}^{\prime}$ as a preamble. (As we actually choose to
do.)

Follow-up work. We will show in a separate paper that $m$-th roots of
nonnegative Lorenz hulls under the product introduced by this paper are unique
when such roots exist, i.e., “uniqueness of roots”.

Organization. We have tried to keep the paper friendly to non-mathematicians
(in the hopes of accommodating computer scientists in particular555Note that
products of two-dimensional Lorenz diagrams, in the sense defined by this
paper, crop up whenever the distinguishability or “divergence” (according to
any standard metric) of two vectors $(X_{1},\dots,X_{k})$,
$(X_{1}^{\prime},\dots,X_{k}^{\prime})$, each consisting of independently
sampled random variables, comes under discussion: The two-dimensional Lorenz
hull for the pair of measures induced by the pair
$((X_{1},\dots,X_{k}),(X_{1}^{\prime},\dots,X_{k}^{\prime}))$ is the product
of the $k$ two-dimensional Lorenz hulls associated to the respective pairs
$(X_{1},X_{1}^{\prime})$, $\ldots$, $(X_{k},X_{k}^{\prime})$.), resulting in
preliminary background sections on measure theory and convex analysis. We also
include a separate introduction to signed measures in Section 4. Familiar
readers should be able to start in Section 5 and skim backwards as necessary
to find the definitions of nonstandard notations.

##  2 Measure Theory

We recall the standard elements of measure theory. More details may be found
in Durrett [6] and Rudin [31].

######  Definition 1.

A _measurable space_ is a pair $(S,\mathcal{F})$ where $S$ is a set and
$\mathcal{F}$ is a $\sigma$-algebra on $S$, i.e., $\mathcal{F}$ is a set of
subsets of $S$ such that

  1. (i)

$S\in\mathcal{F}$,

  2. (ii)

if $A\in\mathcal{F}$ then $S\backslash A\in\mathcal{F}$, and

  3. (iii)

$\bigcup_{i=1}^{\infty}A_{i}\in\mathcal{F}$ for any countable collection
$\\{A_{i}\\}_{i\in\mathbb{N}}$ of elements of $\mathcal{F}$.

######  Definition 2.

A _measure_ on a measurable space $(S,\mathcal{F})$ is a function
$\mu:\mathcal{F}\rightarrow\mathbb{R}\cup\\{\infty\\}$ such that

  1. (i)

$\mu(A)\geq 0$ for all $A\in\mathcal{F}$, and

  2. (ii)

$\mu\big{(}\bigcup_{i=1}^{\infty}A_{i}\big{)}=\sum_{i=1}^{\infty}\mu(A_{i})$
for any collection $\\{A_{i}\\}_{i\in\mathbb{N}}$ of pairwise disjoint
elements of $\mathcal{F}$.

Moreover, $\mu$ is _finite_ if $\mu(S)<\infty$ and is _$\sigma$ -finite_ if
there exists a sequence $A_{1}$, $A_{2}$, $\ldots$ of elements of
$\mathcal{F}$ such that $\mu(A_{n})<\infty$ and $\bigcup_{n}A_{n}=S$.

######  Definition 3.

A _measure space_ is a triple $(S,\mathcal{F},\mu)$ where $(S,\mathcal{F})$ is
a measurable space and $\mu$ is a measure on $(S,\mathcal{F})$.

######  Definition 4.

Let $(S,\mathcal{F})$ and $(S^{\prime},\mathcal{F}^{\prime})$ be two
measurable spaces. A function $g:S\rightarrow S^{\prime}$ is _measurable (with
respect to the $\sigma$-algebras $\mathcal{F}$ and $\mathcal{F}^{\prime}$)_ if
for any $A\in\mathcal{F}^{\prime}$, $g^{-1}(A)\coloneqq\\{s\in S\,:\,g(s)\in
A\\}\in\mathcal{F}$.

######  Definition 5.

A set system $\mathcal{G}$ of subsets of $S$ _generates_ a $\sigma$-algebra
$\mathcal{F}$ of $S$ if $\mathcal{F}$ is the smallest $\sigma$-algebra of $S$
containing $\mathcal{G}$.

As commonly pointed out, the notion of a “smallest” $\sigma$-algebra appearing
in Definition 5 is well-defined since the intersection of an arbitrary
collection of $\sigma$-algebras of $S$ is a $\sigma$-algebra of $S$.

######  Definition 6.

The _product_ $(S,\mathcal{F})\times(T,\mathcal{G})$ of two measurable spaces
$(S,\mathcal{F})$, $(T,\mathcal{G})$ is the measurable space $(S\times
T,\mathcal{H})$ where $\mathcal{H}$ is the $\sigma$-algebra on $S\times T$
generated by sets of the form $F\times G$, $F\in\mathcal{F}$,
$G\in\mathcal{G}$.

Given a set $S$, a measurable space $(T,\mathcal{G})$, and a function $f:S\to
T$, one can check that
$f^{-1}(\mathcal{G})\coloneqq\\{f^{-1}(B)\,:\,B\in\mathcal{G}\\}$ is a
$\sigma$-algebra on $S$. Moreover, if a set system $\mathcal{A}$ of subsets of
$T$ generates $\mathcal{G}$, then $f^{-1}(\mathcal{A})$ generates
$f^{-1}(\mathcal{G})$. The next lemma follows from this observation.

######  Lemma 1.

Let $(S,\mathcal{F})$ and $(T,\mathcal{G})$ be two measurable spaces. Let
$\mathcal{A}$ be a subset of $\mathcal{G}$ which generates $\mathcal{G}$. If a
function $f:S\to T$ is such that $f^{-1}(A)\in\mathcal{F}$ for any
$A\in\mathcal{A}$, then $f$ is measurable with respect to $\mathcal{F}$ and
$\mathcal{G}$.

As an application of Lemma 1, continuous functions are measurable with respect
to Borel (see below) sets. For another example that will be used later, if
functions $g_{i}$ from $(S,\mathcal{F})$ to $(T_{i},\mathcal{G}_{i})$, $1\leq
i\leq n$, are measurable, then $f:S\to\bigtimes_{i=1}^{n}T_{i}$ defined by
$f(s)=(g_{1}(s),g_{2}(s),\ldots,g_{n}(s))$ for $s\in S$ is measurable with
respect to the product $\sigma$-algebra in the sense of Definition 6 on
$\bigtimes_{i=1}^{n}T_{i}$, since for any product set
$A=\bigtimes_{i=1}^{n}A_{i}$ where $A_{i}\in\mathcal{G}_{i}$, $1\leq i\leq n$,
one has $f^{-1}(A)=\bigcap_{i=1}^{n}g_{i}^{-1}(A_{i})\in\mathcal{F}$.

The next lemma establishes the existence and uniqueness of a measure on a
product space that is naïvely compatible with the measures on the component
spaces.

######  Lemma 2.

Let $(S\times T,\mathcal{H})$ be defined as in Definition 6 and let $\mu$,
$\nu$ be $\sigma$-finite measures on $(S,\mathcal{F})$ and $(T,\mathcal{G})$,
respectively. Then there is a unique measure $\lambda$ on $(S\times
T,\mathcal{H})$ such that $\lambda(F\times G)=\mu(F)\nu(G)$ for all
$F\in\mathcal{F},G\in\mathcal{G}$. Moreover, $\lambda$ is $\sigma$-finite.

We write $\mu\times\nu$ for the measure $\lambda$ of Lemma 2.

We note that the proof of Lemma 2 uses the following lemma, that we will
reuse, together with Lemma 2, when it comes time to extend the definition of
product measures to signed measures (cf. Proposition 10 in Section 6); this
next lemma can be proved via the $\pi$-$\lambda$ theorem (cf. Durrett [6]):

######  Lemma 3.

If finite signed measures $\alpha$, $\alpha^{\prime}$ on $(S,\mathcal{F})$
agree on $\mathcal{G}\subseteq\mathcal{F}$ where $\mathcal{G}$ generates
$\mathcal{F}$, where $\mathcal{G}$ is closed under intersection, and where
there exists a sequence of $G_{i}\in\mathcal{G}$ such that
$S=\bigcup_{i=1}^{\infty}G_{i}$, then $\alpha$, $\alpha^{\prime}$ agree on
$\mathcal{F}$.

The set of _extended real numbers_ is the set
$\overline{\mathbb{R}}=\mathbb{R}\,\cup\,\\{\infty,-\infty\\}$ where $\infty$,
(also written “$+\infty$”), $-\infty$ are designated symbols.
$\overline{\mathbb{R}}$ is endowed with a standard topology and arithmetic
[30]. In particular, the topology of $\overline{\mathbb{R}}$ is isomorphic to
the topology of the closed interval $[-1,1]$ via the 1-to-1 mapping from
$\overline{\mathbb{R}}$ to $[-1,1]$ given by

$x\rightarrow\begin{cases}x/(1+|x|)&\textrm{if }x\in\mathbb{R},\\\
1&\textrm{if }x=\infty,\\\ -1&\textrm{if }x=-\infty.\end{cases}$

Arithmetic-wise, one defines $0\cdot\pm\infty=0$, whereas $\infty-\infty$,
$\pm\infty/\pm\infty$ as well as $a/0$ are undefined for all
$a\in\overline{\mathbb{R}}$. One can also note that $\overline{\mathbb{R}}$ is
totally ordered.

The _Borel $\sigma$-algebra_ of $\overline{\mathbb{R}}$ (resp. $\mathbb{R}$)
is the $\sigma$-algebra generated by all open subsets of
$\overline{\mathbb{R}}$ (resp. $\mathbb{R}$). Without ambiguity, the symbol
$\mathcal{B}$ will denote the Borel $\sigma$-algebra of either
$\overline{\mathbb{R}}$ or $\mathbb{R}$. Given a measurable space
$(S,\mathcal{F})$, a function $f:S\rightarrow\overline{\mathbb{R}}$ is
_measurable_ if it is a measurable function from $(S,\mathcal{F})$ to
$(\overline{\mathbb{R}},\mathcal{B})$ in the sense of Definition 4. The Borel
$\sigma$-algebra on $\mathbb{R}^{n}$ is similarly defined as the
$\sigma$-algebra generated by all open subset of $\mathbb{R}^{n}$. It also
coincides with the $\sigma$-algebra on $\mathbb{R}^{n}$ generated by
“rectangles”, i.e., sets of the form $\bigtimes_{1\leq i\leq n}A_{i}$ where
each $A_{i}$ is a Borel subset of $\mathbb{R}$.

Given a measure space $(S,\mathcal{F},\mu)$, a measurable function
$f:S\rightarrow\overline{\mathbb{R}}$, and $A\in\mathcal{F}$, the expression
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\int_{A}f\,\mathrm{d}\mu$

denotes the Lebesgue integral of $f$ on $A$ with respect to the measure $\mu$.
One may also write this integral as aaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\int_{A}f(s)\,\mu(\mathrm{d}s)$

which offers the possibility of specifying the function $f$ on the fly in
terms of an algebraic expression of $s$.

The value of the Lebesgue integral is an element of $\overline{\mathbb{R}}$,
or else is undefined. We recall that the Lebesgue integral is defined as the
difference of its positive and negative parts, being undefined if and only if
the positive and negative parts are both infinite. In particular, the integral
is guaranteed to exist if $f$ is nonnegative, and is guaranteed to exist and
to be finite if $f$ is bounded666A function
$f:S\rightarrow\overline{\mathbb{R}}$ is _bounded_ if there exists an $M>0$,
$M\neq\infty$, such that $|f(s)|\leq M$ for all $s\in S$. and $\mu(S)<\infty$.
Furthermore, the integral is linear (provided the linear-combination-of-
functions evaluates to a well-defined function from $S$ to
$\overline{\mathbb{R}}$ and the linear-combination-of-integrals evaluates to a
well-defined element of $\overline{\mathbb{R}}$), and

$\mu(A)=\int_{S}\mathbf{1}_{A}\,\mathrm{d}\mu=\int_{A}\mathrm{d}\mu$

for any $A\in\mathcal{F}$, where $\mathbf{1}_{A}$ is the indicator function of
$A$ on $S$.

######  Definition 7.

Given a measure $\mu$ on $(S,\mathcal{F})$ and a boolean property $P$ of
elements of $S$ we say $P$ holds _$\mu$ -almost everywhere_ if the set of
$s\in S$ for which $P(s)$ is false is contained in a set $N\in\mathcal{F}$
such that $\mu(N)=0$.

######  Lemma 4.

_(Theorem 1.40, Rudin[31])_ Let $\mu$ be a finite measure on $(S,\mathcal{F})$
and let $f:S\to\mathbb{R}$ be such that $\int_{S}|f|\,\mathrm{d}\mu<\infty$.
Let $E$ be a closed subset of $\mathbb{R}$. If

$\frac{1}{\mu(A)}\int_{A}f\,\mathrm{d}\mu\in E$

for every $A\in\mathcal{F}$ such that $\mu(A)>0$, then $f\in E$ $\mu$-almost
everywhere on $S$.

The following lemma, commonly known as “Fubini’s theorem”, singles out
sufficient conditions under which an integral over a product space can be
evaluated via iterated integration. This will be central to our work:

######  Lemma 5.

_(Fubini’s theorem)_ Let $(S,\mathcal{F},\mu)$ and $(T,\mathcal{G},\nu)$ be
two $\sigma$-finite measure spaces. If $f\geq 0$ or $\int_{S\times
T}|f|\,\mathrm{d}(\mu\times\nu)<\infty$ then

$\int_{S}\int_{T}f(s,t)\,\nu(\mathrm{d}t)\,\mu(\mathrm{d}s)=\int_{S\times
T}f\,\mathrm{d}(\mu\times\nu)=\int_{T}\int_{S}f(s,t)\,\mu(\mathrm{d}s)\,\nu(\mathrm{d}t).$

We note that the condition $\int_{S\times
T}|f|\,\mathrm{d}(\mu\times\nu)<\infty$ of Lemma 5 is automatically fulfilled
if $\mu$, $\nu$ are finite and $f$ is bounded.

######  Lemma 6.

_(dominated convergence theorem)_ Let $\mu$ be a $\sigma$-finite measure on
$(S,\mathcal{F})$. Let $f$ and $f_{n}$, $n\geq 1$ be measurable functions from
$S$ to $\mathbb{R}$ such that $f_{n}\to f$ $\mu$-almost everywhere. If there
exists a measurable function $g$ from $S$ to $\mathbb{R}$ such that
$|f_{n}|\leq g$ for all $n$ and such that $\int_{S}g\,\mathrm{d}\mu<\infty$,
then $\int_{S}f_{n}\,\mathrm{d}\mu\to\int_{S}f\,\mathrm{d}\mu$.

A measure $\nu$ on $(S,\mathcal{F})$ is said to be _absolutely continuous_
with respect to a measure $\mu$ on the same measurable space if $\nu(A)=0$ for
any $A$ such that $\mu(A)=0$. We also say that _$\mu$ dominates $\alpha$_. The
following classical theorem (c.f. [6], Theorem A.4.8 on Page 417) shows that a
measure $\nu$ that is absolutely continuous with respect to $\mu$ can be
expressed in terms of integration with respect to $\mu$.

######  Lemma 7.

_(Radon-Nikodym theorem)_ Let $\mu$, $\nu$ be $\sigma$-finite measures on
$(S,\mathcal{F})$. If $\nu$ is absolutely continuous with respect to $\mu$,
then there exists a measurable $g:S\to[0,\infty)$ such thataaaaaaaaaaaaaa

$\nu(A)=\int_{A}g\,\mathrm{d}\mu$

for all $A\in\mathcal{F}$. Moreover, if $h$ is another such function then
$h=g$ $\mu$-almost everywhere.

The notationaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\frac{\mathrm{d}{\nu}}{\mathrm{d}{\mu}}$

is used to denote an arbitrary choice of the function $g$ described in Lemma 7
for $\mu$, $\nu$, and is called _the Radon-Nikodym derivative of $\nu$ with
respect to $\mu$_. We will introduce a similar notation after Lemma 10 in
Section 4.

The Radon-Nikodym derivative satisfies the following key property,
strengthening the equation in Lemma 7:

######  Lemma 8.

If $\mu$, $\nu$ are $\sigma$-finite measures on a measurable space
$(S,\mathcal{F})$ such that $\nu$ is absolutely continuous with respect to
$\mu$, then

$\int_{S}f\,\mathrm{d}{\nu}=\int_{S}f\cdot\frac{\mathrm{d}{\nu}}{\mathrm{d}{\mu}}\,\mathrm{d}{\mu}$

for any measurable $f:S\to\mathbb{R}$ such that $f>0$, or such that either one
of $\int_{S}|f|\,\mathrm{d}\nu<\infty$ or
$\int_{S}\big{|}f\cdot\frac{\mathrm{d}{\nu}}{\mathrm{d}{\mu}}\big{|}\,\mathrm{d}{\mu}<0$
holds.

##  3 Convex Analysis

Elements of $\mathbb{R}^{n}$ are written in bold font and are interpreted as
column vectors. We write ${\bm{x}}^{T}$ for the row vector transpose of the
column vector ${\bm{x}}\in\mathbb{R}^{n}$. In particular,
${{\bm{x}}}{}^{T}{{\bm{y}}}$ becomes the inner product of vectors ${\bm{x}}$,
${\bm{y}}\in\mathbb{R}^{n}$, written as a matrix product. We write
$\|{\bm{x}}\|$ for the Euclidean norm $\sqrt{{{\bm{x}}}{}^{T}{{\bm{x}}}}$ of
${\bm{x}}\in\mathbb{R}^{n}$.

We take for granted basic notions of point-set topology in $\mathbb{R}^{n}$,
including open and closed sets, as well as closures, interiors, and boundary
points. The closure and interior of a set $C\subseteq\mathbb{R}^{n}$ are
written $\overline{C}$ and ${C}^{\circ}$, respectively.

