# Choquet operators associated to vector capacities

Sorin G. Gal  Department of Mathematics and Computer Science
University of Oradea
University Street No. 1, Oradea, 410087, Romania<EMAIL_ADDRESS>and
Constantin P. Niculescu  Department of Mathematics, University of Craiova
Craiova 200585, Romania<EMAIL_ADDRESS>
(Date: May 28, 2020)

###### Abstract.

The integral representation of Choquet operators defined on a space $C(X)$ is
established by using the Choquet-Bochner integral of a real-valued function
with respect to a vector capacity.

###### Key words and phrases:

Choquet integral, Choquet-Bochner integral, comonotonic additivity, monotonic
operator, vector capacity

###### 2000 Mathematics Subject Classification:

47H07, 28B15, 28C15, 46B42, 46G10,

##  1\. Introduction

Choquet’s theory of integrability (as described by Denneberg [10] and Wang and
Klir [30]) leads to a new class of nonlinear operators called in [15] _Choquet
operators_ because they are defined by a mix of conditions representative for
Choquet’s integral. Its technical definition is detailed as follows.

Given a Hausdorff topological space $X,$ we will denote by $\mathcal{F}(X)$
the vector lattice of all real-valued functions defined on $X$ endowed with
the pointwise ordering. Two important vector sublattices of it are

$C(X)=\left\\{f\in\mathcal{F}(X):\text{ }f\text{ continuous}\right\\}$

and

$C_{b}(X)=\left\\{f\in\mathcal{F}(X):\text{ }f\text{ continuous and
bounded}\right\\}.$

With respect to the sup norm, $C_{b}(X)$ becomes a Banach lattice. See the
next section for details concerning the ordered Banch spaces.

As is well known, all norms on the $N$-dimensional real vector space
$\mathbb{R}^{N}$ are equivalent. See Bhatia [3], Theorem 13, p. 16\. When
endowed with the sup norm and the coordinate wise ordering, $\mathbb{R}^{N}$
can be identified (algebraically, isometrically and in order) with the space
$C\left(\left\\{1,...,N\right\\}\right)$, where $\left\\{1,...,N\right\\}$
carries the discrete topology.

Suppose that $X$ and $Y$ are two Hausdorff topological spaces and $E$ and $F$
are respectively ordered vector subspaces of $\mathcal{F}(X)$ and
$\mathcal{F}(Y).$ An operator $T:E\rightarrow F$ is said to be a _Choquet
operator_(respectively a _Choquet functional when_ $F=\mathbb{R}$) if it
satisfies the following three conditions:

  1. (Ch1)

(_Sublinearity_) $T$ is subadditive and positively homogeneous, that is,

$T(f+g)\leq T(f)+T(g)\quad\text{and}\quad T(af)=aT(f)$

for all $f,g$ in $E$ and $a\geq 0;$

  2. (Ch2)

(_Comonotonic additivity_) $T(f+g)=T(f)+T(g)$ whenever the functions $f,g\in
E$ are comonotone in the sense that

$(f(s)-f(t))\cdot(g(s)-g(t))\geq 0\text{ for all }s,t\in X;$

  3. (Ch3)

(_Monotonicity_) $f\leq g$ in $E$ implies $T(f)\leq T(g).$

The linear Choquet operators acting on ordered Banach spaces are nothing but
the linear and positive operators acting on these spaces; see Corollary 1.
While they are omnipresent in the various fields of mathematics, the nonlinear
Choquet operators are less visible, their study beginning with the seminal
papers of Schmeidler [28], [29] in the 80’s. An important step ahead was done
by the contributions of Zhou [31], Marinacci and Montrucchio [17] and Cerreia-
Vioglio, Maccheroni, Marinacci and Montrucchio [4], [5], which led to the
study of vector-valued Choquet operators in their own. See [14] and [15].

Interestingly, the condition of comonotonic additivity (the substitute for
additivity) lies at the core of many results concerning the real analysis.
Indeed, its meaning in the context of real numbers, can be easily understood
by identifying each real number $x$ with the affine function
$\alpha_{x}(t)=tx$, $t\in\mathbb{R}.$ As a consequence, two real numbers $x$
and $y$ are comonotone if and only if the functions $\alpha_{x}$ and
$\alpha_{y}$ are comonotone, equivalently, if either both $x$ and $y$ are
nonnegative or both are nonpositive. This yields the simplest example of
Choquet functional from $\mathbb{R}~{}$into itself which is not linear, the
function $x\rightarrow x^{+}.$ At the same time one can indicate a large
family of nonlinear Choquet operators from $C([-1,1])$ into an arbitrary
ordered Banach space $E,$

$T_{\varphi,U}(f)=U\left(\int_{-1}^{1}f^{+}(tx)\varphi(x)\mathrm{d}x\right),$

where $\varphi\in C([-1,1])$ is any nonnegative function such that
$\varphi(0)=0$ and $U:C([-1,1])\rightarrow E$ is any monotonic linear
operator.

Based on previous work done by Zhou [31], Cerreia-Vioglio, Maccheroni,
Marinacci and Montrucchio [4], proved that a larger class of functionals
defined on a space $C(X)$ (where $X$ is a Hausdorff compact space) admit a
Choquet analogue of the Riesz representation theorem. The aim of our paper is
to further extend their results to the case of operators by developing a
Choquet-Bochner theory of integration relative to monotone set functions
taking values in ordered Banach spaces.

Section 2 is devoted to a quick review of some basic facts from the theory of
ordered Banach spaces. While the particular case of Banach lattices is nicely
covered by a series of textbooks such as those by Meyer-Nieberg [18] and
Schaefer [26], the general theory of ordered Banach spaces is still waiting to
become the subject of an authoritative book.

In Section 3 we develop the theory of Choquet-Bochner integral associated to a
vector capacity (that is, to a monotone set function $\boldsymbol{\mu}$ taking
values in an ordered Banach space such that $\boldsymbol{\mu}(\emptyset)=0).$
As is shown in Theorem 1, this integral has all nice features of the Choquet
integral: monotonicity, positive homogeneity and comonotonic additivity. The
transfer of properties from vector capacities to their integrals also works in
a number of important cases such as the upper/lower continuity and
submodularity. See Theorem 1. In the case of submodular vector capacities with
values in a Banach lattice, the integral analogue of the modulus inequality
also holds. See Theorem 2.

Section 4 deals with the integral representation of the Choquet operators
defined on spaces $C(X)$ ($X$ being compact and Hausdorff) and taking values
in a Banach lattice with order continuous norm. The main result, Theorem 3,
shows that each such operator is the Choquet-Bochner integral associated to a
suitable upper continuous vector capacity. In Section 5, this representation
is generalized to the framework of comonotonic additive operators with bounded
variation. See Theorem 4. The basic ingredient is Lemma 12, which shows that
every comonotonic additive operator with bounded variation can be written as
the difference of two positively homogeneous, translation invariant and
monotone operators.

The paper ends with a short list of open problems.

##  2\. Preliminaries on ordered Banach spaces

An _ordered vector space_ is a real vector space $E$ endowed with an order
relation $\leq$ such that the following two conditions are verified:

$\displaystyle x$ $\displaystyle\leq y\text{ implies }x+z\leq y+z\text{ for
all }x,y,z\in E;\text{ and}$ $\displaystyle x$ $\displaystyle\leq y\text{
implies }\lambda x\leq\lambda y\text{ for }x,y\in E\text{ and
}\lambda\in\mathbb{R}_{+}=[0,\infty).$

In this case the set $E_{+}=\left\\{x\in E:x\geq 0\right\\}$ is a convex cone,
called the _positive cone_. A real Banach space endowed with an order relation
that makes it an ordered vector space is called an _ordered Banach space_ if
the norm is monotone on the positive cone, that is,

$0\leq x\leq y\text{ implies }\left\|x\right\|\leq\left\|y\right\|.$

Note that in this paper we will consider only ordered Banch spaces $E$ whose
positive cones are closed (in the norm topology), _proper_ $(-E_{+}\cap
E_{+}=\left\\{0\right\\})$ and _generating_ $(E=E_{+}-E_{+})$.

A convenient way to emphasize the properties of ordered Banach spaces is that
described by Davies in [8]. According to Davies, a real Banach space $E$
endowed with a closed and generating cone $E_{+}$ such that

$\left\|x\right\|=\inf\left\\{\left\|y\right\|:y\in E,\text{ }-y\leq x\leq
y\right\\}\text{\quad for all }x\in E,$

is called a _regularly ordered Banach space_.__ Examples are the Banach
lattices and some other spaces such as $\operatorname*{Sym}(n,\mathbb{R)}$,
the ordered Banach space of all $n\times n$-dimensional symmetric matrices
with real coefficients. The norm of a symmetric matrix $A$ is defined by the
formula

$\left\|A\right\|=\sup_{\left\|x\right\|\leq 1}\left|\langle
Ax,x\rangle\right|,$

and the positive cone $\operatorname*{Sym}^{+}(n,\mathbb{R})$ of
$\operatorname*{Sym}(n,\mathbb{R})$ consists of all symmetric matrices $A$
such that $\langle Ax,x\rangle\geq 0$ for all $x.$

######  Lemma 1.

Every ordered Banach space can be renormed by an equivalent norm to become a
regularly ordered Banach space.

For details, see Namioka [19]. Some other useful properties of ordered Banach
spaces are listed below.

######  Lemma 2.

Suppose that $E$ is a regularly ordered Banach space. Then:

$(a)$ There exists a constant $C>0$ such that every element $x\in E$ admits a
decomposition of the form $x=u-v$ where $u,v\in E_{+}$ and
$\left\|u\right\|,\left\|v\right\|\leq C\left\|x\right\|.$

$(b)$ The dual space of $E,$ $E^{\ast},$ when endowed with the dual cone

$E_{+}^{\ast}=\left\\{x^{\ast}\in E^{\ast}:x^{\ast}(x)\geq 0\text{ for all
}x\in E_{+}\right\\}$

is a regularly ordered Banach space.

$(c)\ x\leq y$ in $E$ is equivalent to $x^{\ast}(x)\leq x^{\ast}(y)$ for all
$x^{\ast}\in E_{+}^{\ast}.$

$(d)$ $\left\|x\right\|=\sup\left\\{x^{\ast}(x):x^{\ast}\in
E_{+}^{\ast},\text{ }\left\|x^{\ast}\right\|\leq 1\right\\}$ for all $x\in
E_{+}.$

$(e)$ If $(x_{n})_{n}$ is a decreasing sequence of positive elements of $E$
which converges weakly to $0,$ then $\left\|x_{n}\right\|\rightarrow 0.$

The assertion $(e)$ is a generalization of Dini’s lemma in real analysis; see
[7], p. 173.

###### Proof.

The assertion $(a)$ follows immediately from Lemma 1. For $(b)$, see Davies
[8], Lemma 2.4. The assertion $(c)$ is an easy consequence of the Hahn-Banach
separation theorem; see [24], Theorem 2.5.3, p. 100.

The assertion $(d)$ is also a consequence of the Hahn-Banach separation
theorem; see [27], Theorem 4.3, p. 223. ∎

######  Corollary 1.

Every ordered Banach space $E$ can be embedded into a space $C(X)$, where $X$
is a suitable compact space.

###### Proof.

According to the Alaoglu theorem, the set $X=\left\\{x^{\ast}\in
E_{+}^{\ast}:\left\|x^{\ast}\right\|\leq 1\right\\}$ is compact relative to
the $w^{\ast}$ topology. Taking into account the assertions $(c)$ and $(d)$ of
Lemma 2 one can easily conclude that $E$ embeds into $C(X)$ (algebraically,
isometrically and in order) via the map

$\Phi:E\rightarrow
C(X),\text{\quad}\left(\Phi(x)\right)(x^{\ast})=x^{\ast}(x).$

∎

The following important result is due to V. Klee [16]. A simple proof of it is
available in [23].

######  Lemma 3.

Every positive linear operator $T:E\rightarrow F$ acting on ordered Banach
spaces is continuous.

Sometimes, spaces with a richer structure are necessary.

A vector lattice is any ordered vector space $E$ such that $\sup\\{x,y\\}$ and
$\inf\\{x,y\\}$ exist for all $x,y\in E.$ In this case for each $x\in E$ we
can define $x^{+}=\sup\left\\{x,0\right\\}$ (the positive part of $x$),
$x^{-}=\sup\left\\{-x,0\right\\}$ (the negative part of $x$) and
$\left|x\right|=\sup\left\\{-x,x\right\\}$ (the modulus of $x$). We have
$x=x^{+}-x^{-}$ and $\left|x\right|=x^{+}+x^{-}.$ A vector lattice endowed
with a norm $\left\|\cdot\right\|$ such that

$\left|x\right|\leq\left|y\right|\text{ implies }x\leq\left\|y\right\|$

is called a normed vector lattice; it is called a Banach lattice when in
addition it is metric complete.

