# On continuity of state-dependent utilities

Edoardo Berton<EMAIL_ADDRESS>Alessandro Doldi  Marco Maggis

###### Abstract

State-dependent preferences for a general Savage’s state space were shown in
Wakker and Zank (1999) to admit a numerical representation in the form of the
integral of a state-dependent utility, as soon as pointwise continuity of the
preference ordering is assumed. In this paper we prove that such a state-
dependent function inherits pointwise continuity from the preference ordering,
providing in this way a positive answer to a conjecture posed in the
aforementioned seminal work.
We further apply this result to obtain an explicit representation of
conditional Chisini means in the form of a conditional certainty equivalent.

###### keywords:

pointwise continuity , state-dependent expected utility , conditional
certainty equivalent , nonlinear expectation

###### MSC:

[2020] 91B06 , 91B08 , 60A05

††journal: Journal of Mathematical Economics

[inst2]organization=Università Cattolica del Sacro Cuore,addressline=Largo
Gemelli 1, city=Milano, postcode=20123, country=Italy

[inst3]organization=Università degli Studi di Firenze,addressline=Via delle
Pandette 9, city=Firenze, postcode=50127, country=Italy

[inst1]organization=Università degli Studi di Milano,addressline=Via Saldini
50, city=Milano, postcode=20127, country=Italy

##  1 Introduction

For a general state space $(\Omega,{\mathcal{F}})$, Wakker and Zank [16]
provide a characterization of those preference relations $\succeq$ on the
space of bounded random acts which admit a numerical representation through an
additively decomposable functional ([16], Theorem 11). This result extends to
an infinite dimensional setup the theory developed by Debreu [8]. By adopting
the stronger notion of pointwise continuity of $\succeq$, Theorem 11 in [16]
can be specialized to the following integral representation form.

######  Theorem 1.1 ([16], Theorem 12 and [4], Theorem 5)

Assume that ${\mathcal{F}}$ contains at least three disjoint non-null events.
Then the following two statements are equivalent:

  1. (i)

There exists a countably additive finite measure $\mu$ on $\Omega$ and a
function $\mathbb{u}(\omega,\cdot):\mathbb{R}\to\mathbb{R}$ strictly
increasing for every $\omega\in\Omega$, such that $\succeq$ is represented by
the pointwise continuous integral

$f\to\int_{\Omega}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mu(\omega).$

  2. (ii)

$\succeq$ is a preference order satisfying strict monotonicity, the sure-thing
principle and pointwise continuity 111See page (SM) below for further details
on the preference axioms involved.

The previous theorem leaves an interesting issue unsolved as declared by the
authors [16], page 29:

“[…] Pointwise continuity is not overly restrictive because continuity of each
state-dependent utility implies pointwise continuity. Whether the reversed
implication holds, i.e., whether continuity of the functions $\mathbb{u}$, in
Statement (i) of Theorem 12 holds (after appropriate modification on null
events), is an open question to us. ”

To the best of our knowledge this question did not find an answer in the
subsequent literature. Among the principal aims of the present paper, Theorem
3.1 stated below provides the construction of a pointwise continuous version
of the state dependent utility $\mathbb{u}$. Such a result has the immediate
benefit of clarifying the role of continuity in the application of the Theorem
1.1. For example, in Appendix A of [13] a somewhat cumbersome notion of
continuity is introduced to overcome some technical hurdles in the proof of
the main result. Guaranteeing the existence of a continuous $\mathbb{u}$
allows to circumvent issues of this kind.
As an important byproduct, we provide a representation of the conditional
Chisini mean $\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)$ previously
introduced and studied in [9]. The conditional Chisini mean is a nonlinear
extension of the standard concept of conditional expectation, which takes
inspiration from the original Chisini’s intuition [6] that the notion of
“expected value” should be introduced as the solution of a functional
equation. For a sub-$\sigma$-algebra ${\mathcal{G}}$ of ${\mathcal{F}}$,
representing intermediate information available, we show that
$\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)=\mathbb{u}_{\mathcal{G}}^{-1}(\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(f)\middle|{\mathcal{G}}\right])$,
where $\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(f)\middle|{\mathcal{G}}\right]$
is the conditional expectation of the random outcome
$\omega\mapsto\mathbb{u}(\omega,f(\omega))$ and
$\mathbb{u}_{\mathcal{G}}(\omega,x)=\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(x)\middle|{\mathcal{G}}\right](\omega)$
is the ${\mathcal{G}}$-measurable projection of the state dependent utility
$\mathbb{u}$. This representation can be easily compared to the general
formulation proved in [5], Lemma 5.2, under additional law invariance
assumptions, which traces back to the celebrated Nagumo, de Finetti,
Kolmogorov Theorem (see [11] for a review). It is also worth mentioning that
in this case, $\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)$ takes the form
of a conditional certainty equivalent based on state-dependent utilities (see
[10]).