######  Definition 8.

Let ${\bm{x}}_{1},\ldots,{\bm{x}}_{k}\in\mathbb{R}^{n}$. A _convex combination
of ${\bm{x}}_{1},\ldots,{\bm{x}}_{k}$_ is a vector of the
formaaaaaaaaaaaaaaaaaaaa

$\lambda_{1}{\bm{x}}_{1}+\dots+\lambda_{k}{\bm{x}}_{k}$

where $\lambda_{1},\dots,\lambda_{k}$ are nonnegative real numbers such that
$\lambda_{1}+\dots+\lambda_{k}=1$.

######  Definition 9.

A set $C\subseteq\mathbb{R}^{n}$ is _convex_ if every convex combination of
vectors in $C$ is in $C$.

One can check that a set $C\subseteq\mathbb{R}^{n}$ is convex if and only if
$\lambda{\bm{x}}+(1-\lambda){\bm{y}}\in C$ for all ${\bm{x}}$, ${\bm{y}}\in C$
and all $0\leq\lambda\leq 1$. (I.e., closure under convex combinations of size
two suffices.) Moreover, the closure and interior of a convex set are convex
and an arbitrary intersection of convex sets is convex.

######  Definition 10.

Let $A\subseteq\mathbb{R}^{n}$. The _convex hull of $A$_, written
$\mathsf{conv.hull}(A)$, is the intersection of all convex sets containing
$A$.

It is often more practical to characterize $\mathsf{conv.hull}(A)$ as the set
of all convex combinations of elements of $A$. (Since this set is convex,
contains $A$, and is contained in every convex set containing $A$.) In fact,
convex combinations of a definite size suffice by the following famous result
of Carathéodory (that can be used, e.g., to simplify the proof of (2) or
Proposition 5 below, though, in truth, the previous observation suffices just
as well):

######  Lemma 9.

_(Carathéodory)_ Let $A\subseteq\mathbb{R}^{n}$. Then
${\bm{x}}\in\mathsf{conv.hull}(A)$ if and only if ${\bm{x}}$ is a convex
combination of $n+1$ points in $A$.

We note that since the closure of a convex set is convex, the closure-of-a-
convex-hull is convex; on the other hand, it is not true in general that the
convex hull of a closed set is convex, nor, perforce, that the convex-hull-of-
a-closure is closed. (Take the closed set
$\\{(x,0):x\in\mathbb{R}\\}\cup\\{(0,1)\\}$ in $\mathbb{R}^{2}$.) However:

######  Proposition 1.

$\mathsf{conv.hull}(\overline{C})\subseteq\overline{\mathsf{conv.hull}(C)}$
for all $C\subseteq\mathbb{R}^{n}$, with equality if $C$ is bounded.

Containment relations between closed, convex sets (bounded or unbounded,
though we shall only be concerned with the bounded case) may be obtained in a
“divide and conquer” approach, comparing how far two given sets reach on a
direction-by-direction basis, as per the supremum appearing in (1). We find it
convenient to develop a notational shorthand for the related supremum:

######  Definition 11.

Let $C\subseteq\mathbb{R}^{n}$. The _reach function of $C$_, written
$\|\cdot\|_{C}$, is the function from $\mathbb{R}^{n}$ to
$\overline{\mathbb{R}}$ defined by
$\|{\bm{x}}^{*}\|_{C}\coloneqq\sup\\{{{\bm{x}}^{*}}{}^{T}{{\bm{x}}}\,:\,{\bm{x}}\in
C\\}$.

It is possible to check that $\|\cdot\|_{C}$ is continuous and convex777In the
sense of a _function_ from $\mathbb{R}^{n}$ to $\overline{\mathbb{R}}$ being
convex, not needed for this work. for every nonempty
$C\subseteq\mathbb{R}^{n}$. We also note that despite the suggestive notation,
$\|\cdot\|_{C}$ is not in general888As the incantation goes, $\|\cdot\|_{C}$
is a norm if and only if the convex hull of $C$ is bounded, has nonempty
interior, and is centrally symmetric around ${\bm{0}}\in\mathbb{R}^{n}$. a
norm.

We writeaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\|\cdot\|_{C}\leq\|\cdot\|_{D},\qquad\qquad\|\cdot\|_{C}=\|\cdot\|_{D}$

if $\|{\bm{x}}^{*}\|_{C}\leq\|{\bm{x}}^{*}\|_{D}$, respectively
$\|{\bm{x}}^{*}\|_{C}=\|{\bm{x}}^{*}\|_{D}$, for all
${\bm{x}}^{*}\in\mathbb{R}^{n}$.

It is not hard to see thataaaaaaaaaaaaaaaaaaaaa

$\|\cdot\|_{C}=\|\cdot\|_{\mathsf{conv.hull}(C)},\qquad\|\cdot\|_{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\overline{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}=\|\cdot\|_{\overline{C}}$
(2)

for all $C\subseteq\mathbb{R}^{n}$, where the second identity follows by
continuity of the inner product ${{\bm{x}}^{*}}{}^{T}{{\bm{x}}}$ as a function
of ${\bm{x}}\in\mathbb{R}^{n}$. Moreover, taking closures and taking the
convex hull are the only operations that do not enlarge the reach, in the
sense of the following proposition:

######  Proposition 2.

$\|\cdot\|_{D}\leq\|\cdot\|_{C}$ if and only if
$D\subseteq\overline{\mathsf{conv.hull}(C)}$ for all
$C,D\subseteq\mathbb{R}^{n}$.

If $C$ is convex and closed then $\overline{\mathsf{conv.hull}(C)}=C$,
naturally, so:

######  Proposition 3.

If $C\subseteq\mathbb{R}^{n}$ convex and closed then $D\subseteq C$ if and
only if $\|\cdot\|_{D}\leq\|\cdot\|_{C}$.

It should be noted that Proposition 2 relies on—indeed, is equivalent to—a
“separating hyperplane theorem”, one of the deeper tools in convex analysis.
(See, e.g., Rockafellar [29], Theorem 11.3.)

Our containment results will be obtained by way of Proposition 3. In so doing,
it is often convenient to restrict the comparison between two reach functions
to aaaaaaaaaaaaaaaaaaaaa

$\mathsf{Sph}_{\mathbb{R}^{n}}\coloneqq\\{{\bm{x}}^{*}\in\mathbb{R}^{n}\,:\,\|{\bm{x}}^{*}\|=1\\}$

the unit sphere in $\mathbb{R}^{n}$, since a reach function is positively
homogeneous.

The following elementary propositions are also recorded for completeness:

######  Proposition 4.

$\mathsf{conv.hull}(\mathsf{clf}(B,{\bm{x}}^{*}))\subseteq\mathsf{clf}(\mathsf{conv.hull}(B),{\bm{x}}^{*})$
for all $B\subseteq\mathbb{R}^{n}$,
${\bm{x}}^{*}\in\mathsf{Sph}_{\mathbb{R}^{n}}$, with equality if $B$ is
bounded.

Let $A+B$ denote the Minkowski sum of sets $A$, $B\subseteq\mathbb{R}^{n}$,
i.e.,

$A+B\coloneqq\\{{\bm{x}}+{\bm{y}}\,:\,{\bm{x}}\in A,{\bm{y}}\in B\\}.$

We write ${\bm{x}}+A$ to denote $\\{{\bm{x}}\\}+A$ for
${\bm{x}}\in\mathbb{R}^{n}$, $A\subseteq\mathbb{R}^{n}$ for convenience.
Moreover, letaaaaaaaaaaaaaaaaaaaaaaaaa

$-A\coloneqq\\{-{\bm{x}}\,:\,{\bm{x}}\in A\\}$

for $A\in\mathbb{R}^{n}$ by convention.

######  Proposition 5.

Let $A$, $B\subseteq\mathbb{R}^{n}$. Then
$\mathsf{conv.hull}(A+B)=\mathsf{conv.hull}(A)+\mathsf{conv.hull}(B)$,
$\mathsf{conv.hull}(-A)=-\mathsf{conv.hull}(A)$.

##  4 Signed Vector Measures

######  Definition 12.

A _signed measure_ on a measurable space $(S,\mathcal{F})$ is a function
$\alpha:\mathcal{F}\to\overline{\mathbb{R}}$ such that
aaaaaaaaaaaaaaaaaaaaaaaaaa

$\alpha\Big{(}\bigcup_{i=1}^{\infty}A_{i}\Big{)}=\sum_{i=1}^{\infty}\alpha(A_{i})$
(3)

for any collection $\\{A_{i}\\}_{i\in\mathbb{N}}$ of pairwise disjoint
elements of $\mathcal{F}$.

By contrast, a measure defined as in Definition 2 is sometimes called a
_positive measure_ to emphasize it being a special kind of signed measure.
Just as in the case of (positive) measures, (11) in the definition implies
$\alpha(\emptyset)=0$. It is also noted that the sum in (11) converges
absolutely, since any rearrangement of the series converges to the measure of
the same union. If the range of $\alpha$ does not include $\infty$ or
$-\infty$ then $\alpha$ is called _finite_. Our work will only concern finite
signed measures.

######  Definition 13.

The _total variation_ of a finite signed measure $\alpha$ on $(S,\mathcal{F})$
is the function $|\alpha|:\mathcal{F}\to\overline{\mathbb{R}}$ defined by
aaaaaaaaaaaaaaaaaaaaaaaaaa

$|\alpha|(A)=\sup\sum_{i=1}^{\infty}|\alpha(A_{i})|$

for all $A\in\mathcal{F}$, where the supremum is taken over all collections
$\\{A_{i}\\}_{i\in\mathbb{N}}$ of pairwise disjoint elements of $\mathcal{F}$
of union $A$.

It is easy to check that $|\alpha|$ is a positive measure on $(S,\mathcal{F})$
for every signed measure $\alpha$ on $(S,\mathcal{F})$.

Similarly to the case when $\alpha$ is positive and $\sigma$-finite, a signed
measure $\alpha$ on $(S,\mathcal{F})$ is said to be _absolutely continuous_
with respect to a positive measure $\mu$ on the same measurable space if
$\alpha(A)=0$ for any $A$ such that $\mu(A)=0$, and we also say that _$\mu$
dominates $\alpha$_. Obviously, a finite $\alpha$ is absolutely continuous
with respect to $|\alpha|$, and it is easy to show that $\mu$ dominates
$|\alpha|$ if and only if $\mu$ dominates $\alpha$. As the analogue of Lemma 7
one has the following lemma (c.f. the traditional theorem of Lebesgue-Radon-
Nikodym for complex measures, 6.10 of Rudin [31]):

######  Lemma 10.

Let $\alpha$ be a finite signed measure on $(S,\mathcal{F})$ and let $\mu$ be
a $\sigma$-finite positive measure on $(S,\mathcal{F})$. If $\mu$ dominates
$\alpha$, then there exists a measurable function $g:S\to\mathbb{R}$ such that

$\alpha(A)=\int_{A}g\,\mathrm{d}\mu$

for all $A\in\mathcal{F}$. Moreover, if $h$ is another such function, then
$h=g$ $\mu$-almost everywhere.

Accordingly, aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}$

denotes an arbitrary choice of $g$ for $\alpha$, $\mu$ as in Lemma 10, and is
referred to as _the Radon-Nikodym derivative of $\alpha$ with respect to
$\mu$_. When $\alpha$ is positive in addition to finite, this definition
coincides with the definition made after Lemma 7 and won’t cause any
ambiguity.

The following lemma is a summary of some useful facts that can be found in
Chapter 6 of Rudin [31]:

######  Lemma 11.

The total variation $|\alpha|$ of a finite signed measure $\alpha$ on
$(S,\mathcal{F})$ is a finite positive measure on $(S,\mathcal{F})$. There
exists measurable $h:S\to\\{-1,1\\}$ such that
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\alpha(A)=\int_{A}h\,\mathrm{d}|\alpha|$

for all $A\in\mathcal{F}$. Moreover, if aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\alpha(A)=\int_{A}g\,\mathrm{d}\mu$

for all $A\in\mathcal{F}$ for some measurable $g:S\to\mathbb{R}$ and positive
measure $\mu$ on $(S,\mathcal{F})$, then aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$|\alpha|(A)=\int_{A}|g|\,\mathrm{d}\mu$

for all $A\in\mathcal{F}$.

In particular, the range of the total variation of a finite signed measure,
thus also the range of the finite signed measure, is actually bounded.

######  Proposition 6.

Let $\alpha$ be a finite signed measure on $(S,\mathcal{F})$. Let $\mu$,
$\mu^{\prime}$ be $\sigma$-finite positive measures on $(S,\mathcal{F})$ such
that $\mu^{\prime}$ dominates $\alpha$, $\mu$ dominates $\mu^{\prime}$. Then
$\mu$ dominates $\alpha$ and

$\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}=\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu^{\prime}}}\frac{\mathrm{d}{\mu^{\prime}}}{\mathrm{d}{\mu}}$

$\mu$-almost everywhere.

###### Proof.

The fact that $\mu$ dominates $\alpha$ is obvious. For the rest, one just note
that

$\displaystyle\alpha(A)$
$\displaystyle=\int_{A}\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu^{\prime}}}\,\mathrm{d}\mu^{\prime}$
$\displaystyle=\int_{A}\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu^{\prime}}}\frac{\mathrm{d}{\mu^{\prime}}}{\mathrm{d}{\mu}}\,\mathrm{d}\mu$

for all $A\in\mathcal{F}$, where the second equality follows by Lemma 8, for
which the fact that aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\int_{A}\bigg{|}\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu^{\prime}}}\bigg{|}\,\mathrm{d}\mu^{\prime}=|\alpha|(A)<\infty$

follows by Lemma 11. ∎

Given finite signed measure $\alpha$ on $(S,\mathcal{F})$, $\sigma$-finite
positive measure $\mu$ dominating $\alpha$ on $(S,\mathcal{F})$, and
$f:S\to\mathbb{R}$ such that $\int_{S}|f|\,\mathrm{d}|\alpha|<\infty$, one has

$\int_{A}\bigg{|}f\cdot\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}\bigg{|}\,\mathrm{d}\mu=\int_{A}|f|\cdot\bigg{|}\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}\bigg{|}\,\mathrm{d}\mu=\int_{A}|f|\,\mathrm{d}|\alpha|<\infty$

for any $A\in\mathcal{F}$, where the last equality follows by Lemma 11 and
Lemma 8, so that (note that $\mu$ dominates $|\alpha|$)

$\int_{A}f\cdot\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}\,\mathrm{d}\mu=\int_{A}f\cdot\frac{\mathrm{d}{\alpha}}{\mathrm{d}{|\alpha|}}\cdot\frac{\mathrm{d}{|\alpha|}}{\mathrm{d}{\mu}}\,\mathrm{d}\mu=\int_{A}f\cdot\frac{\mathrm{d}{\alpha}}{\mathrm{d}{|\alpha|}}\,\mathrm{d}|\alpha|$

for any $A\in\mathcal{F}$, where the first equality follows by Proposition 6
and where the second equality follows by Lemma 8. Thus the Lebesgue integrals
with respect to finite signed measures in the following are well-defined:

######  Definition 14.

Let $\alpha$ be a finite signed measure on $(S,\mathcal{F})$ and let
$f:S\to\mathbb{R}$ be a measurable function such that
$\int_{S}|f|\,\mathrm{d}|\alpha|<\infty$. _The Lebesgue integral of $f$ with
respect to $\alpha$_, written as $\int_{S}f\,\mathrm{d}\alpha$, is defined by

$\int_{S}f\,\mathrm{d}\alpha=\int_{S}f\cdot\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}\,\mathrm{d}\mu$

where $\mu$ is any $\sigma$-finite positive measure dominating $\alpha$ on
$(S,\mathcal{F})$.

The integral defined by Definition 14 possesses similar properties that
ordinary Lebesgue integrals hold such as linearity. However we will not list
those properties here because we will do all the computations about such
integrals by applying the definition and manipulating ordinary Lebesgue
integrals with respect to a $\sigma$-finite positive measure.

######  Definition 15.

An $n$-dimensional signed measure on a measurable space $(S,\mathcal{F})$ is a
function ${\bm{\alpha}}:\mathcal{F}\to\overline{\mathbb{R}}^{n}$ so that

${\bm{\alpha}}(A)=(\alpha_{1}(A),\alpha_{2}(A),\ldots,\alpha_{n}(A))$

for all $A\in\mathcal{F}$, where each $\alpha_{i}(A)$, $1\leq i\leq n$, is a
signed measure on $(S,\mathcal{F})$.

We write ${\bm{\alpha}}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ to denote
that ${\bm{\alpha}}$ is defined as in Definition 15, and call $\alpha_{i}$,
$1\leq i\leq n$, the $i$-th component of ${\bm{\alpha}}$. An $n$-dimensional
signed measure ${\bm{\alpha}}$ is _finite_ if each component of
${\bm{\alpha}}$ is finite.

If each component of an $n$-dimensional signed measure ${\bm{\mu}}$ is
positive, then ${\bm{\mu}}$ is simply called an _$n$ -dimensional measure_, or
an _$n$ -dimensional positive measure_ for emphasis. A $1$-dimensional signed
or positive measure reduces to a signed or positive measure, respectively. We
will also call an $n$-dimensional (finite, signed or positive) measure a
(finite, signed or positive) vector measure when the dimension is not
emphasized.