Examples of Banach lattice are numerous: the discrete spaces $\mathbb{R}^{n},$
$c_{0},$ $c$ and $\ell^{p}$ for $1\leq p\leq\infty$ (endowed with the
coordinate-wise order), and the function spaces $C(K)$ (for $K$ a compact
Hausdorff space) and $L^{p}(\mu)$ with $1\leq p\leq\infty$ (endowed with
pointwise order). Of a special interest are the Banach lattices with _order
continuous norm_ , that is, the Banach lattices for which every monotone and
order bounded sequence is convergent in the norm topology. So are
$\mathbb{R}^{n},$ $c_{0}$ and $L^{p}(\mu)$ for $1\leq p<\infty$.

######  Lemma 4.

Every monotone and order bounded sequence of elements in a Banach lattice $E$
with order continuous norm admits a supremum and an infimum and all closed
order intervals in $E$ are weakly compact.

For details, see Meyer-Nieberg [18], Theorem 2.4.2, p. 86.

##  3\. The Choquet-Bochner Integral

This section is devoted to the extension of Choquet’s theory of integrability
to the framework with respect to a monotone set function with values in the
positive cone of a regularly ordered Banach space $E$. This draws a parallel
to the real-valued case already treated in full details by Denneberg [10] and
Wang and Klir [30].

Given a nonempty set $X,$ by a _lattice_ of subsets of $X$ we mean any
collection $\Sigma$ of subsets that contains $\emptyset$ and $X$ and is closed
under finite intersections and unions. A lattice $\Sigma$ is an _algebra_ if
in addition it is closed under complementation. An algebra which is closed
under countable unions and intersections is called a $\sigma$-algebra.

Of a special interest is the case where $X$ is a compact Hausdorff space and
$\Sigma$ is either the lattice $\Sigma_{up}^{+}(X),$ of all _upper contour_
_closed sets_ $S=\left\\{x\in X:f(x)\geq t\right\\},$ or the lattice
$\Sigma_{up}^{-}(X)$ of all _upper contour open sets_ $S=\left\\{x\in
X:f(x)>t\right\\},$ associated to pairs $f\in C(X)$ and $t\in\mathbb{R}.$

When $X$ is a compact metrizable space, $\Sigma_{up}^{+}(X)$ coincides with
the lattice of all closed subsets of $X$ (and $\Sigma_{up}^{-}(X)$ coincides
with the lattice of all open subsets of $X)$.

In what follows $\Sigma$ denotes a lattice of subsets of an abstract set $X$
and $E$ is a regularly ordered Banach space.

######  Definition 1.

A set function $\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$ is called a vector
capacity if it verifies the following two conditions:

$(C1)$ $\boldsymbol{\mu}(\emptyset)=0;$ and

$(C2)~{}\boldsymbol{\mu}(A)\leq\boldsymbol{\mu}(B)$ for all $A,B\in\Sigma$
with $A\subset B$.

Notice that any vector capacity $\boldsymbol{\mu}$ is positive and takes
values in the order interval $[0,$ $\boldsymbol{\mu}(X)].$

An important class of vector capacities is that of _additive_ (respectively
$\sigma$-_additive) vector measures with positive values_ , that is, of
capacities $\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$ with the property

$\boldsymbol{\mu}\left({\displaystyle\bigcup\nolimits_{n}}A_{n}\right)={\displaystyle\sum\nolimits_{n=1}^{\infty}}\boldsymbol{\mu}(A_{n}),$

for every finite (respectively infinite) sequence $A_{1},A_{2},A_{3},...$ of
disjoint sets belonging to $\Sigma$ such that $\cup_{n}A_{n}\in\Sigma.$

Some other classes of capacities exhibiting various extensions of the property
of additivity are listed below.

A vector capacity $\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$ is called
_submodular_ if

(3.1) $\boldsymbol{\mu}(A\cup B)+\boldsymbol{\mu}(A\cap
B)\leq\boldsymbol{\mu}(A)+\boldsymbol{\mu}(B)\text{\quad for all
}A,B\in\Sigma$

and it is called _supermodular_ when the inequality (3.1) works in the
reversed way. Every additive measure taking values in $E_{+}$ is both
submodular and supermodular.

A vector capacity $\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$ is called _lower
continuous_(or continuous by ascending sequences)__ if

$\lim_{n\rightarrow\infty}\boldsymbol{\mu}(A_{n})=\boldsymbol{\mu}({\displaystyle\bigcup\nolimits_{n=1}^{\infty}}A_{n})$

for every nondecreasing sequence $(A_{n})_{n}$ of sets in $\Sigma$ such that
$\cup_{n=1}^{\infty}A_{n}\in\Sigma;$ $\boldsymbol{\mu}$ is called _upper
continuous_(or continuous by descending sequences) if

$\lim_{n\rightarrow\infty}\boldsymbol{\mu}(A_{n})=\boldsymbol{\mu}\left(\cap_{n=1}^{\infty}A_{n}\right)$

for every nonincreasing sequence $(A_{n})_{n}$ of sets in $\Sigma$ such that
$\cap_{n=1}^{\infty}A_{n}\in\Sigma.$ If $\boldsymbol{\mu}$ is an additive
capacity defined on a $\sigma$-algebra, then its upper/lower continuity is
equivalent to the property of $\sigma$-additivity.

When $\Sigma$ is an algebra of subsets of $X,$ then to each vector capacity
$\boldsymbol{\mu}$ defined on $\Sigma$, one can attach a new vector capacity
$\overline{\boldsymbol{\mu}}$, the _dual_ of $\boldsymbol{\mu},$ which is
defined by the formula

$\overline{\boldsymbol{\mu}}(A)=\boldsymbol{\mu}(X)-\boldsymbol{\mu}(X\setminus
A).$

Notice that
$\overline{\left(\overline{\boldsymbol{\mu}}\right)}=\boldsymbol{\mu}.$

The dual of a submodular (supermodular) capacity is a supermodular
(submodular) capacity. Also, dual of a lower continuous (upper continuous)
capacity is an upper continuous (lower continuous) capacity.

######  Example 1.

There are several standard procedures to attach to a $\sigma$-additive vector
measure $\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$ certain not necessarily
additive capacities. So is the case of _distorted measures,_
$\nu(A)=T(\boldsymbol{\mu}(A)),$ obtained from $\boldsymbol{\mu}$ by applying
to it a continuous nondecreasing distortion
$T:[0,\boldsymbol{\mu}(X)]\rightarrow[0,\boldsymbol{\mu}(X)].$ The vector
capacities $\nu$ so obtained are both upper and lower continuous.

For example, this is the case when $E=\operatorname*{Sym}(n,\mathbb{R})$,
$\boldsymbol{\mu}(X)=\mathrm{I}$ $($the identity of $\mathbb{R}^{n}),$ and
$T:[0,\mathrm{I}]\rightarrow[0,\mathrm{I}]$ is the distortion defined by the
formula $T(A)=A^{2}.$

Taking into account the assertions $(c)$ and $(d)$ of Lemma 2, some aspects
(but not all) of the theory of vector capacities are straightforward
consequences of the theory of $\mathbb{R}_{+}$-valued capacities.

######  Lemma 5.

A set function $\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$ is a submodular _(_
supermodular, lower continuous, upper continuous _)_ vector capacity if and
only if $x^{\ast}\circ\boldsymbol{\mu}$ is a submodular _(_ supermodular,
lower continuous, upper continuous _)_ $\mathbb{R}_{+}$-valued capacity
whenever $x^{\ast}\in E_{+}^{\ast}.$

When $E=\mathbb{R}^{n},$ this assertion can be formulated via the components
$\mu_{k}=\operatorname*{pr}_{k}\circ\boldsymbol{\mu}$ of $\mathbf{\mu}.$

In what follows the term of (upper) _measurable_ _function_ refers to any
function $f:X\rightarrow\mathbb{R}$ whose all upper contour sets $\left\\{x\in
X:f(x)\geq t\right\\}$ belong to $\Sigma$. When $\Sigma$ is a
$\sigma$-algebra, this notion of measurability is equivalent to the Borel
measurability.

We will denote by $B(\Sigma)$ the set of all bounded measurable functions
$f:X\rightarrow\mathbb{R}.$ In general $B(\Sigma)$ is not a vector space
(unless the case when $\Sigma$ is a $\sigma$-algebra). However, even when
$\Sigma$ is only an algebra, the set $B(\Sigma)$ plays some nice properties of
stability: if $f,g\in B(\Sigma)$ and $\alpha,\beta\in\mathbb{R},$ then

$\inf\\{f,g\\},\text{ }\sup\left\\{f,g\right\\}\text{ and }\alpha+\beta f$

also belong to $B(\Sigma).$ See [17], Proposition 15.

Given a capacity $\mu:\Sigma\rightarrow\mathbb{R}_{+},$ the _Choquet integral_
of a measurable function $f:X\rightarrow\mathbb{R}$ on a set $A\in\Sigma$ is
defined as the sum of two Riemann improper integrals,

(3.2) $\displaystyle(\operatorname*{C})\int_{A}f\mathrm{d}\mu$
$\displaystyle=\int_{0}^{+\infty}\mu\left(\\{x\in A:f(x)\geq
t\\}\right)\mathrm{d}t$
$\displaystyle+\int_{-\infty}^{0}\left[\mu\left(\\{x\in A:f(x)\geq
t\\}\right)-\mu(A)\right]\mathrm{d}t.$

Accordingly, $f$ is said to be _Choquet integrable_ on $A$ if both integrals
above are finite. See the seminal paper of Choquet [6].

If $f\geq 0$, then the last integral in the formula (3.2) is 0. When $\Sigma$
is a $\sigma$-algebra, the inequality sign $\geq$ in the above two integrands
can be replaced by $>;$ see [30], Theorem 11.1,__ p. 226.

Every bounded measurable function is Choquet integrable. The Choquet integral
coincides with the Lebesgue integral when the underlying set function $\mu$ is
a $\sigma$-additive measure defined on a $\sigma$-algebra.

The theory of Choquet integral is available from numerous sources including
the books of Denneberg [10], and Wang and Klir [30].

The concept of integrability of a measurable function with respect to a vector
capacity $\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$ can be introduced by a
fusion between the Choquet integral and the Bochner theory of integration of
vector-valued functions.

Recall that a function $\psi:\mathbb{R}\rightarrow E$ is _Bochner integrable_
with respect to the Lebesgue measure on $\mathbb{R}$ if there exists a
sequence of step functions $\psi_{n}:\mathbb{R}\rightarrow E$ such that

$\lim_{n\rightarrow\infty}\psi_{m}(t)=\psi(t)\text{ almost everywhere and
}\int_{\mathbb{R}}\left\|\psi-\psi_{n}\right\|\mathrm{d}t\rightarrow 0.$

In this case, the (Bochner) integral of $\psi$ is defined by

$\int_{\mathbb{R}}\psi\mathrm{d}t=\lim_{n\rightarrow\infty}\int_{\mathbb{R}}\psi_{n}\mathrm{d}t.$

Notice that if $T:E\rightarrow F$ is a bounded linear operator, then

(3.3)
$T\left(\int_{\mathbb{R}}\psi\mathrm{d}t\right)=\int_{\mathbb{R}}T\circ\psi\mathrm{d}t.$

Details about Bochner integral are available in the books of Diestel and Uhl
[11] and Dinculeanu [12].

######  Definition 2.

A measurable function $f:X\rightarrow\mathbb{R}$ is called Choquet-Bochner
integrable with respect to the vector capacity
$\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$ on the set $A\in\Sigma$ if for
every $A\in\Sigma$ the functions $t\rightarrow\boldsymbol{\mu}\left(\\{x\in
A:f(x)\geq t\\}\right)$ and $t\rightarrow\boldsymbol{\mu}\left(\\{x\in
A:f(x)\geq t\\}\right)-\boldsymbol{\mu}(A)$ are Bochner integrable
respectively on $[0,\infty)$ and $(-\infty,0].$ Under these circumstances, the
Choquet-Bochner integral over $A$ is defined by the formula

$\displaystyle(\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}$
$\displaystyle=\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x\in A:f(x)\geq
t\\}\right)\mathrm{d}t$ $\displaystyle+$
$\displaystyle\int_{-\infty}^{0}\left[\boldsymbol{\mu}\left(\\{x\in A:f(x)\geq
t\\}\right)-\mu(A)\right]\mathrm{d}t.$

According to the formula (3.3), if $f$ is Choquet-Bochner integrable, then

(3.4)
$x^{\ast}\left((\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}\right)=(\operatorname*{C})\int_{A}f\mathrm{d}(x^{\ast}\circ\mu),$

for every positive linear functional $x^{\ast}\in E^{\ast}.$

A large class of Choquet-Bochner integrable is indicated below.

######  Lemma 6.

If $f:X\rightarrow\mathbb{R}$ is a bounded measurable function, then it is
Choquet-Bochner integrable on every set $A\in\Sigma$.

###### Proof.

Suppose that $f$ takes values in the interval $[0,M].$ Then the function
$\varphi(t)=\boldsymbol{\mu}\left(\\{x\in A:f(x)\geq t\\}\right)$ is positive
and nonincreasing on the interval $[0,M]$ and null outside this interval. As a
consequence, the function $\varphi$ is the uniform limit of the sequence of
step functions defined as follows:

$\displaystyle\varphi_{n}(t)$ $\displaystyle=0\text{ if }t\notin[0,M]$
$\displaystyle\varphi_{n}(t)$ $\displaystyle=\varphi(t)\text{ if }t=M$

and

$\varphi_{n}(t)=\varphi(\frac{k}{n}M)$

if $t\in[\frac{k}{n}M,\frac{k+1}{n}M)$ and $k=0,...,n-1.$ A simple computation
shows that

$\displaystyle\int_{0}^{+\infty}\left\|\varphi(t)-\varphi_{n}(t)\right\|\mathrm{d}t$
$\displaystyle=\int_{0}^{M}\left\|\varphi(t)-\varphi_{n}(t)\right\|\mathrm{d}t$
$\displaystyle\leq{\displaystyle\sum\nolimits_{l=0}^{n-1}}\int_{t_{k}}^{t_{k+1}}\left\|\varphi(t)-\varphi_{n}(t)\right\|\mathrm{d}t$
$\displaystyle\leq\frac{2M}{n}\boldsymbol{\mu}(X)\rightarrow 0$

as $n\rightarrow\infty,$ which means the Bochner integrability of the function
$\varphi.$ See [11], p. 44.

The other cases, when $f$ takes values in the interval $[M,0]$ or in an
interval $[m,M]$ with $m<0<M,$ can be treated in a similar way. ∎

The next lemma collects a number of simple (but important) properties of the
Choquet-Bochner integral.

######  Lemma 7.

Suppose that $E$ is an ordered Banach space,
$\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$ is a vector capacity and
$A\in\Sigma$.

$(a)$ If $f$ and $g$ are Choquet-Bochner integrable functions, then

$\displaystyle f\geq 0\text{ implies
}(\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}\geq 0\text{
\quad\emph{(}positivity\emph{)}}$ $\displaystyle f\leq g\text{ implies
}\left(\operatorname*{CB}\right)\int_{A}f\mathrm{d}\boldsymbol{\mu}\leq\left(\operatorname*{CB}\right)\int_{A}g\mathrm{d}\boldsymbol{\mu}\text{
\quad\emph{(}monotonicity\emph{)}}$
$\displaystyle\left(\operatorname*{CB}\right)\int_{A}af\mathrm{d}\boldsymbol{\mu}=a\cdot\left(\operatorname*{CB}\right)\int_{A}f\mathrm{d}\boldsymbol{\mu}\text{\quad
for all }a\geq 0\text{ \quad\emph{(}positive\emph{ }homogeneity\emph{)}}$
$\displaystyle\left(\operatorname*{CB}\right)\int_{A}1\cdot\mathrm{d}\boldsymbol{\mu}=\boldsymbol{\mu}(A)\text{\quad\emph{(}calibration\emph{).}}$

$(b)$ In general, the Choquet-Bochner integral is not additive but if $f$ and
$g$ are Choquet-Bochner integrable functions and also comonotonic in the sense
of Dellacherie _[9]) (_that is,
$(f(\omega)-f(\omega^{\prime}))\cdot(g(\omega)-g(\omega^{\prime}))\geq 0$, for
all $\omega,\omega^{\prime}\in X$_),_ then

$\left(\operatorname*{CB}\right)\int_{A}(f+g)\mathrm{d}\boldsymbol{\mu}=\left(\operatorname*{CB}\right)\int_{A}f\mathrm{d}\boldsymbol{\mu}+\left(\operatorname*{CB}\right)\int_{A}g\mathrm{d}\boldsymbol{\mu}.$

In particular, the Choquet-Bochner integral is translation invariant,

$\left(\operatorname*{CB}\right)\int_{A}(f+c)\mathrm{d}\boldsymbol{\mu}=\left(\operatorname*{CB}\right)\int_{A}f\mathrm{d}\boldsymbol{\mu}+c\boldsymbol{\mu}(A),$

for all Choquet-Bochner integrable functions $f$, all sets $A\in\Sigma$ and
all numbers $c\in\mathbb{R}$.

###### Proof.

According to Lemma 2 $(c)$, and formula (3.3), applied in the case of an
arbitrary functional $x^{\ast}\in E_{+}^{\ast},$ the proof of both assertions
$(a)$ and $(b)$ reduce to the case of capacities with values in
$\mathbb{R}_{+},$ already covered by Proposition 5.1 in [10], pp. 64-65. ∎

######  Corollary 2.

The equality

$\left(\operatorname*{CB}\right)\int_{A}(\alpha
f+c)\mathrm{d}\boldsymbol{\mu}=\alpha\left(\left(\operatorname*{CB}\right)\int_{A}f\mathrm{d}\boldsymbol{\mu}\right)+c\cdot\boldsymbol{\mu}(A),$

holds for all Choquet-Bochner integrable functions $f$, all sets $A\in\Sigma$
and all numbers $\alpha\in\mathbb{R}_{+}$ and $c\in\mathbb{R}$.

The next result describes how the special properties of a vector capacity
transfer to the Choquet-Bochner integral.

######  Theorem 1.

$(a)$ If $\boldsymbol{\mu}$ is an upper continuous capacity, then the Choquet-
Bochner integral is an upper continuous operator, that is,

$\lim_{n\rightarrow\infty}\left\|\left(\operatorname*{CB}\right)\int_{A}f_{n}\mathrm{d}\boldsymbol{\mu}-\left(\operatorname*{CB}\right)\int_{A}f\mathrm{d}\boldsymbol{\mu}\right\|=0,$

whenever $(f_{n})_{n}$ is a nonincreasing sequence of Choquet-Bochner
integrable functions that converges pointwise to the Choquet-Bochner
integrable function $f$ and $A\in\Sigma$.

$(b)$ If $\boldsymbol{\mu}$ is a lower continuous capacity, then the Choquet-
Bochner integral is lower continuous in the sense that

$\lim_{n\rightarrow\infty}\left\|\left(\operatorname*{CB}\right)\int_{A}f_{n}\mathrm{d}\boldsymbol{\mu}-\left(\operatorname*{CB}\right)\int_{A}f\mathrm{d}\boldsymbol{\mu}\right\|=0$

whenever $(f_{n})_{n}$ is a nondecreasing sequence of Choquet-Bochner
integrable functions that converges pointwise to the Choquet-Bochner
integrable function $f$ and $A\in\Sigma$.

$(c)$ If $\Sigma$ is an algebra and $\boldsymbol{\mu}:\Sigma\rightarrow E_{+}$
is a submodular capacity, then the Choquet-Bochner integral is a submodular
operator in the sense that

$\left(\operatorname*{CB}\right)\int_{A}\sup\left\\{f,g\right\\}\mathrm{d}\boldsymbol{\mu}+\left(\operatorname*{CB}\right)\int_{A}\inf\\{f,g\\}\mathrm{d}\boldsymbol{\mu}\leq\left(\operatorname*{CB}\right)\int_{A}f\mathrm{d}\mu+(\operatorname*{CB})\int_{A}g\mathrm{d}\boldsymbol{\mu}$

whenever $f$ and $g$ are Choquet-Bochner integrable and $A\in\Sigma$ .

###### Proof.

$(a)$ Since $\mathbf{\mu}$ is an upper continuous capacity and $(f_{n})_{n}$
is a nonincreasing sequence of measurable functions that converges pointwise
to the measurable function $f$ it follows that

$\boldsymbol{\mu}\left(\\{x\in A:f_{n}(x)\geq
t\\}\right)\searrow\boldsymbol{\mu}\left(\\{x\in A:f(x)\geq t\\}\right)$

in the norm topology. Taking into account the property of monotonicity of the
Choquet-Bochner integral (already noticed in Lemma 7 $(a))$ we have

$(\operatorname*{CB})\int_{A}f_{n}\mathrm{d}\boldsymbol{\mu}\geq(\operatorname*{CB})\int_{A}f_{n}\mathrm{d}\boldsymbol{\mu}\geq\mathbf{\cdots}\geq(\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}\mathbf{,}$

so by Bepo Levi’s monotone convergence theorem from the theory of Lebesgue
integral (see [12], Theorem 2, p. 133) it follows that

$x^{\ast}\left((\operatorname*{CB})\int_{A}f_{n}\mathrm{d}\boldsymbol{\mu}\right)=(\operatorname*{C})\int_{A}f_{n}\mathrm{d}\mu^{\ast}\rightarrow(\operatorname*{C})\int_{A}f\mathrm{d}\mu^{\ast}=x^{\ast}\left((\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}\right)$

for all $x^{\ast}\in E_{+}^{\ast}$. The conclusion of the assertion $(c)$ is
now a direct consequence of the generalized Dini’s lemma (see Lemma 2 $(e)$).

$(b)$ The argument is similar to that used to prove the assertion $(a)$.

$(c)$ Since $\Sigma$ is an algebra, both functions $\inf\\{f,g\\}$ and
$\sup\left\\{f,g\right\\}$ are measurable. The fact that $\mathbf{\mu}$ is
submodular implies

$\boldsymbol{\mu}\left(\left\\{x:\sup\\{f,g\\}(x)\geq
t\right\\}\right)+\boldsymbol{\mu}\left(\left\\{x:\inf\\{f,g\\}(x)\geq
t\right\\}\right)\\\ =\boldsymbol{\mu}\left(\left\\{x:f(x)\geq
t\right\\}\cup\left\\{x:g(x)\geq
t\right\\}\right)+\boldsymbol{\mu}\left(\left\\{x:f(x)\geq
t\right\\}\cap\left\\{x:g(x)\geq t\right\\}\right)\\\
\leq\boldsymbol{\mu}\left(\left\\{x:f(x)\geq
t\right\\}\right)+\boldsymbol{\mu}\left(\left\\{x:g(x)\geq t\right\\}\right),$

and the same works when $\boldsymbol{\mu}$ is replaced by
$\mu^{\ast}=x^{\ast}\circ\boldsymbol{\mu},$ where $x^{\ast}\in E_{+}^{\ast}$
is arbitrarily fixed. Integrating side by side the last inequality it follows
that

$\left(\operatorname*{C}\right)\int_{A}\sup\left\\{f,g\right\\}\mathrm{d}\mu^{\ast}+\left(\operatorname*{C}\right)\int_{A}\inf\\{f,g\\}\mathrm{d}\mu^{\ast}\leq\left(\operatorname*{C}\right)\int_{A}f\mathrm{d}\mu^{\ast}+(\operatorname*{C})\int_{A}g\mathrm{d}\mu^{\ast},$

which yields (via Lemma 2 $(c))$ the submodularity of the Choquet-Bochner
integral. ∎

The property of subadditivity of the Choquet-Bochner integral makes the
objective of the following result:

######  Theorem 2.

_(The Subadditivity Theorem)_ If $\boldsymbol{\mu}$ is a submodular capacity,
then the associated Choquet-Bochner integral is subadditive, that is,

$\left(\operatorname*{CB}\right)\int_{A}(f+g)\mathrm{d}\boldsymbol{\mu}\leq\left(\operatorname*{CB}\right)\int_{A}f\mathrm{d}\boldsymbol{\mu}+\left(\operatorname*{CB}\right)\int_{A}g\mathrm{d}\boldsymbol{\mu}$

whenever $f$, $g$ and $f+g$ are Choquet-Bochner integrable functions and
$A\in\Sigma$.

In addition, when $E$ is a Banach lattice and $f$, $g,$ $f-g$ and $g-f$ are
Choquet-Bochner integrable functions, then the following integral analog of
the modulus inequality holds true:

$\left|(\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}-(\operatorname*{CB})\int_{A}g\mathrm{d}\boldsymbol{\mu}\right|\leq(\operatorname*{CB})\int_{A}|f-g|\mathrm{d}\mu$

for all $A\in\Sigma$. In particular,

$\left|(\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}\right|\leq(\operatorname*{CB}\int_{A}|f|\mathrm{d}\boldsymbol{\mu}$

whenever $f$ and $-f$ are Choquet-Bochner integrable functions and
$A\in\Sigma.$

###### Proof.

According to Lemma 5, if $\boldsymbol{\mu}$ is submodular, then every real-
valued capacity $\mu^{\ast}=x^{\ast}\circ\boldsymbol{\mu}$ also is submodular,
whenever $x^{\ast}\in E_{+}^{\ast}$. Then

$\left(\operatorname*{C}\right)\int_{A}f\mathrm{d}\mu^{\ast}+\left(\operatorname*{C}\right)\int_{A}g\mathrm{d}\mu^{\ast}\leq\left(\operatorname*{C}\right)\int_{A}(f+g)\mathrm{d}\mu^{\ast},$

as a consequence of Theorem __ 6.3, p. 75, in [10]. Therefore the first
inequality in the statement of Theorem 2 is now a direct consequence of Lemma
2 $(c)$ and the formula (3.4).