For an $n$-dimensional finite signed measure
${\bm{\alpha}}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ on
$(S,\mathcal{F})$, we define aaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$|{\bm{\alpha}}|\coloneqq\sum_{i=1}^{n}|\alpha_{i}|$

to be the total variation of ${\bm{\alpha}}$ (with respect to the $1$-norm).
It is easy to check that $|{\bm{\alpha}}|$ is a measure on $(S,\mathcal{F})$,
and that $\alpha_{i}$ is absolutely continuous with respect to
$|{\bm{\alpha}}|$, $1\leq i\leq n$. Moreover, similar as when ${\bm{\alpha}}$
is $1$-dimensional, a positive measure $\mu$ on $(S,\mathcal{F})$ dominates
$|{\bm{\alpha}}|$ if and only if $\mu$ dominates ${\bm{\alpha}}$ (i.e.,
dominates each $\alpha_{i}$), and there exists
$\frac{\mathrm{d}{{\bm{\alpha}}}}{\mathrm{d}{\mu}}:S\to\mathbb{R}^{n}$ where

$\frac{\mathrm{d}{{\bm{\alpha}}}}{\mathrm{d}{\mu}}=\Big{(}\frac{\mathrm{d}{\alpha_{1}}}{\mathrm{d}{\mu}},\frac{\mathrm{d}{\alpha_{2}}}{\mathrm{d}{\mu}},\ldots,\frac{\mathrm{d}{\alpha_{n}}}{\mathrm{d}{\mu}}\Big{)}$

if $\mu$ is in addition $\sigma$-finite. We note that
$\frac{\mathrm{d}{{\bm{\alpha}}}}{\mathrm{d}{\mu}}$ is measurable by the
discussion following Lemma 1. For a shorthand, and to more clearly signify the
presence of a vector, we use aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

${\bm{D}}^{{\bm{\alpha}}}_{\mu}$

to denote $\frac{\mathrm{d}{{\bm{\alpha}}}}{\mathrm{d}{\mu}}$. We also apply
the notation of integrals of vector-valued functions: given $\sigma$-finite
positive measure $\mu$ on $(S,\mathcal{F})$, let

$\int_{S}{\bm{f}}\,\mathrm{d}\mu\coloneqq\Big{(}\int_{S}f_{1}\,\mathrm{d}\mu,\int_{S}f_{2}\,\mathrm{d}\mu,\ldots,\int_{S}f_{n}\,\mathrm{d}\mu\Big{)}$
(4)

for ${\bm{f}}=(f_{1},f_{2},\dots,f_{n}):S\to\mathbb{R}^{n}$ where each
$f_{i}:S\to\mathbb{R}$ either is non-negative or satisfies
$\int_{S}|f_{i}|\,\mathrm{d}\mu<\infty$. By definition of
${\bm{D}}^{{\bm{\alpha}}}_{\mu}$, one has

${\bm{\alpha}}(A)=\int_{A}{\bm{D}}^{{\bm{\alpha}}}_{\mu}\,\mathrm{d}\mu$

for all $A\in\mathcal{F}$, if $\mu$ dominates ${\bm{\alpha}}$.

Moreover, for $n$-dimensional finite signed measure ${\bm{\alpha}}$ on
$(S,\mathcal{F})$, we define

$\int_{S}f\,\mathrm{d}{\bm{\alpha}}\coloneqq\int_{S}f\cdot{\bm{D}}^{{\bm{\alpha}}}_{\mu}\,\mathrm{d}\mu$
(5)

for all measurable function $f:S\to\mathbb{R}$ such that
$\int_{S}|f|\,\mathrm{d}|{\bm{\alpha}}|<\infty$ (which is especially true when
$f$ is bounded), where $\mu$ is any $\sigma$-finite positive measure
dominating ${\bm{\alpha}}$ on $(S,\mathcal{F})$. This definition is valid for
the same reason that Definition 14 is valid for, from which one also has

$\int_{S}f\,\mathrm{d}{\bm{\alpha}}=\Big{(}\int_{S}f\,\mathrm{d}\alpha_{1},\ldots,\int_{S}f\,\mathrm{d}\alpha_{n}\Big{)}$

for ${\bm{\alpha}}=(\alpha_{1},\ldots,\alpha_{n})$.

######  Definition 16.

Given a signed measure $\alpha$ on $(S,\mathcal{F})$, a set $A\in\mathcal{F}$
is an _atom_ of $\alpha$ if $\alpha(A)\neq 0$ and if for any $B\in\mathcal{F}$
either $\alpha(A\cap B)=\alpha(A)$ or $\alpha(A\cap B)=0$. An $n$-dimensional
signed measure ${\bm{\alpha}}$ is _non-atomic_ (or _continuous_) if none of
its components has atoms.

We note that in particular the Lebesgue measure on $\mathbb{R}^{n}$ is non-
atomic. Moreover, we claim the following proposition without its elementary
proof:

######  Proposition 7.

For finite signed measure $\alpha$ and $n$-dimensional finite signed measure
${\bm{\alpha}}=(\alpha_{1},\ldots,\alpha_{n})$ on $(S,\mathcal{F})$, the
following properties hold:

  1. (i)

Atoms of $\alpha$ are atoms of $|\alpha|$, and vice versa;

  2. (ii)

For $i\in[n]$, each atom $A$ of $|{\bm{\alpha}}|$ is an atom of $\alpha_{i}$
if $|\alpha_{i}|(A)>0$;

  3. (iii)

For $i\in[n]$, each atom of $\alpha_{i}$ contains an atom of
$|{\bm{\alpha}}|$;

  4. (iv)

If $f_{k}:S\to\mathbb{R}$ is measurable for each $k$, $1\leq k\leq m$, and if
$A$ is an atom of $\alpha$, then there exists ${\bm{c}}\in\mathbb{R}^{m}$ such
that ${\bm{f}}(s)={\bm{c}}$ for $s\in A$ $|\alpha|$-almost everywhere, where
${\bm{f}}=(f_{1},\ldots,f_{m})$.

Given an $n$-dimensional finite signed measure ${\bm{\alpha}}$, define two
atoms $A_{1}$, $A_{2}$ of $|{\bm{\alpha}}|$ to be in the same class if
$|{\bm{\alpha}}|(A_{1}\cap A_{2})>0$. There are at most countably many
different classes since $|{\bm{\alpha}}|$ is bounded. Let
$\\{A_{i}\\}_{i\in\mathbb{N}}$ be a collection of atoms such that there is an
$A_{i}$ in each class and such that $A_{i}$, $A_{j}$ belong to different
classes for $i\neq j$. We can moreover assume that the collection of $A_{i}$
are mutually disjoint. Let aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$A=\bigcup_{i=1}^{\infty}A_{i},\quad B=S\backslash A$

The restriction of ${\bm{\alpha}}$ to $\\{E\cap B\,:\,E\in\mathcal{F}\\}$ is
therefore non-atomic by property (iii) in Proposition 7. We call this
restriction of ${\bm{\alpha}}$ _the non-atomic part of ${\bm{\alpha}}$_ (with
respect to the particular set $\\{A_{i}\\}_{i\in\mathbb{N}}$ of chosen atoms),
and call the restriction of ${\bm{\alpha}}$ to $\\{E\cap
A\,:\,E\in\mathcal{F}\\}$ _the purely atomic part of ${\bm{\alpha}}$_. If
$|{\bm{\alpha}}|(B)=0$, then changing the choices of $A_{i}$ will not alter
this fact, and we just call ${\bm{\alpha}}$ _purely atomic_ in this case.

The following lemma can be easily derived from results by Halmos [11]:

######  Lemma 12.

Let ${\bm{\alpha}}$ be a non-atomic finite signed vector measure on
$(S,\mathcal{F})$. Then for any $A\in\mathcal{F}$ there exists a map $\varphi$
from $[0,1]$ to subsets of $A$ in $\mathcal{F}$ such that
$\varphi(0)=\emptyset$, $\varphi(1)=A$, $\varphi(a)\subseteq\varphi(b)$ if
$a<b$, and such that ${\bm{\alpha}}(\varphi(\lambda))=\lambda{\bm{\alpha}}(A)$
for all $\lambda\in[0,1]$.

##  5 The Reach of Lorenz Hulls

######  Definition 17.

The _Lorenz hull_ $\mathsf{LH}({\bm{\alpha}})$ of an $n$-dimensional finite
signed measure ${\bm{\alpha}}$ is the convex hull of the range of
${\bm{\alpha}}$.

We also call the range of ${\bm{\alpha}}$ the _Lorenz skeleton_ of
${\bm{\alpha}}$, notated $\mathsf{LS}({\bm{\alpha}})$, wherefrom
$\mathsf{LH}({\bm{\alpha}})=\mathsf{conv.hull}(\mathsf{LS}({\bm{\alpha}}))$.
Classical results (ref. Halmos [11]) show that $\mathsf{LS}({\bm{\alpha}})$ is
a centrally symmetric, closed and bounded set containing ${\bm{0}}$, and that
if ${\bm{\alpha}}$ is non-atomic then $\mathsf{LS}({\bm{\alpha}})$ is moreover
convex, i.e., $\mathsf{LH}({\bm{\alpha}})=\mathsf{LS}({\bm{\alpha}})$.
Consequently, $\mathsf{LH}({\bm{\alpha}})$ is a centrally symmetric, closed,
bounded and convex set containing ${\bm{0}}$, where in particular the
closedness follows from Proposition 1.

For an $n$-dimensional finite signed measure ${\bm{\alpha}}$ on
$(S,\mathcal{F})$ and $\sigma$-finite positive measure $\mu$ dominating
${\bm{\alpha}}$ on $(S,\mathcal{F})$, let

$[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}\star{0}\hskip
0.43057pt]\\!]\coloneqq\\{s\in
S\,:\,{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}(s)}\star 0\\}$

for ${\bm{x}}^{*}\in\mathbb{R}^{n}$, $\star\in\\{>,\geq,=,\leq,<\\}$. We note
that
$[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}\star{0}\hskip
0.43057pt]\\!]\in\mathcal{F}$ for any ${\bm{x}}^{*}$ and $\star$ since
$\\{{\bm{z}}\in\mathbb{R}^{n}\,:\,{{\bm{x}}^{*}}{}^{T}{{\bm{z}}}\star 0\\}$ is
either a hyperplane or a (closed or open) half-space in $\mathbb{R}^{n}$ and
since ${\bm{D}}^{{\bm{\alpha}}}_{\mu}$ is measurable.

######  Proposition 8.

Let ${\bm{\alpha}}$ be an $n$-dimensional finite signed measure on
$(S,\mathcal{F})$, let $\mu$ be a $\sigma$-finite positive measure dominating
${\bm{\alpha}}$ on $(S,\mathcal{F})$, and let ${\bm{x}}^{*}\in\mathbb{R}^{n}$,
$A\in\mathcal{F}$. If $\mu(A)>0$, then
${{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}(A)}>0$,
${{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}(A)}=0$,
${{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}(A)}<0$ if
$A\subseteq[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}>{0}\hskip
0.43057pt]\\!]$,
$A\subseteq[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}={0}\hskip
0.43057pt]\\!]$,
$A\subseteq[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}<{0}\hskip
0.43057pt]\\!]$, respectively. If $\mu(A)=0$ then
${{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}(A)}=0$.

###### Proof.

If $\mu(A)=0$ then ${\bm{\alpha}}(A)=0$ since $\mu$ dominates ${\bm{\alpha}}$,
and then ${{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}(A)}=0$. For when $\mu(A)>0$, we
will only discuss the situation where
$A\subseteq[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}>{0}\hskip
0.43057pt]\\!]$ and omit the proof for the other situations. Let

$A_{\varepsilon}=\\{s\in
A\,:\,{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}(s)}>\varepsilon\\}$

for $\varepsilon>0$. Then $\mu(A_{\varepsilon})=\delta>0$ for some
$\varepsilon>0$ since $A=\bigcup_{k=1}^{\infty}A_{1/k}$. Thus

$\displaystyle{{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}(A)}$
$\displaystyle={{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}(A_{\varepsilon})}+{{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}(A\backslash
A_{\varepsilon})}$
$\displaystyle=\int_{A_{\varepsilon}}{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}(s)}\,\mu(\mathrm{d}s)+\int_{A\backslash
A_{\varepsilon}}{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}(s)}\,\mu(\mathrm{d}s)$
$\displaystyle\geq\varepsilon\delta+0>0,$

which is the desired result. ∎

######  Proposition 9.

Let ${\bm{\alpha}}$ be an $n$-dimensional finite signed measure on
$(S,\mathcal{F})$ and let $\mu$ be a $\sigma$-finite positive measure
dominating ${\bm{\alpha}}$ on $(S,\mathcal{F})$. Then
$\|{\bm{x}}^{*}\|_{\mathsf{LH}({\bm{\alpha}})}={{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}([\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}>{0}\hskip
0.43057pt]\\!])}={{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}([\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}\geq{0}\hskip
0.43057pt]\\!])}$.

###### Proof.

Proposition 8 implies that
${{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}([\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}>{0}\hskip
0.43057pt]\\!])}\geq{{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}(B)}$ for all
$B\in\mathcal{F}$. Thus
$\|{\bm{x}}^{*}\|_{\mathsf{LS}({\bm{\alpha}})}={{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}([\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}>{0}\hskip
0.43057pt]\\!])}$ since
${\bm{\alpha}}([\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}>{0}\hskip
0.43057pt]\\!])\in\mathsf{LS}({\bm{\alpha}})$. But
$\|\cdot\|_{\mathsf{LS}({\bm{\alpha}})}=\|\cdot\|_{\mathsf{LH}({\bm{\alpha}})}$
by the left-hand of (2). The same argument holds replacing
$[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}>{0}\hskip
0.43057pt]\\!]$ by
$[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}\geq{0}\hskip
0.43057pt]\\!]$. ∎

##  6 The Lorenz Product

Recalling the definition of product of $\sigma$-finite measures from Lemma 2,
we have the following proposition as an analogue of Lemma 2 for finite signed
measures:

######  Proposition 10.

Let $\alpha$, $\beta$ be finite signed measures on $(S,\mathcal{F})$,
$(T,\mathcal{G})$, respectively, let $\mu$, $\nu$ be $\sigma$-finite positive
measures dominating $\alpha$, $\beta$ on $(S,\mathcal{F})$, $(T,\mathcal{G})$,
respectively, and let $\mathcal{H}$ be the $\sigma$-algebra such that
$(S\times T,\mathcal{H})=(S,\mathcal{F})\times(T,\mathcal{G})$. The function
$\omega:\mathcal{H}\to\mathbb{R}$ defined by

$\omega(E)=\int_{E}\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}(s)\frac{\mathrm{d}{\beta}}{\mathrm{d}{\nu}}(t)\,(\mu\times\nu)(\mathrm{d}(s,t))$

for all $E\in\mathcal{H}$ is then a finite signed measure on $(S\times
T,\mathcal{H})$ that satisfies

$\omega(A\times B)=\alpha(A)\beta(B)$ (6)

for all $A\in\mathcal{F}$, $B\in\mathcal{G}$.

###### Proof.

Referring back to Definition 12 let $\\{E_{i}\\}_{i\in\mathbb{N}}$ be a
collection of pairwise disjoint elements of $\mathcal{H}$ and let
$E=\bigcup_{i\in\mathbb{N}}E_{i}$. Let $f_{i}:S\times T\to\mathbb{R}$ be
defined by

$f_{i}(s,t)=\sum_{k=1}^{i}\mathbf{1}_{E_{k}}(s,t)\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}(s)\frac{\mathrm{d}{\beta}}{\mathrm{d}{\nu}}(t)$

for all $(s,t)\in S\times T$, $i\in\mathbb{N}$, and let $f:S\times
T\to\mathbb{R}$ be defined by

$f(s,t)=\mathbf{1}_{E}(s,t)\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}(s)\frac{\mathrm{d}{\beta}}{\mathrm{d}{\nu}}(t)$

for all $(s,t)\in S\times T$. Then

$\sum_{k=1}^{i}\omega(E_{k})=\int_{S\times
T}f_{i}\,\mathrm{d}(\mu\times\nu),\quad\omega(E)=\int_{S\times
T}f\,\mathrm{d}(\mu\times\nu)$

and $f_{i}$ approaches $f$ everywhere on $S\times T$ as $i\to\infty$. Since

$\displaystyle\int_{S\times T}|f_{i}|\,\mathrm{d}(\mu\times\nu)$
$\displaystyle\leq\int_{S\times
T}\bigg{|}\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}(s)\bigg{|}\bigg{|}\frac{\mathrm{d}{\beta}}{\mathrm{d}{\nu}}(t)\bigg{|}\,(\mu\times\nu)(\mathrm{d}(s,t))$
$\displaystyle=\int_{S}\bigg{|}\frac{\mathrm{d}{\alpha}}{\mathrm{d}{\mu}}(s)\bigg{|}\int_{T}\bigg{|}\frac{\mathrm{d}{\beta}}{\mathrm{d}{\nu}}(t)\bigg{|}\,\nu(\mathrm{d}t)\mu(\mathrm{d}s)$
$\displaystyle=|\alpha|(S)|\beta|(T)<\infty$ (7)

where the first equality follows by Fubini’s theorem (Lemma 5) and where the
second equality follows by Lemma 11,

$\omega(E)=\lim_{i\to\infty}\sum_{k=1}^{i}\omega(E_{k})$

by the dominated convergence theorem (Lemma 6). Thus $\omega$ is a signed
measure. The fact that $|\omega(E)|<\infty$ for all $E\in\mathcal{H}$ follows
by (7) with $f_{i}$ replaced by $f$. Lastly, (6) follows by a direct
computation by definition of $\omega$ and by Fubini’s theorem Lemma 5, again
using the finiteness of the second integral in (7). ∎

By (6) and Lemma 3, the measure $\omega$ defined in Proposition 10 is the
unique finite signed measure on $(S\times T,\mathcal{H})$ for which (6) holds
for all $A\in\mathcal{F}$, $B\in\mathcal{G}$, so that the choices of $\mu$ and
$\nu$ in the definition of $\omega$ do not matter. Extending the notation
already in place for $\sigma$-finite positive measures we let

$\alpha\times\beta$

denote the measure defined by Proposition 10.

It should be noted that Proposition 10, when taken as an observation on the
structure of $\alpha\times\beta$ and not as the grounds for definition
thereof, affords the following rephrasing:

######  Proposition 11.

Let $\alpha$, $\beta$ be finite signed measures and let $\mu$, $\nu$ be
$\sigma$-finite positive measures dominating $\alpha$, $\beta$ respectively.
Then $\alpha\times\beta$ is finite, $\mu\times\nu$ dominates
$\alpha\times\beta$, and
${\bm{D}}^{\alpha\times\beta}_{\mu\times\nu}={\bm{D}}^{\alpha}_{\mu}{\bm{D}}^{\beta}_{\nu}$.

###### Proof.

The finiteness of $\alpha\times\beta$ has already been established in
Proposition 10, under different name. The fact that $\mu\times\nu$ dominates
$\alpha\times\beta$ also follows from the first equation of Proposition 10.
The last claim follows since
${\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}={\mathrm{d}(\alpha\times\beta)\over\mathrm{d}(\mu\times\nu)}$
can be taken to be any function that satisfies the selfsame equation as the
coefficient of $(\mu\times\nu)(\mathrm{d}(s,t))$ for all $E\in\mathcal{H}$,
per the definition following Lemma 10. ∎

######  Proposition 12.

The product operation on finite signed measures is associative.

###### Proof.

Let $\alpha$, $\beta$, $\gamma$ be finite signed measures with respective
dominating measures $\mu$, $\nu$, $\eta$. Then
$(\mu\times\nu)\times\eta=\mu\times(\nu\times\eta)$ dominates both
$(\alpha\times\beta)\times\gamma$ and $\alpha\times(\beta\times\gamma)$ by
Proposition 11, and

${\bm{D}}^{(\alpha\times\beta)\times\gamma}_{\mu\times\nu\times\eta}={\bm{D}}^{\alpha\times\beta}_{\mu\times\nu}{\bm{D}}^{\gamma}_{\eta}={\bm{D}}^{\alpha}_{\mu}{\bm{D}}^{\beta}_{\nu}{\bm{D}}^{\gamma}_{\eta}={\bm{D}}^{\alpha}_{\mu}{\bm{D}}^{\beta\times\gamma}_{\nu\times\eta}={\bm{D}}^{\alpha\times(\beta\times\gamma)}_{\mu\times\nu\times\eta}$

by repeated applications of Proposition 11. The statement then follows by
Lemma 10. ∎

For $n$-dimensional finite signed measures
${\bm{\alpha}}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$,
${\bm{\beta}}=(\beta_{1},\beta_{2},$ $\ldots,\beta_{n})$ on $(S,\mathcal{F})$,
$(T,\mathcal{G})$, respectively, we let

${\bm{\alpha}}\times{\bm{\beta}}\coloneqq(\alpha_{1}\times\beta_{1},\alpha_{2}\times\beta_{2},\ldots,\alpha_{n}\times\beta_{n})$

be the _coordinate-wise product_ of ${\bm{\alpha}}$ and ${\bm{\beta}}$, which
is an $n$-dimensional finite signed measure on
$(S,\mathcal{F})\times(T,\mathcal{G})$, as underscored by the first claim of
Proposition 11. No confusion will arise from this generalization of the symbol
“$\times$” since ${\bm{\alpha}}\times{\bm{\beta}}$ reduces to
$\alpha\times\beta$ for $1$-dimensional ${\bm{\alpha}}=\alpha$,
${\bm{\beta}}=\beta$.

We also let ${\odot}^{n}$ denote the coordinate-wise product function on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$, i.e.,

${\odot}^{n}({\bm{x}},{\bm{y}})=(x_{1}y_{1},\ldots,x_{n}y_{n})$

for ${\bm{x}}=(x_{1},\ldots,x_{n})$,
${\bm{y}}=(y_{1},\ldots,y_{n})\in\mathbb{R}^{n}$. We also extend this notation
to the case where the coordinates are function of range $\mathbb{R}$, in the
natural way. This notation is useful for extending the previous result to
vector measures:

######  Proposition 13.

Let ${\bm{\alpha}}$, ${\bm{\beta}}$ be $n$-dimensional finite signed measures
and let $\mu$, $\nu$ be $\sigma$-finite positive measures dominating
${\bm{\alpha}}$, ${\bm{\beta}}$, respectively. Then $\mu\times\nu$ dominates
${\bm{\alpha}}\times{\bm{\beta}}$ and
${\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}={\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{\mu},{\bm{D}}^{{\bm{\beta}}}_{\nu})$.

###### Proof.

By definition $\mu$ dominates ${\bm{\alpha}}$ if and only if $\mu$ dominates
each coordinate of ${\bm{\alpha}}$ and likewise for $\nu$ and ${\bm{\beta}}$
and for $\mu\times\nu$ and ${\bm{\alpha}}\times{\bm{\beta}}$. The proposition
thus directly follows from the coordinate-wise application of Proposition 11.
∎

The following proposition is the technical heart of the paper, and the
culmination of the “trivial” machinery established thus far:

######  Proposition 14.

_( Theorem 4)_ Let ${\bm{\alpha}}$, ${\bm{\alpha}}^{\prime}$, ${\bm{\beta}}$,
${\bm{\beta}}^{\prime}$ be $n$-dimensional finite signed measures such that
$\mathsf{LH}({\bm{\alpha}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime})$,
$\mathsf{LH}({\bm{\beta}})\subseteq\mathsf{LH}({\bm{\beta}}^{\prime})$. Then
$\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$.

###### Proof.

We can restrict our attention to the case ${\bm{\beta}}={\bm{\beta}}^{\prime}$
as the general case will then follow by a symmetric argument. Let $\mu$,
$\mu^{\prime}$ and $\nu$ be $\sigma$-finite positive measures dominating
${\bm{\alpha}}$, ${\bm{\alpha}}^{\prime}$ and ${\bm{\beta}}$, respectively, on
their respective spaces. (E.g., $\mu=|{\bm{\alpha}}|$, etc.) Then
$\mu\times\nu$ dominates ${\bm{\alpha}}\times{\bm{\beta}}$ and
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

${\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}={\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{\mu},{\bm{D}}^{{\bm{\beta}}}_{\nu})$
(8)

by Proposition 13.

Noting that

${{\bm{x}}}{}^{T}{({\odot}^{n}({\bm{y}},{\bm{z}}))}=\sum_{i=1}^{n}x_{i}y_{i}z_{i}={({\odot}^{n}({\bm{x}},{\bm{z}}))}{}^{T}{{\bm{y}}}$
(9)

for all ${\bm{x}}=(x_{1},x_{2},\ldots,x_{n})$,
${\bm{y}}=(y_{1},y_{2},\ldots,y_{n})$,
${\bm{z}}=(z_{1},z_{2},\ldots,z_{n})\in\mathbb{R}^{n}$, we define
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

${\bm{x}}^{*}_{t}:={\odot}^{n}({\bm{x}}^{*},{\bm{D}}^{{\bm{\beta}}}_{\nu}(t))$

for all $t\in T$, ${\bm{x}}^{*}\in\mathbb{R}^{n}$, so that

${{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}(s,t)}={{\bm{x}}^{*}_{t}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}(s)}$
(10)

for all $(s,t)\in S\times T$, ${\bm{x}}^{*}\in\mathbb{R}^{n}$, by (8), (9).
Then

$\displaystyle\|{\bm{x}}^{*}\|_{\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})}$
$\displaystyle=\vphantom{\int_{[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}}}>{0}\hskip
0.3014pt]\\!]}}{{\bm{x}}^{*}}{}^{T}{({\bm{\alpha}}\times{\bm{\beta}})([\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}}}>{0}\hskip
0.43057pt]\\!])}$
$\displaystyle=\vphantom{\int_{[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}}}>{0}\hskip
0.3014pt]\\!]}}\int_{[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}}}>{0}\hskip
0.3014pt]\\!]}{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}(s,t)}\,(\mu\times\nu)(\mathrm{d}(s,t))$
$\displaystyle=\vphantom{\int_{[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}}}>{0}\hskip
0.3014pt]\\!]}}\int_{T}\int_{[\\![{{{\bm{x}}^{*}_{t}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}>{0}\hskip
0.3014pt]\\!]}{{\bm{x}}^{*}_{t}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}(s)}\,\mu(\mathrm{d}s)\,\nu(\mathrm{d}t)$
$\displaystyle=\vphantom{\int_{[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}}}>{0}\hskip
0.3014pt]\\!]}}\int_{T}{{\bm{x}}^{*}_{t}}{}^{T}{{\bm{\alpha}}([\\![{{{\bm{x}}^{*}_{t}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}>{0}\hskip
0.43057pt]\\!])}\,\nu(\mathrm{d}t)$
$\displaystyle=\vphantom{\int_{[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}}}>{0}\hskip
0.3014pt]\\!]}}\int_{T}\|{\bm{x}}^{*}_{t}\|_{\mathsf{LH}({\bm{\alpha}})}\,\nu(\mathrm{d}t)$
$\displaystyle\leq\vphantom{\int_{[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}}}>{0}\hskip
0.3014pt]\\!]}}\int_{T}\|{\bm{x}}^{*}_{t}\|_{\mathsf{LH}({\bm{\alpha}}^{\prime})}\,\nu(\mathrm{d}t)$
$\displaystyle=\vphantom{\int_{[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}}}>{0}\hskip
0.3014pt]\\!]}}\|{\bm{x}}^{*}\|_{\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}})}$

for all ${\bm{x}}^{*}\in\mathbb{R}^{n}$, where the first and fifth equalities
follow by Proposition 9, where the second and fourth equalities follow by
linearity of the integral, where the third equality follows by (10) and by
Fubini’s theorem (Lemma 5), and where the inequality follows by Proposition 3.
Thus, by Proposition 3 again,
$\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}})$.
∎

As a consequence of Proposition 14,
$\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})$ is uniquely determined by
$\mathsf{LH}({\bm{\alpha}})$ and $\mathsf{LH}({\bm{\beta}})$, in the sense of
the following corollary:

######  Corollary 1.

_( Theorem 1)_ Let ${\bm{\alpha}}$, ${\bm{\alpha}}^{\prime}$, ${\bm{\beta}}$,
${\bm{\beta}}^{\prime}$ be $n$-dimensional finite signed measures. If
$\mathsf{LH}({\bm{\alpha}})=\mathsf{LH}({\bm{\alpha}}^{\prime})$,
$\mathsf{LH}({\bm{\beta}})=\mathsf{LH}({\bm{\beta}}^{\prime})$, then
$\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})=\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$.

Let $H_{1}=\mathsf{LH}({\bm{\alpha}})$, $H_{2}=\mathsf{LH}({\bm{\beta}})$ for
some $n$-dimensional finite signed measures ${\bm{\alpha}}$, ${\bm{\beta}}$.
We define the _Lorenz product_ of $H_{1}$ and $H_{2}$, denoted by
${H_{1}}{H_{2}}$, to be the Lorenz hull
$H=\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})$. This product is well-defined
by Corollary 1. We note that Theorem 3 of the introduction is a direct
corollary of Proposition 14. The Lorenz product is commutative since
$\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}})=\mathsf{LS}({\bm{\beta}}\times{\bm{\alpha}})$
for all finite signed vector measures ${\bm{\alpha}}$, ${\bm{\beta}}$, even
while ${\bm{\alpha}}\times{\bm{\beta}}$ and ${\bm{\beta}}\times{\bm{\alpha}}$
have underlying ground sets that are reversed Cartesian products. The
associativity of the Lorenz product follows by associativity of the product of
finite signed vector measures, itself obtained by coordinate-wise application
of Proposition 12.

##  7 Sums and Distributivity

In this section we note that the Minkowski sum of Lorenz hulls is a Lorenz
hull (which is not a new observation, since the set of all zonoids and all
Lorenz hulls coincides), and show that the Lorenz product is distributive over
such sums. The distributivity follows by an analogous property of products of
“disjoint sums” of measures.

We note that since the Lorenz product is commutative the distributivity is
naturally both-sided, though it might be better to view the left- and right-
distributivity as independent corollaries of the analogous identities for
measures, since the measure product may not be commutative in more general
cases—e.g., quaternion-valued measures, as suggested by the material in
Section 8.

Let $(S,\mathcal{F})$, $(T,\mathcal{G})$ be measurable spaces with $S$, $T$
disjoint. We let $\mathcal{F}\oplus\mathcal{G}$ denote the set
aaaaaaaaaaaaaaaaaaaaaaaa

$\\{A\cup B\,:\,A\in\mathcal{F},B\in\mathcal{G}\\},$

that one can easily check is a $\sigma$-algebra on $S\cup T$. and call it _the
union-sum of $\mathcal{F}$ and $\mathcal{G}$_. For $n$-dimensional finite
signed measures ${\bm{\alpha}}$ and ${\bm{\beta}}$ on $(S,\mathcal{F})$ and
$(T,\mathcal{G})$, respectively, we also let ${\bm{\alpha}}\oplus{\bm{\beta}}$
denote the $n$-dimensional finite signed measure on $(S\cup
T,\mathcal{F}\oplus\mathcal{G})$ defined by

$({\bm{\alpha}}\oplus{\bm{\beta}})(A\cup B)={\bm{\alpha}}(A)+{\bm{\beta}}(B)$

for all $A\in\mathcal{F}$, $B\in\mathcal{G}$.

######  Proposition 15.

Let ${\bm{\alpha}}$, ${\bm{\beta}}$ be $n$-dimensional finite signed measures
on $(S,\mathcal{F})$, $(T,\mathcal{G})$, respectively, with $S$, $T$ disjoint.
Then
$\mathsf{LH}({\bm{\alpha}}\oplus{\bm{\beta}})=\mathsf{LH}({\bm{\alpha}})+\mathsf{LH}({\bm{\beta}})$.

###### Proof.

It is easy to check that
$\mathsf{LS}({\bm{\alpha}}\oplus{\bm{\beta}})=\mathsf{LS}({\bm{\alpha}})+\mathsf{LS}({\bm{\beta}})$.
The statement thus follows by the first part of Proposition 5. ∎

The product of finite signed vector measures is distributive over the direct
sum operation “$\oplus$”:

######  Proposition 16.

Let ${\bm{\alpha}}$, ${\bm{\beta}}$ be $n$-dimensional finite signed measures
on $(S,\mathcal{F})$, $(T,\mathcal{G})$, respectively, with $S$, $T$ disjoint.
Let ${\bm{\tau}}$ be an $n$-dimensional finite signed measure on
$(\Omega,\mathcal{H})$. Then
$({\bm{\alpha}}\oplus{\bm{\beta}})\times{\bm{\tau}}=({\bm{\alpha}}\times{\bm{\tau}})\oplus({\bm{\beta}}\times{\bm{\tau}})$.

###### Proof.

It is easy to check that the two measures in question have the same domain,
i.e., that the union-sum of the $\sigma$-algebras of
$(S,\mathcal{F})\times(\Omega,\mathcal{H})$ and
$(T,\mathcal{G})\times(\Omega,\mathcal{H})$ coincides with the
$\sigma$-algebra of $(S\cup
T,\mathcal{F}\oplus\mathcal{G})\times(\Omega,\mathcal{H})$. Moreover, it is
also easy to check that the two measures agree on sets of the form $A\times B$
where $A\in\mathcal{F}\oplus\mathcal{G}$, $B\in\mathcal{H}$. The statement
thus follows by Lemma 3. ∎

######  Proposition 17.

_( Theorem 2)_ ${(H_{1}+H_{2})}{H_{3}}={H_{1}}{H_{3}}+{H_{2}}{H_{3}}$ for any
Lorenz hulls $H_{1}$, $H_{2}$, $H_{3}\subseteq\mathbb{R}^{n}$.

###### Proof.

This is a direct consequence of the last two propositions since for every
$n$-dimensional Lorenz hulls $H_{1}$, $H_{2}$ there exist finite signed vector
measures ${\bm{\alpha}}$, ${\bm{\beta}}$ such that
$\mathsf{LH}({\bm{\alpha}})=H_{1}$, $\mathsf{LH}({\bm{\beta}})=H_{2}$ and such
that ${\bm{\alpha}}\oplus{\bm{\beta}}$ is defined and since
$\mathsf{LH}({\bm{\gamma}})\mathsf{LH}({\bm{\tau}})=\mathsf{LH}({\bm{\gamma}}\times{\bm{\tau}})$
by definition of the Lorenz product for all $n$-dimensional finite signed
vector measures ${\bm{\gamma}}$, ${\bm{\tau}}$. ∎

##  8 The Lorenz Product for Complex Measures

We generalize the above results to complex measures. In comparison with
Definition 12 and Definition 15, we have the following definitions:

######  Definition 18.

A _complex measure_ on a measurable space $(S,\mathcal{F})$ is a function
$\alpha:\mathcal{F}\to\mathbb{C}$ such that aaaaaaaaaaaaaaaaaaaaaaaaaa

$\alpha\Big{(}\bigcup_{i=1}^{\infty}A_{i}\Big{)}=\sum_{i=1}^{\infty}\alpha(A_{i})$
(11)

for any collection $\\{A_{i}\\}_{i\in\mathbb{N}}$ of pairwise disjoint
elements of $\mathcal{F}$.

######  Definition 19.

An $n$-dimensional complex measure on a measurable space $(S,\mathcal{F})$ is
a function ${\bm{\alpha}}:\mathcal{F}\to\mathbb{C}^{n}$ so that

${\bm{\alpha}}(A)=(\alpha_{1}(A),\alpha_{2}(A),\ldots,\alpha_{n}(A))$

for all $A\in\mathcal{F}$, where each $\alpha_{i}(A)$, $1\leq i\leq n$, is a
complex measure on $(S,\mathcal{F})$.

We extend the coordinate-wise product function ${\odot}^{n}$ to be on
$\mathbb{C}^{n}\times\mathbb{C}^{n}$, i.e.,

${\odot}^{n}({\bm{x}},{\bm{y}})=(x_{1}y_{1},\ldots,x_{n}y_{n})$

for ${\bm{x}}=(x_{1},\ldots,x_{n})$,
${\bm{y}}=(y_{1},\ldots,y_{n})\in\mathbb{C}^{n}$.