For the second inequality, notice that the subadditivity property implies

$\displaystyle(\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}$
$\displaystyle=(\operatorname*{CB})\int_{A}\left(f-g+g\right)\mathrm{d}\boldsymbol{\mu}$
$\displaystyle\leq\left(\operatorname*{CB}\right)\int_{A}\left(f-g\right)\mathrm{d}\boldsymbol{\mu}+\left(\operatorname*{CB}\right)\int_{A}g\mathrm{d}\boldsymbol{\mu}\mathbf{\boldsymbol{,}}$

and taking into account that $f-g\leq\left|f-g\right|$ we infer that

$(\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}-\left(\operatorname*{CB}\right)\int_{A}g\mathrm{d}\boldsymbol{\mu}\leq\left(\operatorname*{CB}\right)\int_{A}\left|f-g\right|\mathrm{d}\boldsymbol{\mu}.$

Interchanging $f$ and $g$ we also obtain

$\pm((\operatorname*{CB})\int_{A}f\mathrm{d}\boldsymbol{\mu}-\left(\operatorname*{CB}\right)\int_{A}g\mathrm{d}\boldsymbol{\mu}\mathbf{\boldsymbol{)}}\leq\left(\operatorname*{CB}\right)\int_{A}\left|f-g\right|\mathrm{d}\boldsymbol{\mu}$

and the proof is done. ∎

##  4\. The integral representation of Choquet operators defined on a space
$C(X)$

The special case when $X$ is a compact Hausdorff space and
$\Sigma=\mathcal{B}(X),$ the $\sigma$-algebra of all Borel subsets of $X,$
allows us to shift the entire discussion concerning the Choquet-Bochner
integral from the vector space $B(\Sigma)$ (of all real-valued Borel
measurable functions$)$ to the Banach lattice $C(X)$ (of all real-valued
continuous functions defined on $X$. Indeed, in this case $C(X)$ is a subspace
of $B(\Sigma)$ and all nice properties stated in Lemma 7 and in Theorem 2
remain true when restricting the integral to the space $C(X).$ As a
consequence the integral operator

(4.1) $\operatorname*{CB}\nolimits_{\boldsymbol{\mu}}:C(X)\rightarrow
E,\text{\quad}\operatorname*{CB}\nolimits_{\boldsymbol{\mu}}(f)=(\operatorname*{CB})\int_{X}f\mathrm{d}\boldsymbol{\mu,}$

associated to a vector-valued submodular capacity
$\boldsymbol{\mu}:\mathcal{B}(X)\rightarrow E,$ is a Choquet operator.

The linear and positive functionals defined on $C(X)$ can be represented
either as integrals with respect to a unique regular Borel measure (the Riesz-
Kakutani representation theorem) or as Choquet integrals (see Epstein and Wang
[13]).

######  Example 2.

The space $c$ of all convergent sequences of real numbers can be identified
with the space of continuous functions on
$\mathbb{\hat{N}}=\mathbb{N}\cup\\{\infty\\}$ $($the one-point
compactification of the discrete space $\mathbb{N)}$. The functional

$I:c\rightarrow\mathbb{R},\text{\quad}I(x)=\lim_{n\rightarrow\infty}x(n)$

is linear and monotone _(_ therefore continuous by Lemma _3)_ and its Riesz
representation is

$I(x)=\int_{\mathbb{\hat{N}}}x(n)\mathrm{d}\delta_{\infty}(n),$

where $\delta_{\infty}$ is the Dirac measure concentrated at $\infty,$ that
is, $\delta_{\infty}(A)=1$ if $A\in\mathcal{P}(\mathbb{\hat{N}})$ and
$\\{\infty\\}\in A$ and $\delta_{\infty}(A)=0$ if $\\{\infty\\}\notin A.$
Meantime, $I$ admits the Choquet representation

$I(x)=(\operatorname*{C})\int x(n)d\mu(n)$

where $\mu$ is the capacity defined on the power set
$\mathcal{P}(\mathbb{\hat{N}})$ by $\mu(A)=0$ if $A$ is finite and $\mu(A)=1$
otherwise.

As the Riesz-Kakutani representation theorem also holds in the case of linear
and positive operators $T:C(X)\rightarrow E$, it seems natural to search for
an analogue in the case of Choquet operators. The answer is provided by the
following representation theorem that extend a result due to Epstein and Wang
[13] from functionals to operators:

######  Theorem 3.

Let $X$ be a compact Hausdorff space and $E$ be a Banach lattice with order
continuous norm. Then for every comonotonic additive and monotone operator
$I:C(X)\rightarrow E$ with $I(1)>0$ there exists a unique upper continuous
vector capacity $\boldsymbol{\mu}:\Sigma_{up}^{+}(X)\rightarrow E_{+}$ such
that

(4.2) $I(f)=(\operatorname*{CB})\int_{X}f\mathrm{d}\boldsymbol{\mu}\text{\quad
for all }f\in C(X).$

Moreover, $\boldsymbol{\mu}$ admits a unique extension to $\mathcal{B}(X)$
$($also denoted $\boldsymbol{\mu})$ that fulfils the following two properties
of regularity:

$(R1)$ $\boldsymbol{\mu}(A)=\sup\left\\{\boldsymbol{\mu}(K):K\text{ closed,
}K\subset A\right\\}$ for all $A\in\mathcal{B}(X);$

$(R2)$ $\boldsymbol{\mu}(K)=\inf\left\\{\boldsymbol{\mu}(O):O\text{ open,
}O\supset K\right\\}$ for all closed sets $K.$

Recall that $\Sigma_{up}^{+}(X)$ represents the lattice of upper contour sets
associated to the continuous functions defined on $X.$ This lattice coincides
with the lattice of all closed subsets of $X$ when $X$ is compact and
metrizable.

The proof of Theorem 3 needs several auxiliary results and will be detailed at
the end of this section.

######  Lemma 8.

Let $E$ be an ordered Banach space. Then every monotone and translation
invariant operator $I:C(X)\rightarrow E$ is Lipschitz continuous.

###### Proof.

Given $f,g\in C(X),$ one can choose a decreasing sequence $(\alpha_{n})_{n}$
of positive numbers such that $\alpha_{n}\downarrow\left\|f-g\right\|.$ Then
$f\leq g+\left\|f-g\right\|\leq g+\alpha_{n}\cdot 1,$ which implies

$I(f)\leq I(g+\alpha_{n}\cdot 1)=I(g)+\alpha_{n}I(1)$

due to the properties of monotonicity and translation invariance and positive
homogeneity of $I.$ Since the role of $f$ and $g$ is symmetric, this leads to
the fact that $\left|I(f)-I(g)\right|\leq I(1)\cdot\alpha_{n}$ for all $n,$
whence by passing to the limit as $n\rightarrow\infty,$ we conclude that

$\left|I(f)-I(g)\right|\leq I(1)\cdot\left\|f-g\right\|.$

∎

######  Lemma 9.

Let $E$ be an ordered Banach space. Then every comonotonic additive and
monotone operator $I:C(X)\rightarrow E$ is positively homogeneous.

###### Proof.

Let $f\in C(X)_{+}.$ Since $I$ is comonotonic additive, we get
$I(0)=I(0+0)=2\cdot I(0),$ which implies $I(0)=0$. As a consequence, $I(0\cdot
f)=0=0\cdot I(f).$ Then the same argument shows that $I(2f)=I(f+f)=2I(f)$ and
by mathematical induction we infer that $I(pf)=pI(f)$ for all
$p\in\mathbb{N}$.

Now, consider the case of positive rational numbers $r=p/q,$ where
$p,q\in\mathbb{N}$. Then
$I(f)=I\left(q\cdot\frac{1}{q}f\right)=qI\left(\frac{1}{q}f\right)$, which
implies $I\left(\frac{1}{q}f\right)=\frac{1}{q}\cdot I(f)$. Therefore

$I\left(\frac{p}{q}f\right)=pI\left(\frac{1}{q}f\right)=\frac{p}{q}\cdot
I(f).$

Passing to the case of an arbitrary positive number $\alpha,$ let us choose a
decreasing sequence $(r_{n})_{n}$ of rationals converging to $\alpha$. Then
$r_{n}f\geq\alpha f$ for all $n,$ which yields $r_{n}I(f)=I(r_{n}f)\geq
I(\alpha f)$ $n\in\mathbb{N}$. Passing here to limit (in the norm of $E$) it
follows $\alpha I(f)\geq I(\alpha f)$. On the other hand, considering a
sequence of positive rational numbers $s_{n}\nearrow\alpha$ and reasoning as
above, we easily obtain $\alpha I(f)\leq I(\alpha f)$, which combined with the
previous inequality proves the assertion of Lemma 9 in the case of nonnegative
functions.

When $f\in C(X)$ is arbitrary, one can choose a positive number $\lambda$ such
that $f+\lambda\geq 0.$ By the above reasoning and the property of comonotonic
additivity, for all $\alpha\geq 0,$

$\displaystyle\alpha I(f)+\alpha\lambda I(1)$ $\displaystyle=\alpha
I\left(f+\lambda\right)=I(\alpha\left(f+\lambda\right))$
$\displaystyle=I(\alpha f+\alpha\lambda)=I(\alpha f)+\alpha\lambda I(1),$

which ends the proof of Lemma 9. ∎

######  Lemma 10.

Let $E$ be a Banach lattice with order continuous norm. Then every monotone,
positively homogeneous and translation invariant operator $I:C(X)\rightarrow
E$ is weakly compact and upper continuous.

###### Proof.

The weak compactness of $I$ follows from the fact that $I$ maps the closed
order interval in $C(X)$ (that is, all closed balls) into closed order
intervals in $E$ and all such intervals are weakly compact in $E$. See Lemma
3.

For the property of upper continuity, let $(f_{n})_{n}$ be any nonincreasing
sequence of functions in $C(X)$, which converges pointwise to a continuous
function $f.$ By Dini’s lemma, the sequence $(f_{n})_{n}$ is convergent to $f$
in the norm topology of $C(X).$ Therefore, for each $\varepsilon>0$ there is
an index $N$ such that $f\leq f_{n}<f+\varepsilon$ for all $n\geq N.$ Since
$I$ is monotone it follows that

(4.3) $I(f)\leq I(f_{n})\leq I(f+\varepsilon)=I(f)+\varepsilon\cdot I(1)\text{
for all }n\geq N,$

where $1$ is the unit of $C(X).$ Taking into account that $E$ has order
continuous norm and $I(f_{n})\geq I(f_{n+1})\geq I(f)$ for all $n$ it follows
that the limit $\lim_{n\rightarrow\infty}I(f_{n})$ exists in $E$ and
$\lim_{n\rightarrow\infty}I(f_{n})\geq I(f)$. Combining this fact with (4.3)
we infer that $I(f)\leq\lim_{n\rightarrow\infty}I(f_{n})\leq I(f)+\varepsilon
I(1).$ Since $\varepsilon>0$ was chosen arbitrarily we conclude that
$\lim_{n\rightarrow\infty}I(f_{n})=I(f).$ ∎

The next result, was stated for real-valued functionals in [31], Lemma 1 (and
attributed by him to Masimo Marinacci). For the convenience of the reader we
include here the details.

######  Lemma 11.

Let $E$ be an ordered Banach space.

$(a)$ Suppose that $I:C(X)\rightarrow E$ is a monotone, positively homogeneous
and translation invariant operator. The following two properties are
equivalent:

$(a_{1})$ $\lim_{n\rightarrow\infty}I(f_{n})=I(f)$ for any nonincreasing
sequence $(f_{n})_{n}$ in $C(X)$ that converges pointwise to a function $f$
also in $C(X);$

$(a_{2})$ $\lim_{n\rightarrow\infty}I(f_{n})\leq I(f)$ for any nonincreasing
sequence $(f_{n})_{n}$ in $C(X)$ and any $f$ in $C(X)$ such that for each
$x\in X$ there is an index $n_{x}\in\mathbb{N}$ such that $f_{n}(x)\leq f(x)$
whenever $n\geq n_{x}$.

$(b)$ For any vector capacity $\boldsymbol{\mu}:\mathcal{B}(X)\rightarrow
E_{+}$, the following two properties are equivalent :

$(b_{1})$
$\lim_{n\rightarrow\infty}\boldsymbol{\mu}(A_{n})=\boldsymbol{\mu}(A)$, for
any nonincreasing sequence $(A_{n})_{n}$ of sets in $\mathcal{B}(X)$ such that
$A=\cap_{n=1}^{\infty}A_{n};$

$(b_{2})$
$\lim_{n\rightarrow\infty}\boldsymbol{\mu}(A_{n})\leq\boldsymbol{\mu}(A)$, for
any nonincreasing sequence $(A_{n})_{n}$ of sets in $\mathcal{B}(X)$ and any
$A\in\mathcal{B}(X)$ such that $\cap_{n=1}^{\infty}A_{n}\subset A$.