By replacing $\mathbb{R}^{n}$ with $\mathbb{C}^{n}$, replacing finite signed
measures with complex measures, and replacing absolute values of real numbers
with moduli of complex numbers, Definition 13, the definition of being
absolutely continuous with respect to a positive measure, Definition 17, the
definition of the operation “$\oplus$”, and the definition of integrals of
vector valued functions (4) can be generalized, while Lemma 3, Lemma 5, Lemma
6, Lemma 10, Lemma 11 (the range of $h$ becoming the unit circle of the
complex plane), Proposition 10, Proposition 11, Proposition 13, Proposition
15, Proposition 16 still hold. In particular the range of (the total variation
of) an $n$-dimensional complex measure is bounded, and
${\bm{\alpha}}\times{\bm{\beta}}$ for $n$-dimensional complex measures
${\bm{\alpha}}$, ${\bm{\beta}}$ is well-defined.

For any $n$ without ambiguity, let $\psi:\mathbb{C}^{n}\to\mathbb{R}^{2n}$ be
defined by

$\psi({\bm{z}})=(\mathrm{Re}(z_{1}),\mathrm{Im}(z_{1}),\ldots,\mathrm{Re}(z_{n}),\mathrm{Im}(z_{n}))$

for all ${\bm{z}}=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$, where
$\mathrm{Re}(z)$ and $\mathrm{Im}(z)$ are the real and imaginary part of $z$,
respectively, for $z\in\mathbb{C}$. It is easy to check that $\psi$ is a
bijection, and that both $\psi$ and $\psi^{-1}$ are linear and continuous
(thus measurable).

For any $n$-dimensional complex measure ${\bm{\alpha}}$ on $(S,\mathcal{F})$,
we let $\langle{\bm{\alpha}}\rangle$ be the $2n$-dimensional finite signed
measure on $(S,\mathcal{F})$ defined by

$\langle{\bm{\alpha}}\rangle(A)=\psi({\bm{\alpha}}(A))$

for all $A\in\mathcal{F}$. Obviously, a positive measure $\mu$ dominates
${\bm{\alpha}}$ if and only if $\mu$ dominates $\langle{\bm{\alpha}}\rangle$.
Moreover, it is easy to check that

$\langle{\bm{\alpha}}\rangle(A)=\int_{A}\psi\circ{\bm{D}}^{{\bm{\alpha}}}_{\mu}\,\mathrm{d}\mu$
(12)

for all $A\in\mathcal{F}$, for $\sigma$-finite $\mu$ that dominates
${\bm{\alpha}}$. By the fact that the convex hull of
$A\subseteq\mathbb{C}^{n}$ is the set of convex combinations of elements of
$A$ and that $\psi$ is a linear bijection, one has

$\mathsf{LH}({\bm{\alpha}})=\psi^{-1}(\mathsf{LH}(\langle{\bm{\alpha}}\rangle)),$

which implies that $\mathsf{LH}({\bm{\alpha}})$ is compact since $\psi^{-1}$
is continuous as a function. It is also easy to check that
$\mathsf{LH}({\bm{\alpha}})$ is centrally symmetric and convex while
containing ${\bm{0}}\in\mathbb{C}^{n}$. Moreover, the following proposition
holds:

######  Proposition 18.

Let ${\bm{\alpha}}$, ${\bm{\alpha}}^{\prime}$ be $n$-dimensional complex
measures. Then
$\mathsf{LH}({\bm{\alpha}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime})$ if
and only if
$\mathsf{LH}(\langle{\bm{\alpha}}\rangle)\subseteq\mathsf{LH}(\langle{\bm{\alpha}}^{\prime}\rangle)$.

The isomorphic product function

${\boxtimes}^{2n}:\mathbb{R}^{2n}\times\mathbb{R}^{2n}\to\mathbb{R}^{2n}$

with respect to
${\odot}^{n}:\mathbb{C}^{n}\times\mathbb{C}^{n}\to\mathbb{C}^{n}$ is defined
by

$\displaystyle{\boxtimes}^{2n}({\bm{x}},{\bm{y}})$
$\displaystyle=\psi\big{(}{\odot}^{n}(\psi^{-1}({\bm{x}}),\psi^{-1}({\bm{y}}))\big{)}$
$\displaystyle=(x_{1}y_{1}-x_{2}y_{2},x_{1}y_{2}+x_{2}y_{1},\ldots,$
$\displaystyle\qquad\qquad\qquad
x_{2n-1}y_{2n-1}-x_{2n}y_{2n},x_{2n-1}y_{2n}+x_{2n}y_{2n-1})$

for ${\bm{x}}=(x_{1},x_{2},\ldots,x_{2n-1},x_{2n})$,
${\bm{y}}=(y_{1},y_{2},\ldots,y_{2n-1},y_{2n})\in\mathbb{R}^{2n}$.

######  Proposition 19.

Let ${\bm{\alpha}}$, ${\bm{\beta}}$ be $n$-dimensional complex measures on
$(S,\mathcal{F})$, $(T,\mathcal{G})$, respectively. Let $\mu$, $\nu$ be
$\sigma$-finite positive measures on $(S,\mathcal{F})$, $(T,\mathcal{G})$,
respectively, such that $\mu$ dominates ${\bm{\alpha}}$, $\nu$ dominates
${\bm{\beta}}$. Let $(S\times
T,\mathcal{H})=(S,\mathcal{F})\times(T,\mathcal{G})$. Then

$\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle(E)=\int_{E}{\boxtimes}^{2n}({\bm{D}}^{\langle{\bm{\alpha}}\rangle}_{\mu}(s),{\bm{D}}^{\langle{\bm{\beta}}\rangle}_{\nu}(t))\,(\mu\times\nu)(\mathrm{d}(s,t))$

for all $E\in\mathcal{H}$.

###### Proof.

One has aaaaaaaa

$\displaystyle\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle(E)$
$\displaystyle=\int_{E}\psi\circ{\bm{D}}^{{\bm{\alpha}}\times{\bm{\beta}}}_{\mu\times\nu}\,\mathrm{d}(\mu\times\nu)$
$\displaystyle=\int_{E}\psi\big{(}{\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{\mu}(s),{\bm{D}}^{{\bm{\beta}}}_{\nu}(t))\big{)}\,(\mu\times\nu)(\mathrm{d}(s,t))$
$\displaystyle=\int_{E}{\boxtimes}^{2n}\big{(}\psi({\bm{D}}^{{\bm{\alpha}}}_{\mu}(s)),\psi({\bm{D}}^{{\bm{\beta}}}_{\nu}(t))\big{)}\,(\mu\times\nu)(\mathrm{d}(s,t))$
$\displaystyle=\int_{E}{\boxtimes}^{2n}({\bm{D}}^{\langle{\bm{\alpha}}\rangle}_{\mu}(s),{\bm{D}}^{\langle{\bm{\beta}}\rangle}_{\nu}(t))\,(\mu\times\nu)(\mathrm{d}(s,t))$

for all $E\in\mathcal{H}$, where the first and last equalities follow by (12),
and where the second equality follows by the complex version of Proposition
13. ∎

######  Proposition 20.

Let ${\bm{\alpha}}$, ${\bm{\alpha}}^{\prime}$, ${\bm{\beta}}$,
${\bm{\beta}}^{\prime}$ be $n$-dimensional complex measures. If
$\mathsf{LH}({\bm{\alpha}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime})$,
$\mathsf{LH}({\bm{\beta}})\subseteq\mathsf{LH}({\bm{\beta}}^{\prime})$, then
$\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$.

###### Proof.

Similarly to the proof of Proposition 14, we can assume
${\bm{\beta}}={\bm{\beta}}^{\prime}$. Let ${\bm{\alpha}}$ be on
$(S,\mathcal{F})$, ${\bm{\alpha}}^{\prime}$ be on
$(S^{\prime},\mathcal{F}^{\prime})$, ${\bm{\beta}}$ be on $(T,\mathcal{G})$.
Let $\mu$, $\mu^{\prime}$ and $\nu$ be $\sigma$-finite positive measures
dominating ${\bm{\alpha}}$, ${\bm{\alpha}}^{\prime}$ and ${\bm{\beta}}$,
respectively, on their respective spaces. Then $\mu\times\nu$ dominates
$\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle$ and aaaaaaaaaaaaaaaaaaaaaaaa

${\bm{D}}^{\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle}_{\mu\times\nu}={\boxtimes}^{2n}({\bm{D}}^{\langle{\bm{\alpha}}\rangle}_{\mu},{\bm{D}}^{\langle{\bm{\beta}}\rangle}_{\nu})$
(13)

by Proposition 19.

Noting that

$\displaystyle\quad\ {{\bm{x}}}{}^{T}{({\boxtimes}^{2n}({\bm{y}},{\bm{z}}))}$
$\displaystyle=\sum_{k=1}^{n}x_{2k-1}(y_{2k-1}z_{2k-1}-y_{2k}z_{2k})+x_{2k}(y_{2k-1}z_{2k}+y_{2k}z_{2k-1})$
$\displaystyle=\sum_{k=1}^{n}(x_{2k-1}z_{2k-1}+x_{2k}z_{2k})y_{2k-1}+(-x_{2k-1}z_{2k}+x_{2k}z_{2k-1})y_{2k}$
$\displaystyle={({\boxtimes}^{2n}({\bm{x}},\overline{{\bm{z}}}))}{}^{T}{{\bm{y}}}$
(14)

for all ${\bm{x}}=(x_{1},x_{2},\ldots,x_{2n-1},x_{2n})$,
${\bm{y}}=(y_{1},y_{2},\ldots,y_{2n-1},y_{2n})$,
${\bm{z}}=(z_{1},z_{2},\ldots,$ $z_{2n-1},z_{2n})\in\mathbb{R}^{2n}$, where
$\overline{{\bm{z}}}\coloneqq\psi(\overline{\psi^{-1}({\bm{z}})})=(z_{1},-z_{2},\ldots,z_{2n-1},-z_{2n})$,
we define aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

${\bm{x}}^{*}_{\overline{t}}:={\boxtimes}^{2n}\Big{(}{\bm{x}}^{*},\overline{{\bm{D}}^{\langle{\bm{\beta}}\rangle}_{\nu}(t)}\Big{)}$

for all $t\in T$, ${\bm{x}}^{*}\in\mathbb{R}^{2n}$, so that

${{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle}_{\mu\times\nu}(s,t)}={{\bm{x}}^{*}_{\overline{t}}}{}^{T}{{\bm{D}}^{\langle{\bm{\alpha}}\rangle}_{\mu}(s)}$
(15)

for all $(s,t)\in S\times T$, ${\bm{x}}^{*}\in\mathbb{R}^{2n}$, by (13), (14).
Since
$\mathsf{LH}(\langle{\bm{\alpha}}\rangle)\subseteq\mathsf{LH}(\langle{\bm{\alpha}}^{\prime}\rangle)$
by Proposition 18, one has

$\displaystyle\|{\bm{x}}^{*}\|_{\mathsf{LH}(\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle)}$
$\displaystyle={{\bm{x}}^{*}}{}^{T}{\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle([\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle}_{\mu\times\nu}}}>{0}\hskip
0.43057pt]\\!])}$
$\displaystyle=\int_{[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle}_{\mu\times\nu}}}>{0}\hskip
0.3014pt]\\!]}{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle}_{\mu\times\nu}}(s,t)\,(\mu\times\nu)(\mathrm{d}(s,t))$
$\displaystyle=\int_{T}\int_{[\\![{{{\bm{x}}^{*}_{\overline{t}}}{}^{T}{{\bm{D}}^{\langle{\bm{\alpha}}\rangle}_{\mu}}}>{0}\hskip
0.3014pt]\\!]}{{\bm{x}}^{*}_{\overline{t}}}{}^{T}{{\bm{D}}^{\langle{\bm{\alpha}}\rangle}_{\mu}(s)}\,\mu(\mathrm{d}s)\,\nu(\mathrm{d}t)$
$\displaystyle=\int_{T}{{\bm{x}}^{*}_{\overline{t}}}{}^{T}{\langle{\bm{\alpha}}\rangle([\\![{{{\bm{x}}^{*}_{\overline{t}}}{}^{T}{{\bm{D}}^{\langle{\bm{\alpha}}\rangle}_{\mu}}}>{0}\hskip
0.43057pt]\\!])}\,\nu(\mathrm{d}t)$
$\displaystyle=\int_{T}\|{\bm{x}}^{*}_{\overline{t}}\|_{\mathsf{LH}(\langle{\bm{\alpha}}\rangle)}\,\nu(\mathrm{d}t)$
$\displaystyle\leq\int_{T}\|{\bm{x}}^{*}_{\overline{t}}\|_{\mathsf{LH}(\langle{\bm{\alpha}}^{\prime}\rangle)}\,\nu(\mathrm{d}t)$
$\displaystyle=\|{\bm{x}}^{*}\|_{\mathsf{LH}(\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle)}$

for all ${\bm{x}}^{*}\in\mathsf{Sph}_{\mathbb{R}^{2n}}$, where the first and
fifth equalities follow by Proposition 9, where the second and fourth
equalities follow by linearity of the integral, where the third equality
follows by (15) and by Fubini’s theorem (Lemma 5), and where the inequality
follows by Proposition 3. Thus, by Proposition 3 again,
$\mathsf{LH}(\langle{\bm{\alpha}}\times{\bm{\beta}}\rangle)\subseteq\mathsf{LH}(\langle{\bm{\alpha}}^{\prime}\times{\bm{\beta}}\rangle)$,
so that
$\mathsf{LH}({\bm{\alpha}}\times{\bm{\beta}})\subseteq\mathsf{LH}({\bm{\alpha}}^{\prime}\times{\bm{\beta}})$
by Proposition 18. ∎

It follows that Corollary 1 holds for $n$-dimensional complex measures
${\bm{\alpha}}$, ${\bm{\alpha}}^{\prime}$, ${\bm{\beta}}$,
${\bm{\beta}}^{\prime}$, so that the Lorenz product can also be defined for
Lorenz hulls of complex vector measures, for which inclusion-preservation and
Proposition 17 still hold.

##  9 The Lorenz Product of Lorenz Skeletons

It is a natural question that whether the Lorenz skeleton
$\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}})$ keeps invariant while the
underlying $n$-dimensional finite signed measures ${\bm{\alpha}}$ and
${\bm{\beta}}$ alter in a way that keeps $\mathsf{LS}({\bm{\alpha}})$ and
$\mathsf{LS}({\bm{\beta}})$ unchanged. We give a positive answer to this
question, starting with analysing a discrete case.

######  Definition 20.

Let $p$ denote a chosen norm that is compatible with the Euclidean topology on
$\mathbb{R}^{n}$. Let

$d({\bm{x}},A)=\inf_{{\bm{y}}\in A}p({\bm{x}}-{\bm{y}})$

for all ${\bm{x}}\in\mathbb{R}^{n}$, $A\subseteq\mathbb{R}^{n}$. The
_Hausdorff distance_ $d_{\mathrm{H}}(A,B)$ induced by $p$ between subsets $A$
and $B$ of $\mathbb{R}^{n}$ is defined by

$d_{\mathrm{H}}(A,B)=\max\\{\sup_{{\bm{x}}\in
A}d({\bm{x}},B),\sup_{{\bm{y}}\in B}d({\bm{y}},A)\\}$

for all $A$, $B\subseteq\mathbb{R}^{n}$.

Two closed sets $A$, $B\subseteq\mathbb{R}^{n}$ equal to each other if and
only if $d_{\mathrm{H}}(A,B)=0$.

Let $\|\cdot\|_{1}$ denote the $1$-norm on $\mathbb{R}^{n}$, i.e.,

$\|{\bm{x}}\|_{1}=\sum_{i=1}^{n}|x_{i}|$

for all ${\bm{x}}=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$. All Hausdorff
distance appear in this section should be seen as induced by the $1$-norm.

######  Proposition 21.

Let ${\bm{\alpha}}=(\alpha_{1},\ldots,\alpha_{n})$ be an $n$-dimensional
finite signed measure on $(S,\mathcal{F})$. Then
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\|{\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}\|_{1}=1$

and aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$\bigg{|}\frac{\mathrm{d}{\alpha_{i}}}{\mathrm{d}{|{\bm{\alpha}}|}}\bigg{|}=\frac{\mathrm{d}{|\alpha_{i}|}}{\mathrm{d}{|{\bm{\alpha}}|}}\leq
1$

$|{\bm{\alpha}}|$-almost everywhere.

###### Proof.

The fact

$\bigg{|}\frac{\mathrm{d}{\alpha_{i}}}{\mathrm{d}{|{\bm{\alpha}}|}}\bigg{|}=\frac{\mathrm{d}{|\alpha_{i}|}}{\mathrm{d}{|{\bm{\alpha}}|}}$

follows by Lemma 11 applied with $|{\bm{\alpha}}|$ in place of $\mu$ and
$\frac{\mathrm{d}{\alpha_{i}}}{\mathrm{d}{|{\bm{\alpha}}|}}$ in place of $g$.
That

$\frac{\mathrm{d}{|\alpha_{i}|}}{\mathrm{d}{|{\bm{\alpha}}|}}\leq 1$

follows by Lemma 4 applied with $[0,1]$ in place of $E$ since
$0\leq|\alpha_{i}|(A)\leq|{\bm{\alpha}}|(A)$ for all $A\in\mathcal{F}$. One
then has

$\int_{A}(\|{\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}\|_{1}-1)\,\mathrm{d}|{\bm{\alpha}}|=\sum_{i=1}^{n}\int_{A}\bigg{|}\frac{\mathrm{d}{\alpha_{i}}}{\mathrm{d}{|{\bm{\alpha}}|}}\bigg{|}\,\mathrm{d}|{\bm{\alpha}}|-|{\bm{\alpha}}|(A)=0$

and it follows that $\|{\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}\|_{1}=1$
$|{\bm{\alpha}}|$-almost everywhere by Lemma 4 applied with $\\{0\\}$ in place
of $E$. ∎

Let aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

${\mathrm{U}^{n}_{\scriptscriptstyle{\|\|_{1}}}}\coloneqq\\{{\bm{x}}\in\mathbb{R}^{n}\,:\,\|{\bm{x}}\|_{1}=1\\}$

be the unit sphere in $\mathbb{R}^{n}$ with respect to the $1$-norm. It
follows by Proposition 21 that
${\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s)\in{\mathrm{U}^{n}_{\scriptscriptstyle{\|\|_{1}}}}$
for all $s\in S$ except for $s\in E$ for some $E\in\mathcal{F}$ such that
$|{\bm{\alpha}}|(E)=0$.

One notes the following basic fact: if $z_{1}$, $\ldots$, $z_{N}$ are real
numbers then their exists $I\subseteq[N]$ such that aaaaaaaaaaa

$\Bigg{|}\sum_{k\in I}z_{k}\Bigg{|}\geq\frac{1}{2}\sum_{k=1}^{N}|z_{k}|.$

Then for a finite signed measure $\beta$ on $(T,\mathcal{G})$,

$|\beta|(T)\leq 2\sup_{B\in\mathcal{G}}|\beta(B)|$

by definition of the total variation $|\beta|$ of $\beta$. Consider
$n$-dimensional finite signed measure ${\bm{\beta}}$ on $(T,\mathcal{G})$ and
its total variation $|{\bm{\beta}}|$ with respect to the $1$-norm, one has
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$|{\bm{\beta}}|(T)\leq 2nM$

where $M$ is any nonnegative number such that
$\mathsf{LS}({\bm{\beta}})\subseteq\mathsf{Cube}(M)$ where

$\mathsf{Cube}(M)\coloneqq\\{(z_{1},\ldots,z_{n})\in\mathbb{R}^{n}\,:\,|z_{i}|\leq
M,1\leq i\leq n\\}$

for $M\geq 0$.

######  Proposition 22.

Let ${\bm{\alpha}}$, ${\bm{\alpha}}^{\prime}$, ${\bm{\beta}}$,
${\bm{\beta}}^{\prime}$ be $n$-dimensional finite signed measures on
$(S,\mathcal{F})$, $(S^{\prime},\mathcal{F}^{\prime})$, $(T,\mathcal{G})$,
$(T^{\prime},\mathcal{G}^{\prime})$, respectively, where all four measurable
spaces are countable sets with their discrete $\sigma$-algebras, and where
$\mathsf{LS}({\bm{\alpha}})$, $\mathsf{LS}({\bm{\alpha}}^{\prime})$,
$\mathsf{LS}({\bm{\beta}})$, $\mathsf{LS}({\bm{\beta}}^{\prime})$ are all
included in $\mathsf{Cube}(M)$ for a fixed $M>0$. Let $\varepsilon>0$, then
aaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\big{)}<\varepsilon$

whenever
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}),\mathsf{LS}({\bm{\alpha}}^{\prime})\big{)}<\delta$,
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\beta}}),\mathsf{LS}({\bm{\beta}}^{\prime})\big{)}<\delta$
for $\delta<\varepsilon/4nM$.

###### Proof.

Firstly note that all functions in this proof are measurable since the
underlying spaces are discrete, and note that $|{\bm{\alpha}}|(S)$,
$|{\bm{\alpha}}^{\prime}|(S^{\prime})$, $|{\bm{\beta}}|(T)$,
$|{\bm{\beta}}|(T^{\prime})\leq 2nM$. Let $\delta<\varepsilon/4nM$. If
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}),\mathsf{LS}({\bm{\alpha}}^{\prime})\big{)}<\delta$,
then in particular for any measurable $f:S\to\\{0,1\\}$ there exists a
measurable $f^{\prime}:S^{\prime}\to\\{0,1\\}$ such that aaaaaaaaaaaa

$\bigg{\|}\int_{S}f\,\mathrm{d}{\bm{\alpha}}-\int_{S^{\prime}}f^{\prime}\,\mathrm{d}{\bm{\alpha}}^{\prime}\bigg{\|}_{1}<\delta.$

Now for any $h:S\times T\to\\{0,1\\}$ one can construct
$h^{\prime}:S^{\prime}\times T\to\\{0,1\\}$ such that

$\bigg{\|}\int_{S}h_{t}\,\mathrm{d}{\bm{\alpha}}-\int_{S^{\prime}}h^{\prime}_{t}\,\mathrm{d}{\bm{\alpha}}^{\prime}\bigg{\|}_{1}<\delta.$

for all $t\in T$, where $h_{t}:S\to\\{0,1\\}$,
$h^{\prime}_{t}:S^{\prime}\to\\{0,1\\}$ are defined by

$h_{t}(s)=h(s,t),\quad h^{\prime}_{t}(s^{\prime})=h^{\prime}(s^{\prime},t)$

for all $s\in S$, $s^{\prime}\in S^{\prime}$. Then

$\displaystyle\quad\ \bigg{\|}\int_{S\times
T}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}\times
T}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}})\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}\int_{T}\int_{S}h(s,t){\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))\,|{\bm{\alpha}}|(\mathrm{d}s)\,|{\bm{\beta}}|(\mathrm{d}t)$
$\displaystyle\qquad\qquad\qquad-\int_{T}\int_{S}^{\prime}h(s^{\prime},t){\odot}^{n}({\bm{D}}^{{\bm{\alpha}}^{\prime}}_{|{\bm{\alpha}}^{\prime}|}(s^{\prime}),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))\,|{\bm{\alpha}}^{\prime}|(\mathrm{d}s^{\prime})\,|{\bm{\beta}}|(\mathrm{d}t)\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}\int_{T}{\odot}^{n}\bigg{(}\bigg{(}\int_{S}h_{t}{\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}\,\mathrm{d}|{\bm{\alpha}}|-\int_{S^{\prime}}h^{\prime}_{t}{\bm{D}}^{{\bm{\alpha}}^{\prime}}_{|{\bm{\alpha}}^{\prime}|}\,\mathrm{d}|{\bm{\alpha}}^{\prime}|\bigg{)},{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t)\bigg{)}\,|{\bm{\beta}}|(\mathrm{d}t)\bigg{\|}_{1}$
$\displaystyle\leq\int_{T}\bigg{\|}{\odot}^{n}\bigg{(}\bigg{(}\int_{S}h_{t}\,\mathrm{d}{\bm{\alpha}}-\int_{S^{\prime}}h^{\prime}_{t}\,\mathrm{d}{\bm{\alpha}}^{\prime}\bigg{)},{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t)\bigg{)}\bigg{\|}_{1}\,|{\bm{\beta}}|(\mathrm{d}t)$
$\displaystyle\leq\int_{T}\bigg{\|}\bigg{(}\int_{S}h_{t}\,\mathrm{d}{\bm{\alpha}}-\int_{S^{\prime}}h^{\prime}_{t}\,\mathrm{d}{\bm{\alpha}}^{\prime}\bigg{)}\bigg{\|}_{1}\,|{\bm{\beta}}|(\mathrm{d}t)<2nM\delta$

where the first equality follows by Proposition 13 and by Fubini’s theorem
Lemma 5, where the second equality follows by linearity of integrations, and
where the second inequality follows by Proposition 21. This shows that for
each ${\bm{x}}\in\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}})$ there exists
${\bm{x}}^{\prime}\in\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}})$
such that $\|{\bm{x}}-{\bm{x}}^{\prime}\|_{1}<2nM\delta$.

A symmetric argument shows that for each
${\bm{x}}^{\prime}\in\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}})$
there exists ${\bm{x}}\in\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}})$ such
that $\|{\bm{x}}-{\bm{x}}^{\prime}\|_{1}<2nM\delta$. Then
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}})\big{)}\leq
2nM\delta$ by definition. Similarly,
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\big{)}\leq
2nM\delta$. It follows that
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\big{)}\leq
4nM\delta<\varepsilon$. ∎

In fact, Proposition 22 holds when only $T$ and $S^{\prime}$ are known to be
countable by the above proof. In particular,
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}})\big{)}\leq
2nM\delta$ when $T$ is countable, and
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\big{)}\leq
2nM\delta$ when $S^{\prime}$ is countable.

######  Proposition 23.

Let ${\bm{\alpha}}$, ${\bm{\beta}}$ be $n$-dimensional finite signed measures
on $(S,\mathcal{F})$, $(T,\mathcal{G})$, respectively. Let $A$ be an atom of
$|{\bm{\alpha}}|$ and $B$ be an atom of $|{\bm{\beta}}|$. Let $f:S\times
T\to\mathbb{R}$ be bounded and measurable. Then

$\int_{S\times
B}f\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})={\odot}^{n}\bigg{(}\int_{S}f(s,t_{0})\,{\bm{\alpha}}(\mathrm{d}s),{\bm{\beta}}(B)\bigg{)}$

for some $t_{0}\in B$,

$\int_{A\times
T}f\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})={\odot}^{n}\bigg{(}{\bm{\alpha}}(A),\int_{T}f(s_{0},t)\,{\bm{\beta}}(\mathrm{d}t)\bigg{)}$

for some $s_{0}\in A$, and

$\int_{A\times
B}f\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})=f(s_{0},t_{0})\big{(}{\odot}^{n}({\bm{\alpha}}(A),{\bm{\beta}}(B))\big{)}$

for some $s_{0}\in A$, $t_{0}\in B$.

###### Proof.

Let ${\bm{\alpha}}=(\alpha_{1},\ldots,\alpha_{n})$,
${\bm{\beta}}=(\beta_{1},\ldots,\beta_{n})$. Fubini’s theorem Lemma 5 implies
that the function $h^{S}_{i}:T\to\mathbb{R}$ defined by
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$h^{S}_{i}(t)=\int_{S}f(s,t)\frac{\mathrm{d}{\alpha_{i}}}{\mathrm{d}{|{\bm{\alpha}}|}}(s)\frac{\mathrm{d}{\beta_{i}}}{\mathrm{d}{|{\bm{\beta}}|}}(t)\,|{\bm{\alpha}}|(\mathrm{d}s)$

for all $t\in T$ is measurable for each $i$, $1\leq i\leq n$, so that the
function
$({\bm{h}}^{S},{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}):T\to\mathbb{R}^{2n}$
is $|{\bm{\beta}}|$-almost constant on $B$ by (iv) in Proposition 7, where
${\bm{h}}^{S}:T\to\mathbb{R}^{n}$ is defined by

$\displaystyle{\bm{h}}^{S}(t)$
$\displaystyle=\int_{S}f(s,t)\big{(}{\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))\big{)}\,|{\bm{\alpha}}|(\mathrm{d}s)$
$\displaystyle={\odot}^{n}\bigg{(}\int_{S}f(s,t)\,{\bm{\alpha}}(\mathrm{d}s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t)\bigg{)}$

for all $t\in T$, where the second equality follows by linearity of integrals.
This implies aaaaaaaaaaaaaa

$\displaystyle\quad\ \int_{S\times
B}f\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})$ $\displaystyle=\int_{S\times
B}f\big{(}{\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))\big{)}\,(|{\bm{\alpha}}|\times|{\bm{\beta}}|)(\mathrm{d}(s,t))$
$\displaystyle=\int_{B}\int_{S}f\big{(}{\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))\big{)}\,|{\bm{\alpha}}|(\mathrm{d}s)\,|{\bm{\beta}}|(\mathrm{d}t)$
$\displaystyle=\int_{B}{\bm{h}}^{S}\,\mathrm{d}|{\bm{\beta}}|={\odot}^{n}\bigg{(}\int_{S}f(s,t_{0})\,{\bm{\alpha}}(\mathrm{d}s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t_{0})\bigg{)}|{\bm{\beta}}|(B)$
$\displaystyle={\odot}^{n}\bigg{(}\int_{S}f(s,t_{0})\,{\bm{\alpha}}(\mathrm{d}s),{\bm{\beta}}(B)\bigg{)}$

for some $t_{0}\in B$ for which
$({\bm{h}}^{S},{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|})=({\bm{h}}^{S},{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|})(t_{0})$
$|{\bm{\beta}}|$-almost everywhere, where the second equality follows by
Fubini’s theorem Lemma 5 and by linearity of integrals. The other equations
follow by similar arguments. ∎

######  Proposition 24.

Let ${\bm{\alpha}}$, ${\bm{\beta}}$ be $n$-dimensional finite signed measures
on $(S,\mathcal{F})$, $(T,\mathcal{G})$, respectively. For any
$\varepsilon>0$, there exist $n$-dimensional finite signed measures
${\bm{\alpha}}^{\prime}$, ${\bm{\beta}}^{\prime}$ on discrete
$\sigma$-algebras $\mathcal{F}^{\prime}$, $\mathcal{G}^{\prime}$ of countable
sets $S^{\prime}$, $T^{\prime}$, respectively, such that
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}),\mathsf{LS}({\bm{\alpha}}^{\prime})\big{)}<\varepsilon$,
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\beta}}),\mathsf{LS}({\bm{\beta}}^{\prime})\big{)}<\varepsilon$,
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\big{)}<\varepsilon$.

###### Proof.

Without loss of generality we assume $|{\bm{\alpha}}|(S)|{\bm{\beta}}|(T)>0$,
for other cases can be trivially solved once the general construction is
clear.

We firstly discuss the case where ${\bm{\alpha}}$ and ${\bm{\beta}}$ are both
non-atomic. Fix $\varepsilon_{0}>0$ such that $\varepsilon_{0}<\varepsilon$.
Let $0<\delta<\varepsilon_{0}/4|{\bm{\alpha}}|(S)|{\bm{\beta}}|(T)$. One can
decompose ${\mathrm{U}^{n}_{\scriptscriptstyle{\|\|_{1}}}}$ into a union of
$K$ disjoint measurable sets $U_{1}$, $\ldots$, $U_{K}$ where

$\|{\bm{x}}-{\bm{y}}\|_{1}<\delta$

for all ${\bm{x}}$, ${\bm{y}}\in U_{k}$, $1\leq k\leq K$. Let

$S_{k}=({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|})^{-1}(U_{k}),\quad
T_{k}=({\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|})^{-1}(U_{k})$

for all $k\in[K]$. Then

$|{\bm{\alpha}}|(S\backslash\bigcup_{k=1}^{K}S_{k})=0,\quad|{\bm{\beta}}|(T\backslash\bigcup_{k=1}^{K}T_{k})=0$

by Proposition 21, so that we can assume

$S=\bigcup_{k=1}^{K}S_{k},\quad T=\bigcup_{k=1}^{K}T_{k}$

when calculating integrals. Fix ${\bm{u}}_{k}\in U_{k}$ for each $k\in[K]$.
Simple calculation shows that aaaaaaaaaaaaaaaaaa

$\|{\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))-{\odot}^{n}({\bm{u}}_{p},{\bm{u}}_{q})\|_{1}<2\delta$
(16)

for all $s\in S_{p}$, $t\in T_{q}$, $p$, $q\in[K]$. We also assume without
loss of generality that $|{\bm{\alpha}}|(S_{k})>0$, $|{\bm{\beta}}|(T_{k})>0$
for $k\in[K]$.

Let $N$ be a positive integer such that
$N^{2}>2n|{\bm{\alpha}}|(S)|{\bm{\beta}}|(T)/\varepsilon_{0}$. Let both
$S^{\prime}$, $T^{\prime}$ be the finite set $[K]\times[N]$ and let both
$\mathcal{F}^{\prime}$, $\mathcal{G}^{\prime}$ be the discrete
$\sigma$-algebra on $[K]\times[N]$. Let ${\bm{\alpha}}^{\prime}$,
${\bm{\beta}}^{\prime}$ be defined by

$\displaystyle{\bm{\alpha}}^{\prime}(\\{(k,m)\\})$
$\displaystyle={\bm{a}}_{k}\coloneqq|{\bm{\alpha}}|(S_{k}){\bm{u}}_{k}/N,$
$\displaystyle{\bm{\beta}}^{\prime}(\\{(k,m)\\})$
$\displaystyle={\bm{b}}_{k}\coloneqq|{\bm{\beta}}|(T_{k}){\bm{u}}_{k}/N$

for all $m\in[N]$, $k\in[K]$.

Consider an arbitrary measurable function $h:S\times T\to\\{0,1\\}$. We show
there exists $h^{\prime}:S^{\prime}\times T^{\prime}\to\\{0,1\\}$ (which is
measurable since $S^{\prime}\times T^{\prime}$ is finite) such that
aaaaaaaaaaaaaaaaaaaaaaa

$\bigg{\|}\int_{S\times
T}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}\times
T^{\prime}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}<\varepsilon.$

For this, let aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$r_{p,q}=\frac{\int_{S_{p}\times
T_{q}}h\,\mathrm{d}(|{\bm{\alpha}}|\times|{\bm{\beta}}|)}{|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(T_{q})}\in[0,1]$

and fix $h^{\prime}$ to be such that

$\sum_{i,j\in[N]}h^{\prime}((p,i),(q,j))\leq
r_{p,q}N^{2}<\sum_{i,j\in[N]}h^{\prime}((p,i),(q,j))+1$

for all $p$, $q\in[K]$. Then on one hand

$\displaystyle\quad\
\bigg{\|}r_{p,q}N^{2}({\odot}^{n}({\bm{a}}_{p},{\bm{b}}_{q}))-\int_{(\\{p\\}\times[N])\times(\\{q\\}\times[N])}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}\bigg{(}r_{p,q}N^{2}-\sum_{i,j\in[N]}h^{\prime}((p,i),(q,j))\bigg{)}({\odot}^{n}({\bm{a}}_{p},{\bm{b}}_{q}))\bigg{\|}_{1}$
$\displaystyle\leq\|{\odot}^{n}({\bm{a}}_{p},{\bm{b}}_{q})\|_{1}=\frac{|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(T_{q})}{N^{2}}\|{\odot}^{n}({\bm{u}}_{p},{\bm{u}}_{q})\|_{1}$
$\displaystyle\leq\frac{n}{N^{2}}|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(T_{q})$

for all $p$, $q\in K$. On the other hand

$\displaystyle\quad\ \bigg{\|}\int_{S_{p}\times
T_{q}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-r_{p,q}N^{2}({\odot}^{n}({\bm{a}}_{p},{\bm{b}}_{q}))\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}\int_{S_{p}\times
T_{q}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\bigg{(}\int_{S_{p}\times
T_{q}}h\,\mathrm{d}(|{\bm{\alpha}}|\times|{\bm{\beta}}|)\bigg{)}({\odot}^{n}({\bm{u}}_{p},{\bm{u}}_{q}))\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}\int_{S_{p}\times
T_{q}}h({\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))-{\odot}^{n}({\bm{u}}_{p},{\bm{u}}_{q}))\,(|{\bm{\alpha}}|\times|{\bm{\beta}}|)(\mathrm{d}(s,t))\bigg{\|}_{1}$
$\displaystyle\leq\int_{S_{p}\times
T_{q}}|h|\|{\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))-{\odot}^{n}({\bm{u}}_{p},{\bm{u}}_{q})\|_{1}\,(|{\bm{\alpha}}|\times|{\bm{\beta}}|)(\mathrm{d}(s,t))$
$\displaystyle\leq 2\delta|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(T_{q})$

for all $p$, $q\in K$, where the second equality follows by Proposition 13 and
linearity of integrals, and where the second inequality follows by (16). It
follows that

$\displaystyle\bigg{\|}\int_{S\times
T}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}\times
T^{\prime}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}$
$\displaystyle\qquad\leq\sum_{p,q\in[K]}\Big{(}\frac{n}{N^{2}}+2\delta\Big{)}|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(T_{q})=\Big{(}\frac{n}{N^{2}}+2\delta\Big{)}|{\bm{\alpha}}|(S)|{\bm{\beta}}|(T)<\varepsilon_{0}.$

This means for any ${\bm{x}}\in\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}})$
there exists
${\bm{x}}^{\prime}\in\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$
such that $\|{\bm{x}}-{\bm{x}}^{\prime}\|_{1}<\varepsilon_{0}$.