All the limits above are considered in the norm topology of $E$.

###### Proof.

$(a_{1})\Rightarrow(a_{2})$. Let an arbitrary sequence $(f_{n})_{n}$ in $C(X)$
and $f\in C(X)$, be such that $(f_{n})_{n}$ is nonincreasing and, for all
$x\in X$, there is an $n_{x}$ with $f_{n}(x)\leq f(x)$ for all $n\geq n_{x}$.
Since the sequence $(\max\\{f_{n},f\\})$ is also a nonincreasing sequence in
$C(X)$ and $\lim_{n\rightarrow\infty}\max\\{f_{n}(x),f(x)\\}=f(x)$ for all
$x\in X$, $(a_{i})$ implies
$\lim_{n\rightarrow\infty}I(\max\\{f_{n},f\\})=I(f)$. By the monotonicity of
$I$ it follows
$\lim_{n\rightarrow\infty}I(f_{n})\leq\lim_{n\rightarrow\infty}I(\max\\{f_{n},f\\})=I(f)$.

$(a_{2})\Rightarrow(a_{1})$. Let $f_{n},f\in C(X)$, $n\in\mathbb{N}$, be such
that $(f_{n})_{n}$ is nonincreasing and
$\lim_{n\rightarrow\infty}f_{n}(x)=f(x)$, for all $x\in X$. Fix
$\varepsilon>0$ arbitrary. Since for all $x\in X$, there exists
$n_{x}\in\mathbb{N}$ such that $f_{n}(x)\leq f(x)+\varepsilon$, for all $n\geq
n_{x}$, $(a_{ii})$ implies (from monotonicity, positive homogeneity and
translation invariance) $\lim_{n\rightarrow\infty}I(f_{n})\leq
I(f+\varepsilon\cdot 1)=I(f)+\varepsilon I(1)$. Passing with
$\varepsilon\rightarrow 0$ it follows $\lim_{n\rightarrow\infty}I(f_{n})\leq
I(f)$. But since $(f_{n})_{n}$ is nonincreasing and $I$ is monotone, we also
have $\lim_{n\rightarrow\infty}I(f_{n})\geq I(f)$, which combined with the
previous inequality implies $\lim_{n\rightarrow\infty}I(f_{n})=I(f)$.

The equivalence $(b_{1})\Leftrightarrow(b_{2})$ can be proved in a similar
way. ∎

Recall that in the case of compact Hausdorff space $X$ the lattice
$\Sigma_{up}^{+}(X)$ represents the lattice of upper contour sets associated
to the continuous functions defined on $X.$ This lattice coincides with the
lattice of all closed subsets of $X$ when $X$ is compact and metrizable;
indeed, if $d$ is the metric of $X,$ then every closed subset $A\subset X$
admits the representation $A=\left\\{x:-d(x,A)\geq 0\right\\}.$

###### Proof of Theorem $3$.

Notice first that according to Lemma 9 and Lemma 10 the operator $I$ is also
positively homogeneous and upper continuous.

Every set $K\in\Sigma_{up}^{+}(X)$ admits a representation of the form

$K=\left\\{x:f(x)\geq\alpha\right\\},$

for suitable $f\in C(X)$ and $\alpha\in\mathbb{R}.$ As a consequence its
characteristic function $\chi_{K}$ is the pointwise limit of a nonincreasing
sequence $(f_{n}^{K})_{n}$ of continuous and nonnegative functions. For
example, one may choose

(4.4)
$f_{n}^{K}(x)=1-\inf\left\\{1,n(\alpha-f)^{+}\right\\}=\left\\{\begin{array}[c]{cl}0&\text{if
}f(x)\leq\alpha-1/n\\\ \in(0,1)&\text{if }\alpha-1/n<\varphi(x)<\alpha\\\
1&\text{if }f(x)\geq\alpha\text{ (i.e., }x\in K).\text{ }\end{array}\right.$

See [31], p. 1814. Since $I$ is monotone, the sequence $(I(f_{n}^{K}))_{n}$ is
also nonincreasing and bounded from below by $0$, which implies (due to the
order continuity of the norm of $E)$, that it is also convergent in the norm
topology of $E.$

This allows us to define $\boldsymbol{\mu}$ on the sets
$K\in\Sigma_{up}^{+}(X)$ by the formula

(4.5) $\boldsymbol{\mu}(K)=\lim_{n\rightarrow\infty}I(f_{n}^{K}).$

The definition of $\boldsymbol{\mu}(K)$ is independent of the particular
sequence $(f_{n}^{K})_{n}$ with the aforementioned properties. Indeed, if
$(g_{n}^{K})_{n}$ is another such sequence, fix a positive integer $m$, and
infer from Lemma 11 $(a)$ that $\lim_{n\rightarrow\infty}I(g_{n}^{A})\leq
I(f_{m}^{A})$. Taking the limit as $m\rightarrow\infty$ on the right-hand side
we get

$\lim_{n\rightarrow\infty}I(g_{n}^{A})\leq\lim_{m\rightarrow\infty}I(f_{m}^{A}).$

Then, interchanging $(f_{n}^{K})_{n}$ and $(g_{n}^{K})_{n}$ we conclude that
actually equality holds.

Clearly, the set function $\mathbf{\mu}:\Sigma_{up}^{+}(X)\rightarrow E_{+}$
is a vector capacity and it takes values in the order interval $[0,I(1)].$

We next show that $\boldsymbol{\mu}$ is upper continuous on
$\Sigma_{up}^{+}(X)$, that is,

$\boldsymbol{\mu}(K)=\lim_{n\rightarrow\infty}\boldsymbol{\mu}(K_{n})$

whenever $(K_{n})_{n}$ is a nonincreasing sequence of sets in
$\Sigma_{up}^{+}(X)$ such that $K=\cap_{n=1}^{\infty}K_{n}$. Indeed, using
formula (4.4) one can choose a nonincreasing sequence of continuous function
$g_{n}$ such that $g_{n}\geq\chi_{K_{n}},$ $g_{n}=0$ outside the neighborhood
of radius $1/n$ of $K_{n}$ and $I(g_{n})-\boldsymbol{\mu}(K_{n})\rightarrow
0$. See formula (4.4) and using analogous reasonings with those concerning
relations (7)-(9) in [31], pp. 1814-1815. Then
$\boldsymbol{\mu}(K)=\lim_{n\rightarrow\infty}I(g_{n}),$ which implies the
equality
$\boldsymbol{\mu}(K)=\lim_{n\rightarrow\infty}\boldsymbol{\mu}(K_{n}).$

The next goal is the representation formula (4.2). For this, let $x^{\ast}\in
E_{+}^{\ast}$ be arbitrarily fixed and consider the comonotonic additive and
monotone functional

$x^{\ast}\circ I:C(X)\rightarrow\mathbb{R}.$

It verifies $I^{\ast}(1)=x^{\ast}(I(1))>0,$ so by Theorem 1 in [31] there is a
unique upper continuous capacity
$\nu^{\ast}:\Sigma_{up}^{+}(X)\rightarrow[0,I^{\ast}(1)]$ such that

$\left(x^{\ast}\circ I\right)(f)=(C)\int_{X}f\mathrm{d}\nu^{\ast}\text{ for
all }f\in C(X).$

The capacity $\nu^{\ast}$ is obtained via an approximation process similar to
(4.5). Therefore

$\nu^{\ast}(K)=\lim_{n\rightarrow\infty}x^{\ast}(I(f_{n}^{K}))=x^{\ast}(\lim_{n\rightarrow\infty}[I(f_{n}^{K}))=x^{\ast}(\boldsymbol{\mu}(K))$

for all $K\in\Sigma_{up}^{+}(X),$ which implies

$x^{\ast}(I(f))=(C)\int_{X}f\mathrm{d}(x^{\ast}\circ\boldsymbol{\mu}).$

Since $x^{\ast}\in E_{+}^{\ast}$ was arbitrarily fixed, an appeal to Lemma 2
$(c)$ easily yields the equality

$I(f)=(\operatorname*{CB})\int_{X}f\mathrm{d}\boldsymbol{\mu}.$

For the second part of Theorem 3,since $\boldsymbol{\mu}$ takes values in an
order bounded interval, one can extend it to all Borel subsets of $X$ via the
formula

$\boldsymbol{\mu}(A)=\sup\left\\{\boldsymbol{\mu}(K):K\text{ closed, }K\subset
A\right\\},\text{\quad}A\in\mathcal{B}(X).$

The fact that the resulting set function $\boldsymbol{\mu}$ is a vector
capacity is immediate.

This set function $\boldsymbol{\mu}$ also verifies the regularity condition
$(R2)$. Indeed, given a closed set $K$, we can consider the sequence of open
sets

$O_{n}=\left\\{x\in X:d(x,K)<1/n\right\\}.$

Clearly,
$\boldsymbol{\mu}(K)\leq\boldsymbol{\mu}(O_{n})\leq\boldsymbol{\mu}(K_{n})$
where

$K_{n}=\left\\{x\in X:d(x,K)\leq 1/n\right\\}.$

Since $\boldsymbol{\mu}$ is upper continuous on closed sets, it follows that
$\lim_{n\rightarrow\infty}\boldsymbol{\mu}(K_{n})=\boldsymbol{\mu}(K)$, whence
$\lim_{n\rightarrow\infty}\boldsymbol{\mu}(O_{n})=\boldsymbol{\mu}(K).$
Therefore $\boldsymbol{\mu}$ verifies the regularity condition $(R2)$. The
uniqueness of the extension of $\boldsymbol{\mu}$ to $\mathcal{B}(X)$ is
motivated by the condition $(R1).$ ∎

######  Remark 1.

If the operator $I$ is submodular, that is,

$I(\sup\left\\{{f,g}\right\\})+I(\inf\left\\{{f,g}\right\\})\leq
I(f)+I(g)\text{\quad for all }f,g\in C(X),$

then the vector capacity $\boldsymbol{\mu}:\Sigma_{up}^{+}(X)\rightarrow
E_{+}$ stated by Theorem _3_ is submodular. This is a consequence of Theorem
$13$ $(c)$ in _[4]_. For the convenience of the reader we will recall here the
argument. Let $A,B\in\Sigma_{up}^{+}(X)$ and consider the sequences
$(f_{n}^{A}(x))_{n}$ and $(f_{n}^{B}(x))_{n}$ of continuous functions
associated respectively to $A$ and $B$ by the formula _( 4.5)_. Then

$\boldsymbol{\mu}(A)=\lim_{n\rightarrow\infty}I(f_{n}^{A}),\ \text{
}\boldsymbol{\mu}(B)=\lim_{n\rightarrow\infty}I(f_{n}^{B}),\ \text{
}\boldsymbol{\mu}(A\cup
B)=\lim_{n\rightarrow\infty}I(\sup\left\\{f_{n}^{A},f_{n}^{B}\right\\}\ \text{
}$

and $\boldsymbol{\mu}(A\cap
B)=\lim_{n\rightarrow\infty}I(\inf\left\\{f_{n}^{A},f_{n}^{B}\right\\}$. Since
$I$ is submodular, it follows that

$\boldsymbol{\mu}(A\cup B)+\boldsymbol{\mu}(A\cap
B)\leq\boldsymbol{\mu}(A)+\boldsymbol{\mu}(B),$

and the proof is done.

##  5\. The case of operators with bounded variation

The representation Theorem 3 can be extended outside the framework of monotone
operators by considering the class of operators with bounded variation.

As above, $X$ is a compact Hausdorff space and $E$ is a Banach lattice with
order continuous norm.

######  Definition 3.

An operator $I:C(X)\rightarrow E$ has _bounded variation_ _over an order
interval_ $[f,g]$ if

(5.1)
$\vee_{f}^{g}I=\sup{\displaystyle\sum\nolimits_{k=0}^{n}}\left|I(f_{k})-I(f_{k-1})\right|\text{\quad
exists in }E,$

the supremum being taken over all finite chains $f=f_{0}\leq
f_{1}\leq\cdots\leq f_{n}=g$ of functions in the Banach lattice $C(X).$ The
operator $I$ is said to have _bounded variation_ if it has bounded variations
on all __ order intervals $[f,g]$ in $C(X).$

Clearly, if $I$ is monotone, then $\vee_{f}^{g}I=I(g)-I(f)$ for all $f\leq g$
in $C(X)$ and thus $I$ has bounded variation.

More generally, every operator $I:C(X)\rightarrow E$ which can be represented
as the difference $I=I_{1}-I_{2}$ of two monotone operators
$I_{1},I_{2}:C(X)\rightarrow E$ has bounded variation. This follows from the
order completeness of $E$ and the modulus inequality, which provides an upper
bound for the sums appearing in formula (5.1):

${\displaystyle\sum\nolimits_{k=0}^{n}}\left|I(f_{k})-I(f_{k-1})\right|\leq
I_{1}(g)-I_{1}(f)+I_{2}(g)-I_{2}(f).$

Remarkably, the converse also holds. The basic ingredient is the following
result.