To show the other direction, consider an arbitrary function
$h^{\prime}:S^{\prime}\times T^{\prime}\to\\{0,1\\}$, and let
aaaaaaaaaaaaaaaaaaaa

$r^{\prime}_{p,q}=\frac{1}{N^{2}}\sum_{i,j\in[N]}h^{\prime}((p,i),(q,j))\in[0,1]$

for all $p$, $q\in[K]$. Since ${\bm{\alpha}}$, ${\bm{\beta}}$ are non-atomic,
$|{\bm{\alpha}}|$, $|{\bm{\beta}}|$ are non-atomic by (ii) in Proposition 7,
and one can fix subsets $S^{\prime\prime}_{p,q}\in\mathcal{F}$,
$T^{\prime\prime}_{p,q}\in\mathcal{G}$ of $S_{p}$, $T_{q}$, respectively, such
that

$|{\bm{\alpha}}|(S^{\prime\prime}_{p,q})=(r^{\prime}_{p,q})^{\frac{1}{2}}|{\bm{\alpha}}|(S_{p}),\quad|{\bm{\beta}}|(T^{\prime\prime}_{p,q})=(r^{\prime}_{p,q})^{\frac{1}{2}}|{\bm{\beta}}|(T_{q})$

for each $p$, $q\in[K]$ by Lemma 12. Let $h:S\times T\to\\{0,1\\}$ be defined
by

$h(s,t)=\mathbf{1}_{S^{\prime\prime}_{p,q}\times T^{\prime\prime}_{p,q}}(s,t)$

for all $(s,t)\in S_{p}\times T_{q}$, $p$, $q\in[K]$. Then

$\displaystyle\quad\ \bigg{\|}\int_{S_{p}\times
T_{q}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{(\\{p\\}\times[N])\times(\\{q\\}\times[N])}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}\int_{S_{p}\times
T_{q}}\mathbf{1}_{S^{\prime\prime}_{p,q}\times
T^{\prime\prime}_{p,q}}\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-r^{\prime}_{p,q}N^{2}({\odot}^{n}({\bm{a}}_{p},{\bm{b}}_{q}))\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}\int_{S^{\prime\prime}_{p,q}\times
T^{\prime\prime}_{p,q}}\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-|{\bm{\alpha}}|(S^{\prime\prime}_{p,q})|{\bm{\beta}}|(T^{\prime\prime}_{p,q})({\odot}^{n}({\bm{u}}_{p},{\bm{u}}_{q}))\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}\int_{S^{\prime\prime}_{p,q}\times
T^{\prime\prime}_{p,q}}({\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))-{\odot}^{n}({\bm{u}}_{p},{\bm{u}}_{q}))\,(|{\bm{\alpha}}|\times|{\bm{\beta}}|)(\mathrm{d}(s,t))\bigg{\|}_{1}$
$\displaystyle\leq\int_{S^{\prime\prime}_{p,q}\times
T^{\prime\prime}_{p,q}}\|{\odot}^{n}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}(s),{\bm{D}}^{{\bm{\beta}}}_{|{\bm{\beta}}|}(t))-{\odot}^{n}({\bm{u}}_{p},{\bm{u}}_{q})\|_{1}\,(|{\bm{\alpha}}|\times|{\bm{\beta}}|)(\mathrm{d}(s,t))$
$\displaystyle\leq
2\delta|{\bm{\alpha}}|(S^{\prime\prime}_{p,q})|{\bm{\beta}}|(T^{\prime\prime}_{p,q})\leq
2\delta|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(T_{q})$

for all $p$, $q\in K$, where the third equality follows by Proposition 13 and
linearity of integrals, and where the second inequality follows by (16). It
follows that

$\displaystyle\bigg{\|}\int_{S\times
T}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}\times
T^{\prime}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}$
$\displaystyle\qquad\qquad\qquad\qquad\leq\sum_{p,q\in[K]}2\delta|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(T_{q})=2\delta|{\bm{\alpha}}|(S)|{\bm{\beta}}|(T)<\varepsilon_{0}.$

This means for any
${\bm{x}}^{\prime}\in\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$
there exists ${\bm{x}}\in\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}})$ such
that $\|{\bm{x}}-{\bm{x}}^{\prime}\|_{1}<\varepsilon_{0}$. It finally follows
that

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\big{)}\leq\varepsilon_{0}<\varepsilon$

for our construction of ${\bm{\alpha}}^{\prime}$ and ${\bm{\beta}}^{\prime}$.
The arguments for showing

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}),\mathsf{LS}({\bm{\alpha}}^{\prime})\big{)}<\varepsilon,\quad
d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\beta}}),\mathsf{LS}({\bm{\beta}}^{\prime})\big{)}<\varepsilon$

are similar and simpler, in which we need to let

$0<\delta<\min\\{\varepsilon_{0}/4|{\bm{\alpha}}|(S)|{\bm{\beta}}|(T),\varepsilon_{0}/4|{\bm{\alpha}}|(S),\varepsilon_{0}/4{\bm{\beta}}|(T)\\}$

and aaaaaaaaaaaaaaaaaaaaaaaaaaa

$N>\max\\{(2n|{\bm{\alpha}}|(S)|{\bm{\beta}}|(T)/\varepsilon_{0})^{\frac{1}{2}},2n|{\bm{\alpha}}|(S)/\varepsilon_{0},2n|{\bm{\beta}}|(T)/\varepsilon_{0}\\}$

replace the existing requirement of $\delta$ and $N$.

Calling the above the first part of our proof, we now start to consider the
most general case. By the discussion below Proposition 7, $S$ can be
decomposed as union of disjoint subsets $S_{\rm{con}}$, $S_{\rm{atm}}$ where
the restriction of ${\bm{\alpha}}$ to $S_{\rm{con}}$ is non-atomic and where
$S_{\rm{atm}}$ is union of at most countably many mutually disjoint atoms
$A_{1},A_{2},\ldots$ of $|{\bm{\alpha}}|$. Similarly $T$ can be decomposed as
union of disjoint subsets $T_{\rm{con}}$, $T_{\rm{atm}}$ where the restriction
of ${\bm{\beta}}$ to $T_{\rm{con}}$ is non-atomic and where $T_{\rm{atm}}$ is
union of at most countably many mutually disjoint atoms $B_{1},B_{2},\ldots$
of $|{\bm{\beta}}|$. Let $A_{a}=\emptyset$ for $a>N_{1}$ if $S_{\rm{atm}}$
contains $N_{1}<\infty$ atoms of $|{\bm{\alpha}}|$ and let $B_{b}=\emptyset$
for $b>N_{2}$ if $T_{\rm{atm}}$ contains $N_{2}<\infty$ atoms of
$|{\bm{\beta}}|$. Let ${\bm{\alpha}}_{1}$, ${\bm{\alpha}}_{2}$ be the
respective restrictions of ${\bm{\alpha}}$ to $S_{\rm{con}}$, $S_{\rm{atm}}$,
and let ${\bm{\beta}}_{1}$, ${\bm{\beta}}_{2}$ be the respective restrictions
of ${\bm{\beta}}$ to $T_{\rm{con}}$, $T_{\rm{atm}}$.

Let ${\bm{\alpha}}^{\prime}_{1}$, ${\bm{\beta}}^{\prime}_{1}$ on the discrete
$\sigma$-algebra on $S^{\prime}_{\rm{con}}=T^{\prime}_{\rm{con}}=[K]\times[N]$
be defined respectively for ${\bm{\alpha}}_{1}$, ${\bm{\beta}}_{1}$ as
${\bm{\alpha}}^{\prime}$, ${\bm{\beta}}^{\prime}$ be defined respectively for
${\bm{\alpha}}$, ${\bm{\beta}}$ in the first part of our proof with additional
requirements that $\varepsilon_{0}<\varepsilon/3$,
$N>2n|{\bm{\alpha}}|(S)|{\bm{\beta}}|(T)/\varepsilon_{0}$. Let
${\bm{\alpha}}^{\prime}_{2}$, ${\bm{\beta}}^{\prime}_{2}$ on the discrete
$\sigma$-algebra on $S^{\prime}_{\rm{atm}}=T^{\prime}_{\rm{atm}}=\mathbb{N}$
be defined by

${\bm{\alpha}}^{\prime}_{2}(\\{a\\})={\bm{\alpha}}(A_{a}),\quad{\bm{\beta}}^{\prime}_{2}(\\{b\\})={\bm{\beta}}(B_{b})$

for $a$, $b\in\mathbb{N}$. Finally, let
${\bm{\alpha}}^{\prime}={\bm{\alpha}}^{\prime}_{1}\oplus{\bm{\alpha}}^{\prime}_{2}$,
${\bm{\beta}}^{\prime}={\bm{\beta}}^{\prime}_{1}\oplus{\bm{\beta}}^{\prime}_{2}$.

As the usual, firstly consider an arbitrary measurable function $h:S\times
T\to\\{0,1\\}$, and we claim to find $h^{\prime}:S^{\prime}\times
T^{\prime}\to\\{0,1\\}$ such that

$\bigg{\|}\int_{S\times
T}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}\times
T^{\prime}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}<3\varepsilon_{0},$
(17)

where $h^{\prime}$ is automatically measurable since $S^{\prime}\times
T^{\prime}$ is at most countable. The first part of our proof has already
shown that one can make the restriction of $h^{\prime}$ to
$S^{\prime}_{\rm{con}}\times T^{\prime}_{\rm{con}}$ to be such that

$\bigg{\|}\int_{S_{\rm{con}}\times
T_{\rm{con}}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}_{\rm{con}}\times
T^{\prime}_{\rm{con}}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}<\varepsilon_{0}.$
(18)

By Proposition 23, there exists $t_{p,b}\in B_{b}$ such that

$\int_{S_{p}\times
B_{b}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})={\odot}^{n}\bigg{(}\int_{S_{p}}h(s,t_{p,b})\,{\bm{\alpha}}(\mathrm{d}s),{\bm{\beta}}(B_{b})\bigg{)}$

for all $p\in[K]$, $b\in\mathbb{N}$ such that $B_{b}\neq\emptyset$. Let

$\ell_{p,b}=\frac{\int_{S_{p}}h(s,t_{p,b})\,|{\bm{\alpha}}|(\mathrm{d}s)}{|{\bm{\alpha}}|(S_{p})}\in[0,1]$

and let the restriction of $h^{\prime}$ on $S^{\prime}_{\rm{con}}\times
T^{\prime}_{\rm{atm}}$ be such that

$\sum_{i\in[N]}h^{\prime}((p,i),b)\leq\ell_{p,b}N<\sum_{i\in[N]}h^{\prime}((p,i),b)+1$

for all $p\in[K]$, $b\in\mathbb{N}$ such that $B_{b}\neq\emptyset$, and such
that $h^{\prime}((p,i),b)=0$ for all $(p,i)\in[K]\times[N]$, $b\in\mathbb{N}$
such that $B_{b}=\emptyset$. Then

$\displaystyle\quad\ \bigg{\|}\int_{S_{p}\times
B_{b}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{(\\{p\\}\times[N])\times\\{b\\}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}$
$\displaystyle\leq\bigg{\|}{\odot}^{n}\bigg{(}\int_{S_{p}}h(s,t_{p,b})\,{\bm{\alpha}}(\mathrm{d}s),{\bm{\beta}}(B_{b})\bigg{)}-\ell_{p,b}N{\odot}^{n}({\bm{a}}_{p},{\bm{\beta}}(B_{b}))\bigg{\|}_{1}$
$\displaystyle\qquad\qquad+\bigg{\|}\ell_{p,b}N{\odot}^{n}({\bm{a}}_{p},{\bm{\beta}}(B_{b}))-\sum_{i\in[N]}h^{\prime}((p,i),b){\odot}^{n}({\bm{a}}_{p},{\bm{\beta}}^{\prime}(\\{b\\}))\bigg{\|}_{1}$
$\displaystyle\leq\Big{(}\delta+\frac{n}{N}\Big{)}|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(B_{b})$

for all $p\in[K]$, $b\in\mathbb{N}$ such that $B_{b}\neq\emptyset$ by
calculations similar as in the first part of our proof, and

$\displaystyle\bigg{\|}\int_{S_{\rm{con}}\times
T_{\rm{atm}}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}_{\rm{con}}\times
T^{\prime}_{\rm{atm}}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}$
$\displaystyle\qquad\qquad\qquad\qquad\leq\sum_{b=1}^{\infty}\sum_{p\in[K]}\Big{(}\delta+\frac{n}{N}\Big{)}|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(B_{b})$
$\displaystyle\qquad\qquad\qquad\qquad=\Big{(}\delta+\frac{n}{N}\Big{)}|{\bm{\alpha}}|(S_{\rm{con}})|{\bm{\beta}}|(T_{\rm{atm}})<\varepsilon_{0}.$
(19)

A similar process shows that on can set the restriction of $h^{\prime}$ on
$S^{\prime}_{\rm{atm}}\times T^{\prime}_{\rm{con}}$ to be such that
aaaaaaaaaaaa

$\bigg{\|}\int_{S_{\rm{atm}}\times
T_{\rm{con}}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}_{\rm{atm}}\times
T^{\prime}_{\rm{con}}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}<\varepsilon_{0}.$
(20)

At last, for each $a$, $b\in\mathbb{N}$ such that $A_{a}$,
$B_{b}\neq\emptyset$, their exist $s^{a,b}\in A_{a}$, $t^{a,b}\in B_{b}$ such
that aaaaaaaaaaaaaa

$\int_{A_{a}\times
B_{b}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})=h(s^{a,b},t^{a,b})\big{(}{\odot}^{n}({\bm{\alpha}}(A_{a}),{\bm{\beta}}(B_{b}))\big{)}$

by Proposition 23. Let the restriction of $h^{\prime}$ on
$S^{\prime}_{\rm{atm}}\times T^{\prime}_{\rm{atm}}$ be such that
$h^{\prime}(a,b)=h(s^{a,b},t^{a,b})$. Then

$\displaystyle\int_{A_{a}\times
B_{b}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{\\{a\\}\times\\{b\\}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$
$\displaystyle\qquad\qquad\qquad=h(s^{a,b},t^{a,b})\big{(}{\odot}^{n}({\bm{\alpha}}(A_{a}),{\bm{\beta}}(B_{b}))\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-h^{\prime}(a,b)\big{(}{\odot}^{n}({\bm{\alpha}}^{\prime}(\\{a\\}),{\bm{\beta}}^{\prime}(\\{b\\}))\big{)}$
$\displaystyle\qquad\qquad\qquad=0$

for all $a$, $b\in\mathbb{N}$ such that $A_{a}$, $B_{b}\neq\emptyset$, so that

$\bigg{\|}\int_{S_{\rm{atm}}\times
T_{\rm{atm}}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}_{\rm{atm}}\times
T^{\prime}_{\rm{atm}}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}=0.$
(21)

The inequality (17) then follows by (18), (19), (20) and (21).