######  Lemma 12.

Suppose that $I:C(X)\rightarrow E$ is a comonotonic additive operator with
bounded variation. Then there exist two positively homogeneous, translation
invariant and monotone operators $I_{1},I_{2}:C(X)\rightarrow E$ such that
$I=I_{1}-I_{2}.$

Moreover, if $I$ is upper continuous, then both operators $I_{1}$ and $I_{2}$
can be chosen to be upper continuous.

###### Proof.

The proof is done in the footsteps of Lemma 14 in [4] by noticing first the
following four facts:

$(a)$ According to our hypotheses,

$I(\alpha f+\beta)=\alpha I(f)+I(\beta)$

for all $f\in C(X),$ $\alpha\in\mathbb{R}_{+}$ and $\beta\in\mathbb{R}.$

$(b)$ $\vee_{0}^{f+\alpha}I=\vee_{-\alpha}^{f}I$ for all $f\in C(X)$ and
$\alpha\in\mathbb{R}$ with $f+\alpha\geq 0.$

This follows from the definition of the variation.

$(c)$ $\vee_{0}^{\alpha f}I=\alpha\vee_{0}^{f}I$ for all $f\in C(X)_{+}$ and
$\alpha\in\mathbb{R}_{+}.$

Indeed for every $\varepsilon\in E,$ $\varepsilon>0,$ there exists a chain
$0=f_{0}\leq f_{1}\leq\cdots\leq f_{n}=f$ such that

${\displaystyle\sum\nolimits_{k=0}^{n}}\left|I(f_{k})-I(f_{k-1})\right|\geq\vee_{0}^{f}I-\varepsilon.$

According to fact $(a)$ the chain $0=\alpha f_{0}\leq\alpha
f_{1}\leq\cdots\leq\alpha f_{n}=\alpha f$ verifies

$\vee_{0}^{\alpha f}I\geq{\displaystyle\sum\nolimits_{k=0}^{n}}\left|I(\alpha
f_{k})-I(\alpha f_{k-1})\right|\geq\alpha\vee_{0}^{f}I-\alpha\varepsilon.$

As $\varepsilon>0$ was arbitrarily fixed it follows that $\vee_{0}^{\alpha
f}I\geq\alpha\vee_{0}^{f}I.$ By replacing $\alpha$ to $1/\alpha$ and then $f$
by $\alpha f$, one obtains the reverse inequality, $\vee_{0}^{\alpha
f}I\leq\alpha\vee_{0}^{f}I.$

$(d)$
$\vee_{-\alpha}^{f}I=\vee_{-\alpha}^{0}I+\vee_{0}^{f}I=\vee_{0}^{\alpha}I+\vee_{0}^{f}I$
for all $f\in C(X)_{+}$ and $\alpha\in\mathbb{R}_{+}.$

The fact that
$\vee_{-\alpha}^{f}I\geq\vee_{-\alpha}^{0}I+\vee_{0}^{f}I=\vee_{0}^{\alpha}I+\vee_{0}^{f}I$
is a direct consequence of the definition of variation. For the other
inequality, fix arbitrarily $\varepsilon>0$ in $E$ and choose a chain
$-\alpha=f_{0}\leq f_{1}\leq\cdots\leq f_{n}=f$ such that

${\displaystyle\sum\nolimits_{k=0}^{n}}\left|I(f_{k})-I(f_{k-1})\right|\geq\vee_{-\alpha}^{f}I-\varepsilon.$

Then $-\alpha=-f_{0}^{-}\leq-f_{1}^{-}\leq\cdots\leq-f_{n}^{-}=0$ and
$0=f_{0}^{+}\leq f_{1}^{+}\leq\cdots\leq f_{n}^{+}=f.$ Since all pairs
$-f_{k}^{-},f_{k}^{+}$ are comonotonic,

we have

$\displaystyle\vee_{-\alpha}^{0}I+\vee_{0}^{f}I$
$\displaystyle\geq{\displaystyle\sum\nolimits_{k=0}^{n}}\left|I(-f_{k}^{-})-I(-f_{k-1}^{-})\right|+{\displaystyle\sum\nolimits_{k=0}^{n}}\left|I(f_{k}^{+})-I(f_{k-1}^{+})\right|$
$\displaystyle\geq{\displaystyle\sum\nolimits_{k=0}^{n}}\left|I(-f_{k}^{-})-I(-f_{k-1}^{-})+I(f_{k}^{+})-I(f_{k-1}^{+})\right|$
$\displaystyle={\displaystyle\sum\nolimits_{k=0}^{n}}\left|I(f_{k})-I(f_{k-1})\right|\geq\vee_{-\alpha}^{f}I-\varepsilon$

and it remains to take the supremum over $\varepsilon>0.$

Now we can proceed to the choice of the operators $I_{1}$ and $I_{2}.$ By
definition,

$I_{1}(f)=\vee_{0}^{f}I\text{\quad if }f\in C(X)_{+},$

while if $f\in C(X)$ is an arbitrary function, one choose
$\alpha\in\mathbb{R}_{+}$ such that $f+\alpha\geq 0$ and put

$I_{1}(f)=\vee_{0}^{f+\alpha}I-\alpha\vee_{0}^{1}I.$

The fact that $I_{1}$ is well-defined (that is, independent of $\alpha)$ can
be proved as follows. Suppose that $\alpha,\beta>0$ are two numbers such that
$f+\alpha\geq 0$ and $f+\beta\geq 0.$ Without loss of generality we may assume
that $\alpha<\beta.$ Indeed, according to the facts $(b)-(d)$, we have

$\displaystyle\vee_{0}^{f+\beta}I-\beta\vee_{0}^{1}I$
$\displaystyle=\vee_{0}^{f+\alpha+(\beta-\alpha)}I-\beta\vee_{0}^{1}I\overset{\text{fact
}(b)}{=}\vee_{-(\beta-\alpha)}^{f+\alpha}I-\beta\vee_{0}^{1}I$
$\displaystyle\overset{\text{fact
}(d)}{=}\vee_{-(\beta-\alpha)}^{0}I+\vee_{0}^{f+\alpha}I-\beta\vee_{0}^{1}I$
$\displaystyle\overset{\text{fact
}(b)}{=}\vee_{0}^{\beta-\alpha}I+\vee_{0}^{f+\alpha}I-\beta\vee_{0}^{1}I$
$\displaystyle\overset{\text{fact
}(c)}{=}(\beta-\alpha)\vee_{0}^{1}I+\vee_{0}^{f+\alpha}I-\beta\vee_{0}^{1}I$
$\displaystyle=\vee_{0}^{f+\alpha}I-\alpha\vee_{0}^{1}I.$

By definition,

$I_{2}=I-I_{1}.$

Let $f,g$ be two functions in $C(X)$ such that $f\leq g$ and let $\alpha>0$
such that $f+\alpha\geq 0.$ Since $I$ is monotonic and has bounded variation,

$0\leq
I(g)-I(f)=I(g+\alpha)-I(f+\alpha)\leq\vee_{f+\alpha}^{g+\alpha}I\leq\vee_{0}^{g+\alpha}I-\vee_{0}^{f+\alpha}I=I_{1}(g)-I_{1}(f),$

whence we infer that both operators $I_{1}$ and $I_{2}$ are monotonic.

Due to the fact $(c),$ the operator $I_{1}$ is positively homogeneous.
Therefore the same is true for $I_{2}.$

For the property of translation invariance, let $f\in C(X)$,
$\beta\in\mathbb{R}$ and choose $\alpha>0$ such that $f+\beta+\alpha\geq 0$
and $\beta+\alpha\geq 0.$ Then

$\displaystyle I_{1}(f+\beta)$
$\displaystyle=\vee_{0}^{f+\beta+\alpha}I-\alpha\vee_{0}^{1}I$
$\displaystyle=I_{1}(f+\beta+\alpha)-\alpha\vee_{0}^{1}I$

while from facts $(b)\&(d)$ we infer that
$I_{1}(f)=I_{1}(f+\beta+\alpha)-\left(\beta+\alpha\right)\vee_{0}^{1}I.$
Therefore

$I_{1}(f+\beta)=I_{1}(f)+I_{1}(\beta)$

which proves that indeed $I_{1}$ is translation invariant. The same holds for
$I_{2}=I-I_{1}.$

As concerns the second part of Lemma 12, it suffices to prove that $I_{1}$ is
upper continuous when $I$ has this property.

Let $(f_{n})_{n}$ be a decreasing sequence in $C(X)$ which converges pointwise
to a function $f$ also in $C(X).$ By Dini’s lemma,
$\left\|f_{n}-f\right\|\rightarrow 0$. Since $I_{1}$ is a Lipschitz operator
(see Lemma 8) we conclude that $\left\|I_{1}(f_{n})-I(f)\right\|\rightarrow
0.$

This ends the proof of Lemma 12. ∎

Now it is clear that the representation Theorem 3 can be extended to the
framework of comonotonic additive operators $I:C(X)\rightarrow E$ with bounded
variation by considering Choquet-Bochner integrals associated to differences
of vector capacities (that is, to set functions with bounded variation in the
sense of Aumann and Shapley [1]).

Let $\Sigma$ be a lattice of sets of $X$ and
$\boldsymbol{\mu}:\mathcal{\Sigma}\rightarrow E$ a set function taking values
in a Banach lattice $E$ with order continuous norm.

The _variation_ of the set function $\boldsymbol{\mu}$ is the set function
$\left|\boldsymbol{\mu}\right|$ defined by the formula

$\left|\mathbf{\mu}\right|(A)=\sup{\displaystyle\sum\nolimits_{k=0}^{n}}\left|\boldsymbol{\mu}(A_{k})-\boldsymbol{\mu}(A_{k-1})\right|\text{\quad
for }A\in\Sigma,$

where the supremum is taken over all finite chains $A_{0}=\emptyset\subset
A_{1}\subset\cdots\subset A_{n}\subset A$ of sets in the lattice $\Sigma.$

The space of set functions with bounded variation,

$\operatorname*{bv}(\Sigma,E)=\left\\{\boldsymbol{\mu}:\Sigma\rightarrow
E:\boldsymbol{\mu}(\emptyset)=0\text{ and
}\left|\boldsymbol{\mu}\right|(X)\text{ exists in }E\right\\},$

is a normed vector space when endowed with the norm

$\left\|\boldsymbol{\mu}\right\|_{\operatorname*{bv}}=\left\|\left|\boldsymbol{\mu}\right|(X)\right\|.$

Associated to every set function
$\boldsymbol{\mu}\in\operatorname*{bv}(\Sigma,E)$ are two positive vector-
valued set functions, the _inner upper variation_ of $\mu,$ defined by

$\boldsymbol{\mu}^{+}(A)=\sup{\displaystyle\sum\nolimits_{k=0}^{n}}\left(\boldsymbol{\mu}(A_{k})-\boldsymbol{\mu}(A_{k-1})\right)^{+}\text{\quad
for }A\in\Sigma$

and the _inner lower variation_ of $\boldsymbol{\mu},$ defined by

$\boldsymbol{\mu}^{-}(A)=\sup{\displaystyle\sum\nolimits_{k=0}^{n}}\left(\boldsymbol{\mu}(A_{k})-\boldsymbol{\mu}(A_{k-1})\right)^{-}\text{\quad
for }A\in\Sigma;$

in both cases the supremum is taken over all finite chains
$A_{0}=\emptyset\subset A_{1}\subset\cdots\subset A_{n}\subset A$ of sets in
$\Sigma.$ Notice that

$\boldsymbol{\mu}=\boldsymbol{\mu}^{+}-\boldsymbol{\mu}^{-}\text{ and
}\left|\boldsymbol{\mu}\right|=\boldsymbol{\mu}^{+}+\boldsymbol{\mu}^{-}.$

######  Lemma 13.

We assume that $E$ is a Banach lattice with order continuous norm. The
following conditions are equivalent for a set function
$\boldsymbol{\mu}:\Sigma\rightarrow E:$

$(a)$ $\boldsymbol{\mu}\in\operatorname*{bv}(\Sigma,E);$

$(b)$ $\boldsymbol{\mu}^{+}$ and $\boldsymbol{\mu}^{-}$ are vector capacities;

$(c)$ there exist two vector capacities $\boldsymbol{\mu}_{1}$ and
$\boldsymbol{\mu}_{2}$ on $\Sigma$ such that
$\boldsymbol{\mu}=\boldsymbol{\mu}_{1}-\boldsymbol{\mu}_{2}.$

Moreover, for any such decomposition we have
$\boldsymbol{\mu}^{+}\leq\boldsymbol{\mu}_{1}$ and
$\boldsymbol{\mu}^{-}\leq\boldsymbol{\mu}_{2}$.

The details are similar to those presented in [1], Ch. 1, §4.

We are now in a position to state the main result of this section.

######  Theorem 4.

Let $X$ be a compact Hausdorff space and $E$ be a Banach lattice with order
continuous norm. Then for every comonotonic additive operator
$I:C(X)\rightarrow E$ with bounded variation there exists a unique upper
continuous set function $\boldsymbol{\mu}:\Sigma_{up}^{+}(X)\rightarrow E$
with bounded variation such that

(5.2) $\displaystyle I(f)$
$\displaystyle=\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t$
$\displaystyle+\int_{-\infty}^{0}\left[\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)-\mu(A)\right]\mathrm{d}t.$

for all $f\in C(X).$

###### Proof.

By Lemma 12, there exist two functionals $I_{1},I_{2}:C(X)\rightarrow E$ that
are monotone, translation invariant, positively homogeneous, upper continuous
and such that $I=I_{1}-I_{2}$. Define
$\boldsymbol{\mu}:\Sigma_{up}^{+}(X)\rightarrow E$ by
$\boldsymbol{\mu}=\boldsymbol{\mu}_{1}-\boldsymbol{\mu}_{2},$ where
$\boldsymbol{\mu}_{1},\boldsymbol{\mu}_{2}$ are associated to the functionals
$I_{1}$ and $I_{2}$ via Theorem 3. Then $\boldsymbol{\mu}$ is upper continuous
and by Lemma 13 it also has bounded variation. We now prove that the
representation formula (5.2) holds.

Suppose that $f\in C(X)_{+}.$ The integral

$\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq t\\}\right)\mathrm{d}t$

is well defined as a Bochner integral (see Lemma 6). Given $\varepsilon>0,$
one can choose an equidistant division $0=t_{0}<\cdots<t_{m}=\left\|f\right\|$
of $[0,\left\|f\right\|]$ such that

$\displaystyle\left\|\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t-\sum_{k=0}^{m-1}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t_{k}\\}\right)(t_{k+1}-t_{k})\right\|$ (5.3)
$\displaystyle=\left\|\int_{0}^{\left\|f\right\|}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t-\sum_{k=0}^{m-1}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t_{k}\\}\right)(t_{k+1}-t_{k})\right\|<\varepsilon$

and $\left\|f\right\|/m<\varepsilon$.

Denote $C_{k}=\\{x:f(x)\geq t_{k}\\}$ for $k=0,...,m-1.$ By the definition of
$\boldsymbol{\mu}_{1}$ and $\boldsymbol{\mu}_{2}$ (see the proof of Theorem 3)
one can choose functions $f_{n}^{C_{k}}\in C(X)_{+}$ such that

(5.4)
$\left\|\boldsymbol{\mu}\left(C_{k}\right)-I(f_{n}^{C_{k}})\right\|<\varepsilon/\left\|f\right\|\text{
and }n^{-1}<\left\|f\right\|/m.$

Because the functions $f_{n}^{C_{k}}$ are defined by the formula (4.4) and
$n^{-1}<\left\|f\right\|/m$, it follows that the functions
$f_{n}^{C_{i}}(t_{i+1}-t_{i})$ and
$\sum_{k=i+1}^{m-1}f_{n}^{C_{k}}(t_{k+1}-t_{k})$ are comonotonic for all
indices $i,$ so that

$\displaystyle I(\sum_{k=0}^{m-1}f_{n}^{C_{k}}(t_{k+1}-t_{k}))$
$\displaystyle=I(f_{n}^{C_{0}})(t_{1}-t_{0})+I(\sum_{k=1}^{m-1}f_{n}^{C_{k}}(t_{k+1}-t_{k}))$
$\displaystyle=\cdots=\sum_{k=0}^{m-1}I(f_{n}^{C_{k}})(t_{k+1}-t_{k});$

the property of positive homogeneity of $I$ is assured by Lemma 12.

Notice that

$f(x)\leq\sum_{k=0}^{m-1}f_{n}^{C_{k}}(x)(t_{k+1}-t_{k})\leq
f(x)+2\varepsilon\text{\quad for all }x\in X.$

Since the operators $I_{1},I_{2}$ are monotone and translation invariant, it
follows that

$I_{j}(f)\leq I_{j}(\sum_{k=0}^{m-1}f_{n}^{C_{k}}(t_{k+1}-t_{k}))\leq
I_{j}(f)+2\varepsilon I_{j}(1)\text{\quad for }j\in\\{1,2\\}.$

whence

(5.5)
$\left\|I_{j}(\sum_{k=0}^{m-1}f_{n}^{C_{k}}(t_{k+1}-t_{k}))-I_{j}(f)\right\|\leq
2\varepsilon\left\|I_{j}(1)\right\|\text{\quad for }j\in\\{1,2\\}.$

Therefore

$\displaystyle\left\|\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t-I(f)\right\|$
$\displaystyle\leq\left\|\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t-\sum_{k=0}^{m-1}I(f_{n}^{C_{k}})(t_{k+1}-t_{k})\right\|$
$\displaystyle+\left\|\sum_{k=0}^{m-1}I(f_{n}^{C_{k}})(t_{k+1}-t_{k})-I(f)\right\|$
$\displaystyle\overset{\text{see
}(\ref{cond2})}{\leq}\left\|\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t-\sum_{k=0}^{m-1}I(f_{n}^{C_{k}})(t_{k+1}-t_{k})\right\|$
$\displaystyle+2\varepsilon\left\|I_{1}(1)\right\|+2\varepsilon\left\|I_{2}(1)\right\|$
$\displaystyle\leq\left\|\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t-\sum_{k=0}^{m-1}\boldsymbol{\mu}\left(C_{k}\right)(t_{k+1}-t_{k})\right\|$
$\displaystyle+\left\|\sum_{k=0}^{n-1}\boldsymbol{\mu}\left(C_{k}\right)(t_{k+1}-t_{k})-\sum_{k=0}^{n-1}I(f_{n}^{C_{k}})(t_{k+1}-t_{k})\right\|$
$\displaystyle+2\varepsilon\left\|I_{1}(1)\right\|+2\varepsilon\left\|I_{2}(1)\right\|$
$\displaystyle\overset{\text{see
}(\ref{cond0})\&(\ref{cond1})}{\leq}2\varepsilon\left(1+\left\|I_{1}(1)\right\|+\left\|I_{2}(1)\right\|\right).$

Since $\varepsilon>0$ was arbitrarily fixed, the above reasoning yields
formula (5.2) in the case of positive functions.

If $f\notin C(X)_{+},$ then $f+\left\|f\right\|\in C(X)_{+~{}}$ and by the
preceding considerations we have

$I(f)+\left\|f\right\|I(1)=I(f+\left\|f\right\|)=\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)+\left\|f\right\|\geq
t\\}\right)\mathrm{d}t\\\
=\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t+\int_{-\left\|f\right\|}^{0}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t\\\
=\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t+\int_{-\left\|f\right\|}^{0}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)-\boldsymbol{\mu}(X)\mathrm{d}t+\left\|f\right\|\boldsymbol{\mu}(X)\\\
=\int_{0}^{+\infty}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)\mathrm{d}t+\int_{-\infty}^{0}\boldsymbol{\mu}\left(\\{x:f(x)\geq
t\\}\right)-\boldsymbol{\mu}(X)\mathrm{d}t+\left\|f\right\|I(1).$

The proof of the representation formula (5.2) is now complete.

As concerns the uniqueness of $\boldsymbol{\mu}$, suppose that
$\boldsymbol{\nu}$ is another upper continuous monotone set function with
bounded variations for which the formula (5.2) holds. Given a set
$K=\left\\{x:f(x)\geq t\right\\}\in\Sigma_{up}^{+}(X),$ it is known that the
functions $f_{n}^{K}:X\rightarrow[0,1]$ defined by the formula (4.4) decrease
to$\chi_{K}.$ This implies that $(\left\\{x:f_{n}^{K}(x)\geq t\right\\})_{n}$
is decreasing to $K$ for each $t\in[0,1].$ Consider the sequence of functions

$\varphi_{n}:[0,1]\rightarrow
E,\text{\quad}\varphi_{n}(t)=\boldsymbol{\nu}(\left\\{x:f_{n}^{K}(x)\geq
t\right\\}).$

Notice that all these functions have bounded variation and their variation is
bounded by the variation of $|\boldsymbol{\nu}|(X).$

By Lebesgue dominated convergence,

$\lim_{n\rightarrow\infty}\int_{0}^{1}\varphi_{n}(t)dt=\boldsymbol{\nu}(K).$

On ther hand, by (5.2) and the definition of $\boldsymbol{\mu},$

$\boldsymbol{\mu}(K)=\lim_{n\rightarrow\infty}I(f_{n}^{K})=\lim_{n\rightarrow\infty}\int_{0}^{1}\varphi_{n}(t)dt=\boldsymbol{\nu}(K),$

which ends the proof of the uniqueness. ∎

##  6\. Open problems

We end our paper by mentioning few open problems that might be of interest to
our readers.

######  Problem 1.

Is the order continuity of the norm of $E$ a necessary condition for the
validity of Theorems 3 and 4?

As was noticed by Bartle, Dunford and Schwartz [2], much of the theory of
weakly compact linear operators defined on a space $C(X)$ is dominated by the
concept of absolute continuity. For more recent contributions see [20], [21]
and [22].

Suppose that $\mathcal{A}$ is a $\sigma$-algebra and $E$ is a Banach latttice
with order continuous norm.

A vector capacity $\boldsymbol{\mu}:\mathcal{A}\rightarrow E_{+}$ is called
absolutely continuous with respect to a capacity
$\lambda:\mathcal{A}\rightarrow[0,\infty)$ (denoted $\mu\ll\lambda)$ if for
every $\varepsilon>0$ there is $\delta>0$ such that

$A\in\mathcal{A},\text{
}\lambda(A)<\delta\Longrightarrow\left\|\boldsymbol{\mu}(A)\right\|<\varepsilon.$

The following lemma extends a result proved by Pettis in the context of
$\sigma$-additive measures.

######  Lemma 14.

If $\mu$ is upper continuous and $\lambda$ is upper continuous and
supermodular then the condition $\mu\ll\lambda$ is equivalent to the following
one:

$A\in\mathcal{A}\text{ and }\lambda(A)=0\Longrightarrow\boldsymbol{\mu}(A)=0.$

The proof is immediate, by reductio ad absurdum.

######  Problem 2.

Suppose $\boldsymbol{\mu}:\mathcal{A}\rightarrow E_{+}$ is an upper continuous
vector measure. Does there exist a capacity
$\lambda:\mathcal{A}\rightarrow[0,\infty)$ such that $\mu\ll\lambda?$

If Yes, is it possible to choose $\lambda$ of the form
$\lambda=x^{\ast}\circ\boldsymbol{\mu}$ for a suitable $x^{\ast}\in
E_{+}^{\ast}?$

It would be also interesting the following operator analogue of Problem 2:

######  Problem 3.

Does there exist for each Choquet operator $I:C(X)\rightarrow E$ a functional
$x^{\ast}\in E_{+}^{\ast}$ such that for every $\varepsilon>0$ there is
$\delta>0$ with the property

$\left\|I(f)\right\|\leq\varepsilon\left\|f\right\|+\delta
x^{\ast}\left(\left|f\right|\right)\text{\quad for all }f\in C(X)?$

Declaration of interests : none.

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††thanks: Essay has been awarded an Honorable Mention in the Gravity Essay
Competition 2019 with the Essay published in a Special issue of
IJMPD.††thanks: Corresponding author

# IR Quantum Gravity solves naturally cosmic acceleration and its coincidence
problem

Fotios K. Anagnostopoulos<EMAIL_ADDRESS>National and
Kapodistrian University of Athens, Physics Department, Panepistimioupoli
Zografou, 15772, Athens, Greece  Georgios Kofinas<EMAIL_ADDRESS>Research
Group of Geometry, Dynamical Systems and Cosmology,
Department of Information and Communication Systems Engineering,
University of the Aegean, Karlovassi 83200, Samos, Greece  Vasilios Zarikas
<EMAIL_ADDRESS>School of Engineering, Nazarbayev University, Nur-Sultan
(Astana), Kazakhstan  General Department, University of Thessaly, Lamia,
Greece

###### Abstract

The novel idea is that the undergoing accelerated expansion of the universe
happens due to infrared quantum gravity modifications at intermediate
astrophysical scales of galaxies or galaxy clusters, within the framework of
Asymptotically Safe gravity. The reason is that structures of matter are
associated with a scale-dependent positive cosmological constant of quantum
origin. In this context no extra unproven energy scales or fine-tuning are
used. Furthermore, this model was confronted with the most recent
observational data from a variety of probes, and with aid of Bayesian
analysis, the most probable values of the free parameters were extracted.
Finally, the model proved to be statistically equivalent with $\Lambda$CDM,
and thus being able to resolve naturally the concept of dark energy and its
associated cosmic coincidence problem.