The other direction is again easier. Consider an arbitrary
$h^{\prime}:S^{\prime}\times T^{\prime}\to\\{0,1\\}$. We claim to construct a
measurable $h:S\times T\to\\{0,1\\}$ such that

$\bigg{\|}\int_{S\times
T}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}\times
T^{\prime}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}<3\varepsilon_{0}.$
(22)

Again, the first part of our proof has shown that one can make the restriction
of $h$ to $S_{\rm{con}}\times T_{\rm{con}}$ to be such that

$\bigg{\|}\int_{S_{\rm{con}}\times
T_{\rm{con}}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}_{\rm{con}}\times
T^{\prime}_{\rm{con}}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}<\varepsilon_{0}.$
(23)

Let aaaaaaaaaaaaaaaaaaaa

$\ell^{\prime}_{p,b}=\frac{1}{N}\sum_{i\in[N]}h^{\prime}((p,i),b)\in[0,1]$

for all $p\in[K]$, $b\in\mathbb{N}$ such that $B_{b}\neq\emptyset$. Since
${\bm{\alpha}}$ is non-atomic, one can fix a subset
$S^{\prime\prime\prime}_{p,b}\in\mathcal{F}$ of $S_{p}$ such that

$|{\bm{\alpha}}|(S^{\prime\prime\prime}_{p,b})=\ell^{\prime}_{p,b}|{\bm{\alpha}}|(S_{p})$

for each $p\in[K]$, $b\in\mathbb{N}$ such that $B_{b}\neq\emptyset$ by Lemma
12. Let $h:S_{\rm{con}}\times T_{\rm{atm}}\to\\{0,1\\}$ be defined by
aaaaaaaaaaaaaaaaaaaaaa

$h(s,t)=\mathbf{1}_{S^{\prime\prime\prime}_{p,b}}(s)$

for all $(s,t)\in S_{p}\times B_{b}$, $p\in[K]$, $b\in\mathbb{N}$ such that
$B_{b}\neq\emptyset$. Then

$\displaystyle\quad\ \bigg{\|}\int_{S_{p}\times
B_{b}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{(\\{p\\}\times[N])\times\\{b\\}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}\int_{S_{p}\times
B_{b}}\mathbf{1}_{S^{\prime\prime\prime}_{p,b}\times
B_{b}}\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\ell^{\prime}_{p,b}N\big{(}{\odot}^{n}({\bm{a}}_{p},{\bm{\beta}}^{\prime}(\\{b\\}))\big{)}\bigg{\|}_{1}$
$\displaystyle=\bigg{\|}{\odot}^{n}({\bm{\alpha}}(S^{\prime\prime\prime}_{p,b}),{\bm{\beta}}(B_{b}))-|{\bm{\alpha}}|(S^{\prime\prime\prime}_{p,b})\big{(}{\odot}^{n}({\bm{u}}_{p},{\bm{\beta}}(B_{b}))\big{)}\bigg{\|}_{1}$
$\displaystyle=|{\bm{\beta}}|(B_{b})\bigg{\|}{\odot}^{n}\bigg{(}\int_{S^{\prime\prime\prime}_{p,b}}({\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}-{\bm{u}}_{p})\,\mathrm{d}|{\bm{\alpha}}|,{\bm{v}}_{b}\bigg{)}\bigg{\|}_{1}$
$\displaystyle\leq|{\bm{\beta}}|(B_{b})\int_{S^{\prime\prime\prime}_{p,b}}\|{\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}-{\bm{u}}_{p}\|_{1}\,\mathrm{d}|{\bm{\alpha}}|$
$\displaystyle\leq\delta|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(B_{b})$

for all $p\in K$, $b\in\mathbb{N}$ such that $B_{b}\neq\emptyset$, where
${\bm{v}}_{b}\coloneqq{\bm{\beta}}(B_{b})/|{\bm{\beta}}|(B_{b})\in{\mathrm{U}^{n}_{\scriptscriptstyle{\|\|_{1}}}}$.
It follows that aaaaa

$\displaystyle\bigg{\|}\int_{S_{\rm{con}}\times
T_{\rm{atm}}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}_{\rm{con}}\times
T^{\prime}_{\rm{atm}}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\leq\sum_{b=1}^{\infty}\sum_{p\in[K]}\delta|{\bm{\alpha}}|(S_{p})|{\bm{\beta}}|(B_{b})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad=\delta|{\bm{\alpha}}|(S_{\rm{con}})|{\bm{\beta}}|(T_{\rm{atm}})<\varepsilon_{0}.$
(24)

A similar process shows that on can set the restriction of $h$ on
$S_{\rm{atm}}\times T_{\rm{con}}$ to be such that aaaaaaaaaaaa

$\bigg{\|}\int_{S_{\rm{atm}}\times
T_{\rm{con}}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}_{\rm{atm}}\times
T^{\prime}_{\rm{con}}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}<\varepsilon_{0}.$
(25)

Lastly, for each $a$, $b\in\mathbb{N}$ such that $A_{a}$,
$B_{b}\neq\emptyset$, let the restriction of $h$ on $S_{\rm{atm}}\times
T_{\rm{atm}}$ be such that aaaaaaaaaaaaaa

$h(s,t)=h^{\prime}(a,b)$

for all $(s,t)\in A_{a}\times B_{b}$, $a$, $b\in\mathbb{N}$, $A_{a}$,
$B_{b}\neq\emptyset$. Then it is easy to check that

$\bigg{\|}\int_{S_{\rm{atm}}\times
T_{\rm{atm}}}h\,\mathrm{d}({\bm{\alpha}}\times{\bm{\beta}})-\int_{S^{\prime}_{\rm{atm}}\times
T^{\prime}_{\rm{atm}}}h^{\prime}\,\mathrm{d}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\bigg{\|}_{1}=0.$
(26)

The inequality (22) then follows by (23), (24), (25) and (26).

Finally, by (17) and (22),

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\big{)}\leq
3\varepsilon_{0}<\varepsilon$

for our construction of ${\bm{\alpha}}^{\prime}$ and ${\bm{\beta}}^{\prime}$.
The arguments for showing

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}),\mathsf{LS}({\bm{\alpha}}^{\prime})\big{)}<\varepsilon,\quad
d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\beta}}),\mathsf{LS}({\bm{\beta}}^{\prime})\big{)}<\varepsilon$

are again similar and simpler. ∎

######  Proposition 25.

Let ${\bm{\alpha}}$, ${\bm{\beta}}$ be $n$-dimensional finite signed measures
on $(S,\mathcal{F})$, $(T,\mathcal{G})$, respectively. Then for any
$\varepsilon>0$ there exists $\delta>0$ such that

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\big{)}<\varepsilon$

for any $n$-dimensional finite signed measures ${\bm{\alpha}}^{\prime}$,
${\bm{\beta}}^{\prime}$ for which

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}),\mathsf{LS}({\bm{\alpha}}^{\prime})\big{)}<\delta,\quad
d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\beta}}),\mathsf{LS}({\bm{\beta}}^{\prime})\big{)}<\delta.$

###### Proof.

Let $M>1$ be large enough so that $\mathsf{Cube}(M)$ includes all sets $A$,
$B\subseteq\mathbb{R}^{n}$ such that aaaaaaaaaaaaaaaaaaaaaaaa

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}),A\big{)}<1,\quad
d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\beta}}),B\big{)}<1.$

Choose $\delta<\varepsilon/36nM<\varepsilon/3$ for $\delta\in(0,1)$ and let
${\bm{\alpha}}^{\prime}$, ${\bm{\beta}}^{\prime}$ be arbitrary $n$-dimensional
finite signed measures such that

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}),\mathsf{LS}({\bm{\alpha}}^{\prime})\big{)}<\delta,\quad
d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\beta}}),\mathsf{LS}({\bm{\beta}}^{\prime})\big{)}<\delta.$

By Proposition 24 there exist $n$-dimensional finite signed measures
${\bm{\lambda}}$, ${\bm{\omega}}$, ${\bm{\lambda}}^{\prime}$,
${\bm{\omega}}^{\prime}$ on discrete $\sigma$-algebras on countable sets such
that

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\lambda}}\times{\bm{\omega}})\big{)}<\delta,$
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}),\mathsf{LS}({\bm{\lambda}})\big{)}<\delta,\quad
d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\beta}}),\mathsf{LS}({\bm{\omega}})\big{)}<\delta$

and such that aaaaaaaaaaaaaaaaaaaaaaaa

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime}),\mathsf{LS}({\bm{\lambda}}^{\prime}\times{\bm{\omega}}^{\prime})\big{)}<\delta,$
$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}^{\prime}),\mathsf{LS}({\bm{\lambda}}^{\prime})\big{)}<\delta,\quad
d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\beta}}^{\prime}),\mathsf{LS}({\bm{\omega}}^{\prime})\big{)}<\delta.$

In particular,

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\lambda}}),\mathsf{LS}({\bm{\lambda}}^{\prime})\big{)}<3\delta,\quad
d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\omega}}),\mathsf{LS}({\bm{\omega}}^{\prime})\big{)}<3\delta$

so that aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\lambda}}\times{\bm{\omega}}),\mathsf{LS}({\bm{\lambda}}^{\prime}\times{\bm{\omega}}^{\prime})\big{)}\leq
12nM\delta<\varepsilon/3$

by Proposition 22 since $\mathsf{LS}({\bm{\lambda}})$,
$\mathsf{LS}({\bm{\omega}})$, $\mathsf{LS}({\bm{\lambda}}^{\prime})$,
$\mathsf{LS}({\bm{\omega}}^{\prime})\subseteq\mathsf{Cube}(M)$. It then
follows that aaaaaaaaaaaaaaaaaaaaa

$d_{\mathrm{H}}\big{(}\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}}),\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})\big{)}<\varepsilon/3+\delta+\delta<\varepsilon$

and this concludes the proof. ∎

Taking $\varepsilon\to 0$ in Proposition 25, one has
$\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}})=\mathsf{LS}({\bm{\alpha}}^{\prime}\times{\bm{\beta}}^{\prime})$
if $\mathsf{LS}({\bm{\alpha}})=\mathsf{LS}({\bm{\alpha}}^{\prime})$,
$\mathsf{LS}({\bm{\beta}})=\mathsf{LS}({\bm{\beta}}^{\prime})$. This implies
that we can define _the Lorenz product of Lorenz skeletons_ in the same way as
we define the Lorenz product of Lorenz hulls. I.e., the Lorenz product
$K_{1}K_{2}$ of Lorenz skeletons $K_{1}$, $K_{2}$ is the Lorenz skeleton
$\mathsf{LS}({\bm{\alpha}}\times{\bm{\beta}})$ where
$\mathsf{LS}({\bm{\alpha}})=K_{1}$, $\mathsf{LS}({\bm{\beta}})=K_{2}$.
Moreover, Proposition 25 also implies that the Lorenz product of Lorenz
skeletons is continuous with respect to the Hausdorff distance.

##  10 Appendix: An Equivalent Definition

As a generalization of the first statement of Proposition 8, we have the next
proposition:

######  Proposition 26.

Let ${\bm{\alpha}}$ be an $n$-dimensional finite signed measure on
$(S,\mathcal{F})$, let $\mu$ be a $\sigma$-finite positive measure dominating
${\bm{\alpha}}$ on $(S,\mathcal{F})$, let $f:S\to[0,\infty)$ be a measurable
function such that $\int_{S}f\,\mathrm{d}|{\bm{\alpha}}|<\infty$, and let
$A\in\mathcal{F}$. Then
${{\bm{x}}^{*}}{}^{T}{\int_{A}f\,\mathrm{d}{\bm{\alpha}}}\geq 0$,
${{\bm{x}}^{*}}{}^{T}{\int_{A}f\,\mathrm{d}{\bm{\alpha}}}\leq 0$ if
$A\subseteq[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}\geq{0}\hskip
0.43057pt]\\!]$,
$A\subseteq[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}\leq{0}\hskip
0.43057pt]\\!]$, respectively.

###### Proof.

Use the fact that

$f(s)\cdot{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}(s)}\geq 0,\quad
f(s)\cdot{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}(s)}\leq 0$

for
$s\in[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}\geq{0}\hskip
0.43057pt]\\!]$,
$s\in[\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{\mu}}}\leq{0}\hskip
0.43057pt]\\!]$, respectively. The rest of the proof follows by definition of
$\int_{A}f\,\mathrm{d}{\bm{\alpha}}$ and by linearity of the ordinary Lebesgue
integral. ∎

######  Proposition 27.

Let ${\bm{\alpha}}$ be an $n$-dimensional finite signed measure on
$(S,\mathcal{F})$. Then aaaaaaaaaaaaaaaaaaaa

$\mathsf{LH}({\bm{\alpha}})=\Big{\\{}\int_{S}f\,\mathrm{d}{\bm{\alpha}}\,:\,f:S\to[0,1]\text{
is measurable}\Big{\\}}.$

###### Proof.

Let aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

$V=\Big{\\{}\int_{S}f\,\mathrm{d}{\bm{\alpha}}\,:\,f:S\to[0,1]\text{ is
measurable}\Big{\\}}$

for short. It is easy to see that $V$ is convex and that
$\mathsf{LS}({\bm{\alpha}})\subseteq V$, so that
$\mathsf{LH}({\bm{\alpha}})\subseteq V$. For the reverse containment, it
suffices to show that

$\|{\bm{x}}^{*}\|_{\mathsf{LH}({\bm{\alpha}})}\geq\|{\bm{x}}^{*}\|_{V}$

for all ${\bm{x}}^{*}\in\mathsf{Sph}_{\mathbb{R}^{n}}$, so that
$\overline{V}\subseteq\overline{\mathsf{LH}({\bm{\alpha}})}$ by Proposition 3,
from which $V\subseteq\overline{V}\subseteq\mathsf{LH}({\bm{\alpha}})$ since
$\mathsf{LH}(\alpha)$ is closed. In turn, to establish the inequality, by
Proposition 9 it suffices to show that

${{\bm{x}}^{*}}{}^{T}{{\bm{\alpha}}([\\![{{{\bm{x}}^{*}}{}^{T}{{\bm{D}}^{{\bm{\alpha}}}_{|{\bm{\alpha}}|}}}\geq{0}\hskip
0.43057pt]\\!])}\geq{{\bm{x}}^{*}}{}^{T}{\int_{S}f\,\mathrm{d}{\bm{\alpha}}}$

for all ${\bm{x}}^{*}\in\mathsf{Sph}_{\mathbb{R}^{n}}$ and all measurable
$f:S\to[0,1]$. However, this is easily implied by Proposition 26 since $f\geq
0$, $1-f\geq 0$ on $S$. ∎

######  Corollary 2.

Let ${\bm{\alpha}}$ be an $n$-dimensional finite signed measure on
$(S,\mathcal{F})$ where $\mathcal{F}=\sigma(\mathcal{A})$ for
$\mathcal{A}=\\{A_{1},\ldots,A_{m}\\}$ a disjoint partition of $S$. Then

$\mathsf{LH}({\bm{\alpha}})=\Big{\\{}\sum_{i=1}^{m}\lambda_{i}{\bm{\alpha}}(A_{i})\,:\,\lambda_{i}\in[0,1],1\leq
i\leq m\Big{\\}}.$

A _zonoid_ is the range of a non-atomic $n$-dimensional finite signed measure,
and as discussed at the beginning of Section 5, it is also a Lorenz hull. For
completeness, we show that the reverse statement also holds.999A very brief
proof can be found in Bolker [2] (Theorem 1.6). We hereby present one with
details enough to our own satisfaction.

######  Proposition 28.

For any $n$-dimensional finite signed measure ${\bm{\alpha}}$ on
$(S,\mathcal{F})$, there exists $(S^{\prime},\mathcal{F}^{\prime})$ and non-
atomic $n$-dimensional finite signed measure ${\bm{\alpha}}^{\prime}$ on
$(S^{\prime},\mathcal{F}^{\prime})$ such that
$\mathsf{LS}({\bm{\alpha}}^{\prime})=\mathsf{LH}({\bm{\alpha}})$.

###### Proof.

Recalling the discussion following Proposition 7, let
$\\{A_{i}\\}_{i\in\mathbb{N}}$ be a chosen collection of atoms with respect to
the purely atomic part ${\bm{\alpha}}_{|A}$ of ${\bm{\alpha}}$, and let
${\bm{\alpha}}_{|B}$ denote the non-atomic part of ${\bm{\alpha}}$.

Consider $A^{\prime}=(0,\infty)$ and $\mathcal{B}$ the Borel $\sigma$-algebra
on $A^{\prime}$. Let ${\bm{\beta}}$ be the non-atomic finite vector measure on
$(A^{\prime},\mathcal{B})$ such that
${\bm{\beta}}(E)=\mu(E){\bm{\alpha}}(A_{i})$ for all Borel subsets
$E\subseteq(i-1,i]$, where $\mu$ is the Lebesgue measure on Borel subsets of
$\mathbb{R}$, for all $i\in\mathbb{N}$. Then

$\displaystyle\mathsf{LS}({\bm{\beta}})$
$\displaystyle=\\{{\bm{\beta}}(E)\,:\,E\in\mathcal{B}\\}$
$\displaystyle=\Big{\\{}\sum_{i=1}^{\infty}{\bm{\beta}}(E\cap(i-1,i])\,:\,E\in\mathcal{B}\Big{\\}}$
$\displaystyle=\Big{\\{}\sum_{i=1}^{\infty}\lambda_{i}{\bm{\alpha}}(A_{i})\,:\,\lambda_{i}\in[0,1],i\in\mathbb{N}\Big{\\}}.$

On the other hand,

$\displaystyle\mathsf{LH}({\bm{\alpha}}_{|A})$
$\displaystyle=\Big{\\{}\int_{A}f\,\mathrm{d}{\bm{\alpha}}\,:\,f:S\to[0,1]\text{
is measurable}\Big{\\}}$
$\displaystyle=\Big{\\{}\sum_{i=1}^{\infty}\int_{A_{i}}f\,\mathrm{d}{\bm{\alpha}}\,:\,f:S\to[0,1]\text{
is measurable}\Big{\\}}$
$\displaystyle=\Big{\\{}\sum_{i=1}^{\infty}\lambda_{i}{\bm{\alpha}}(A_{i})\,:\,\lambda_{i}\in[0,1],i\in\mathbb{N}\Big{\\}},$

where the first equality follows by Proposition 27, where the second equality
follows since $|{\bm{\alpha}}|(\bigcup_{i=n}^{\infty}A_{i})\to 0$ as
$n\to\infty$, and where the third equality follows since $f$ is
$|{\bm{\alpha}}|$-almost constant on each atom $A_{i}$. Thus
$\mathsf{LS}({\bm{\beta}})=\mathsf{LH}({\bm{\alpha}}_{|A})$.

Let ${\bm{\alpha}}^{\prime}={\bm{\beta}}\oplus{\bm{\alpha}}_{|B}$. Then
${\bm{\alpha}}^{\prime}$ is non-atomic and

$\mathsf{LS}({\bm{\alpha}}^{\prime})=\mathsf{LS}({\bm{\beta}})+\mathsf{LS}({\bm{\alpha}}_{|B})=\mathsf{LH}({\bm{\alpha}}_{|A})+\mathsf{LH}({\bm{\alpha}}_{|B})=\mathsf{LH}({\bm{\alpha}}),$

where the first and last equality follow by Proposition 15. ∎

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