##  I Introduction

The idea that Quantum Gravity effects can be important at astrophysical and
cosmological distances has recently attracted much attention. In particular
the framework of Exact Renormalization Group (RG) approach for quantum gravity
[1], also found in the literature under the names Asymptotic Safety (AS) or
Quantum Einstein Gravity, has opened the possibility of investigating both the
ultraviolet (UV) and the infrared (IR) sector of gravity in a systematic
manner.

The key element is the Effective Average Action $\Gamma_{k}[g_{\mu\nu}]$, a
Wilsonian coarse-grained free energy, which defines an effective field theory
appropriate for the momentum scale $k$. This $\Gamma_{k}$, evaluated at tree
level, describes appropriately all gravitational phenomena, including all loop
effects. The application to Einstein-Hilbert action generates RG flow
equations [2], which have made possible the consistent study of the scaling
behavior of Newton constant $G$ and cosmological constant $\Lambda$ at high
energies [3], [4]. The initial idea was first demonstrated by Weinberg [5],
where he suggested that a pertubatively divergent theory could be consistently
defined in four dimensions at a nontrivial UV fixed point with the
dimensionless $g(k)=G(k)k^{2}$ non-vanishing in the $k\rightarrow\infty$
limit. In the framework of AS, one should also include $\Lambda$, which
becomes energy dependent and receives quantum contribution form vacuum
fluctuations.

Recent works have also considered matter fields or a growing number of purely
gravitational operators in the action. In particular, truncations involving
quadratic terms in the curvature or higher powers of the Ricci scalar have
been studied [6], [7]. In all the investigations the UV critical surface has
turned out to be finite dimensional ($d_{\rm UV}=3$), implying that the theory
is nonperturbatively renormalizable.

A weakly coupled gravity at high energies is expected to generate important
consequences in several astrophysical and cosmological contexts and in fact
the RG flow of $\Gamma_{k}$, obtained by different truncations of theory
space, has been the basis of various investigations of “RG improved” black
hole spacetimes [8], [9] and early Universe models [10]-[12].

##  II IR point from AS approach

The behavior of AS theory is more complicated at low energies, corresponding
at cosmological or astrophysical scales. The problem arises because the
$\beta$-functions of any local operator of the type $\sqrt{g}R^{n}$ are
singular in the IR due to the presence of a pole at
$\lambda(k)=\Lambda(k)/k^{2}=1/2$. This pole indicates that the Einstein-
Hilbert truncation is not trustable approximation and new relevant operators
emerge in the $k\rightarrow 0$ limit. It was shown in [13] that the dynamical
origin of these strong IR effects is due to the “instability driven
renormalization”, a phenomenon well-known from other physical systems
[14]-[16]. Fortunately, the low energy domain of the theory is regulated by an
IR fixed point which drives the cosmological constant to zero. Out of that,
some first astrophysical consequences and a possible explanation for the
galaxy rotation curves without dark matter appeared [13], [17], [18], although
a detailed analysis based on available experimental data is still missing.

Very recently, authors in [19], [20] showed that the coincidence problem of
cosmic acceleration can be explained naturally and without introducing new
energy scales or fine-tuning in the context of a Swiss cheese cosmological
model which utilizes this IR behavior of AS gravity. The present essay focuses
on this discovery. The importance of these works lies on the fact that no
special “dark energy” field is required and the explanation uses only the
concrete well motivated framework of AS.

More precisely, in the proposed scenario, including all the components of the
metric fluctuation $h_{\mu\nu}$ [4], quantum effects dynamically drive
$\Lambda$ along the RG trajectories generated by the unstable infrared modes
of the gravitational sector. Close to the IR fixed point, $\Lambda$ runs
proportional to $k^{2}$ to avoid the singularity,

$\frac{\Lambda(k)}{k^{2}}=\lambda^{\mathrm{IR}}_{*}+h_{2}k^{2\theta}\,\,,\;\;\;\;k\rightarrow
0\,,$ (1)

where $\lambda^{\mathrm{IR}}_{*}<1/2$ is an infrared fixed point of the
$\lambda$-evolution, $h_{2}$ is a constant related to the eigenvalues of the
stability matrix, and $\theta>0$ is a critical exponent which parametrizes the
subleading terms and ensures that the fixed point is attractive. According to
(1), the renormalized $\Lambda$ vanishes at very large (cosmological)
distances, $\Lambda(k\to 0)=0$, regardless of its bare value. Furthermore,
recent investigations based on a conformal reduction of Einstein gravity
discovered a new IR fixed point suggesting the existence of the counterpart of
the physical IR fixed point present in the full theory [21].

##  III Cosmology by matching local astrophysical metrics with global
cosmological metric

In [19], the recent cosmic acceleration naturally emanated from the recent
formation of structure. A Swiss cheese (Einstein-Strauss) model was
constructed to derive the cosmology. The interior static spherically symmetric
metric, modeling a galaxy or a galaxy cluster, matches smoothly to a
cosmological exterior across a spherical boundary. A quantum improved
Schwarzschild-de Sitter (SdS) interior metric was used, which contains the
appropriate antigravity effect. This model uses dimensionless order one
parameters of AS, the conventional Newton constant $G_{\\!N}$ and the
astrophysical length scale. It provides a recent passage to the sufficient
acceleration, while the freedom of the order one parameters has to be
constrained by observational data. To the best of our knowledge, this is the
first solution of the dark energy problem without using fine-tuning or
introducing add-hoc energy scales.

The exterior FRW metric is

$ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-\kappa
r^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}\\!\theta\,d\varphi^{2}\right)\right]$
(2)

with $a(t)$ the scale factor. The interior metric has the form

$ds^{2}=-\Big{(}1-\frac{2G_{k}M}{R}-\frac{1}{3}\Lambda_{k}R^{2}\Big{)}dT^{2}+\frac{dR^{2}}{1-\frac{2G_{k}M}{R}-\frac{1}{3}\Lambda_{k}R^{2}}+R^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)$ (3)

with the functions $G_{k}=G(k),\Lambda_{k}=\Lambda(k)$ determined by AS. The
matching of the two patches occurs at the boundary with constant
$r=r_{\Sigma}$, which at the same time experiences the universal expansion.
Since astrophysical structures are still large compared to the cosmological
scales, $\Lambda(k)$ is expected to differ slightly from its IR form $k^{2}$,
so a power law $\Lambda_{k}=\gamma k^{b}$, with $\gamma>0$ and $b$ close to
the value 2, is a fair approximation of the running behavior (1).
Additionally, it is set $G_{k}=G_{N}$ at observable macroscopic distances, in
agreement with standard Newton law.

The energy scale $k$ is expected to be associated with a characteristic length
scale $L$, $k=\xi/L$, where $\xi$ is an order one dimensionless number. In the
Swiss cheese approach, only the value $R_{S}=ar_{\Sigma}$ of the Schucking
radius (or $k_{S}$) enters the cosmic evolution. Although a simple option is
$L=R$, a more natural one is to set as $L$ the proper distance

$D(R)=\int^{R}\\!d\mathcal{R}\,\Big{(}1-\frac{2G_{N}M}{\mathcal{R}}-\frac{1}{3}\Lambda_{k}\mathcal{R}^{2}\Big{)}^{-1/2}\,.$
(4)

The new cosmological constant term is
$\frac{1}{3}\Lambda_{k}R^{2}\sim\frac{1}{G_{\\!N}}\big{(}\frac{\sqrt{G_{\\!N}}}{r_{\Sigma}}\big{)}^{\\!b}\,\big{(}\frac{r_{\Sigma}}{D}\big{)}^{\\!b}R^{2}$,
where $\tilde{\gamma}=\gamma G_{\\!N}^{1-\frac{b}{2}}$ a dimensionless order
one number. For $b$ close to the value 2.1, the quantity
$\frac{1}{G_{\\!N}}\big{(}\frac{\sqrt{G_{\\!N}}}{r_{\Sigma}}\big{)}^{\\!b}$ is
very close to the order of magnitude of the standard cosmological constant
$\Lambda\simeq 4.7\times 10^{-84}\text{GeV}^{2}$ of the concordance
$\Lambda$CDM model, while the factor
$\big{(}\frac{r_{\Sigma}}{D}\big{)}^{\\!b}$ contributes only a small distance-
dependent deformation since the today value of $D_{S}$ should be of order
$r_{\Sigma}$ in order to have the correct amount of dark energy. Therefore,
the hard coincidence of the standard $\Lambda\sim H_{0}^{2}$, has been
exchanged with a mild adjustment of the index $b$ close to 2.1.

The Hubble evolution in terms of the redshift $z$ arises by the integration of
the cosmological equations of the model which consider the Israel-Darmois
matching conditions on the boundary radius,

$\frac{H^{2}(z)}{H_{0}^{2}}=\Omega_{m0}(1\\!+\\!z)^{3}+\Big{[}\Omega_{DE0}^{-\frac{1}{b}}-\frac{3^{\frac{1}{b}}}{\xi\tilde{\gamma}^{\frac{1}{b}}}(G_{\\!N}H_{0}^{2})^{\frac{1}{b}}\frac{r_{\Sigma}a_{0}}{\sqrt{G_{\\!N}}}\,\frac{z}{1\\!+\\!z}\Big{]}^{-b}+\Omega_{\kappa
0}(1\\!+\\!z)^{2}\,.$ (5)

In addition, a preliminary analysis has shown that there is no obvious
contradiction between the model discussed and the internal dynamics of the
astrophysical object. The potential and the force due to the varying
cosmological constant term are small percentages of the corresponding
Newtonian potential and force, where the precise values depend on the
considered structure and the considered point at the boundary of the object or
inside.

##  IV Thorough test of the model using observational data

No matter what are the theoretical merits of a given model, it needs to be
confronted with the observational data. In [20], authors used the most recent
observational data sets, namely direct measurements of the Hubble rate $H(z)$
[22], Supernovae Ia (Pantheon data set [23]), Quasi-Stellar-Objects (QSO),
Baryonic Acoustic Oscillations and direct measurements of the CMB shift
parameters, to constraint the free parameters of the AS cosmological model
[19]. It was found that this model is very efficient and in excellent
agreement with observations. The energy density of matter that was calculated
is compatible with the same quantity imposed by concordance cosmology from the
CMB angular power spectrum. Another result is that the AS model supports a
lower value of Hubble constant than the value derived from Cepheids, so the
best fit value for $H_{0}$ is closer to the Planck value.

In addition, after reconstructing the effective dark energy equation-of-state
parameter $w_{DE}(z)$ using the derived values of the free parameters, it was
found that the today’s value $w_{DE0}$ is close to $w=-1$. Both the transition
redshift and the current value of the deceleration parameter are in perfect
agreement with the corresponding values calculated with model-independent
techniques.

Figure 1: The evolution of $w_{DE}(z)$ with the corresponding $1\sigma$ and $2\sigma$ uncertainties. In the upper panel we use the best fit values from the combination $H(z)$/Pantheon/QSO, while in the lower panel we utilize those of $H(z)$/Pantheon/QSO/CMBshift. The dashed line corresponds to $\Lambda$CDM value $w=-1$. Model | $\Omega_{m0}$ | $h$ | $\Omega_{b0}h^{2}$ | $\chi_{\text{min}}^{2}$ | ${\rm AIC}$ |  $\Delta$AIC
---|---|---|---|---|---|---
_$H(z)$ /Pantheon/QSO_
AS model | $0.270_{-0.018}^{+0.019}$ | $0.684_{-0.012}^{+0.013}$ | - | $84.463$ | 92.889 | 0.937
$\Lambda$CDM | $0.281^{+0.016}_{-0.015}$ | $0.686\pm 0.013$ | - | 85.700 | 91.952 | 0
_$H(z)$ /Pantheon/QSO/CMB shift_
AS model | $0.303\pm 0.001$ | $0.685\pm 0.009$ | $0.0223\pm 0.0002$ | 89.774 | 100.419 | 1.653
$\Lambda$CDM | $0.307^{+0.008}_{-0.007}$ | $0.679\pm 0.006$ | $0.0223\pm 0.0001$ | 90.340 | 98.766 | 0
Table 1: The observational constraints on the density parameters
$\Omega_{m0}$, $\Omega_{b0}$ and the statistical criteria values
$\chi^{2}_{\rm min}$, AIC, $\Delta{\rm AIC}$ for the AS cosmology and
$\Lambda$CDM.

Finally, the comparison of $\Lambda$CDM model with the considered AS model in
terms of the fitting properties, using a variety of information criteria,
revealed that the AS model is statistically equivalent with that of
$\Lambda$CDM. This is a significant conclusion since the AS model, unlike the
majority of the cosmological models, does not include new fields in nature or
has fine-tuning problems. Therefore, it must be considered as a viable and
efficient alternative cosmological scenario towards explaining the recent
accelerated expansion of the universe. Interestingly, the two models are
expected to have differences at the perturbation level.

###### Acknowledgements.

G. Kofinas and V. Zarikas acknowledge the support of Orau Grant Number
110119FD4534, “Quantum gravity at astrophysical scales”.

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