##  2 Notations and preliminaries

Throughout this paper $(\Omega,{\mathcal{F}})$ denotes a measurable space,
with $\Omega$ being a general state space and ${\mathcal{F}}$ a
$\sigma$-algebra of subsets of $\Omega$, which we refer to as events. We shall
denote by $\mathbb{R}$ the set of real numbers, which will always be endowed
with the Borel $\sigma$-algebra $\mathbb{B}_{\mathbb{R}}$. By
$\overline{\mathbb{R}}$ we indicate the extended real line
$\left[-\infty,+\infty\right]$ and with $\mathbb{B}_{\overline{\mathbb{R}}}$
the associated Borel $\sigma$-algebra. We use $<,>$ for strict inequalities
between real numbers. Similarly, $\subset,\supset$ are used to indicate proper
subsets. In case equalities are not excluded, we use $\leq,\geq$ and
$\subseteq,\supseteq$, respectively. We consider, as conventional,
$\inf\varnothing=+\infty$. We denote by $\mathcal{L}(\Omega,{\mathcal{F}})$
the space of ${\mathcal{F}}$-measurable functions taking values in
$\mathbb{R}$. We call random variables the elements
$f\in\mathcal{L}(\Omega,{\mathcal{F}})$ and we denote with
$\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ the collection of elements $f$ of
${\mathcal{L}}(\Omega,{\mathcal{F}})$ such that $|f(\omega)|\leq k$ for every
$\omega\in\Omega$ and some $k\geq 0$. We consider on
$\mathcal{L}(\Omega,{\mathcal{F}})$ and
$\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ the natural pointwise order $f\geq
g$ if and only if $f(\omega)\geq g(\omega)$ for all $\omega\in\Omega$. The
space $\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ is a Banach space with
respect to the usual sup norm
$\|f\|_{\infty}=\inf\\{k\in\mathbb{R}\mid\,|f(\omega)|\leq
k\;\forall\,\omega\in\Omega\\}$. Given any $A\in{\mathcal{F}}$, we define
$\mathbf{1}_{A}$ as the element of $\mathcal{L}^{\infty}(\Omega,\mathcal{F})$
that takes value 1 if $\omega\in A$ and 0 otherwise. For any couple of random
variables $f,g\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ we denote with
$f\mathbf{1}_{A}+g\mathbf{1}_{A^{c}}$ the random variable that agrees with $f$
whenever the event $A$ occurs and with $g$ otherwise.
Whenever working with the product space $\Omega\times\mathbb{R}$, we shall
make use of the product $\sigma$-algebra
${\mathcal{F}}\otimes\mathbb{B}_{\mathbb{R}}$ i.e. the $\sigma$-algebra
generated by elements of the form $F\times B$ with $F\in{\mathcal{F}}$ and
$B\in\mathbb{B}_{\mathbb{R}}$.

Given a probability measure $\mathbb{P}$, $(\Omega,\mathcal{F},\mathbb{P})$ is
a probability space. A (finite) signed measure $\nu$ on
$(\Omega,{\mathcal{F}})$ is said to be dominated by $\mathbb{P}$
($\nu\ll\mathbb{P}$) if, for any $A\in{\mathcal{F}}$, $\mathbb{P}(A)=0$
implies $\lvert\nu\rvert(A)=0$, where $\lvert\nu\rvert$ is the total variation
measure of $\nu$. By the Hahn decomposition theorem, $\nu\ll\mathbb{P}$ is
equivalent to: $\mathbb{P}(A)=0$ implies ${\nu}(A)=0$.
For a given probability $\mathbb{P}$ on $(\Omega,{\mathcal{F}})$ we shall
denote the space of integrable functions by
${\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})=\left\\{f\in{\mathcal{L}}(\Omega,{\mathcal{F}}):\mathbb{E}_{\mathbb{P}}\left[|f|\right]<\infty\right\\}$.
For any $f\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ and a sub
$\sigma$-algebra ${\mathcal{G}}$,
$\mathbb{E}_{\mathbb{P}}\left[f\mid{\mathcal{G}}\right]$ represents the
conditional expectation i.e.

$\mathbb{E}_{\mathbb{P}}\left[f\middle|{\mathcal{G}}\right]=\\{g\in{\mathcal{L}}^{1}(\Omega,{\mathcal{G}},\mathbb{P}):\mathbb{E}_{\mathbb{P}}\left[g\mathbf{1}_{A}\right]=\mathbb{E}_{\mathbb{P}}\left[f\mathbf{1}_{A}\right]\;\forall\,A\in{\mathcal{G}}\\}.$
(1)

In the following we shall assume familiarity with the properties of
conditional expectation. We refer the reader to [17], page 88, for a
comprehensive review of such properties.

A preference relation is a binary relation $\succeq$ on
$\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ which satisfies completeness and
transitivity and induces a (strict) preference order $\succ$ by $f\succ g$ if
and only if $f\succeq g$ but $g\nsucceq f$. In addition we set $g\sim f$
whenever $g\succeq f$ and $f\succeq g$. We denote

${\mathcal{N}}_{\succeq}:=\left\\{A\in{\mathcal{F}}:f\mathbf{1}_{A}+g\mathbf{1}_{A^{c}}\sim
g\;\;\forall f,g\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})\right\\}$ (2)

the set of null events (induced by the preference $\succeq$).
We present a few additional properties that the preference relation might
enjoy.

(SM)

$\succeq$ is strictly monotone if $x\mathbf{1}_{A}+f\mathbf{1}_{A^{c}}\succ
y\mathbf{1}_{A}+f\mathbf{1}_{A^{c}}$, for all non null events
$A\in{\mathcal{F}}$, for all $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$
and constant outcomes $x,y\in\mathbb{R}$ with $x>y$;

(ST)

$\succeq$ satisfies the sure-thing principle whenever, considering arbitrary
$f,g,h\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ and $A\in{\mathcal{F}}$
such that $f\mathbf{1}_{A}+h\mathbf{1}_{A^{c}}\succeq
g\mathbf{1}_{A}+h\mathbf{1}_{A^{c}}$, for every
$\tilde{h}\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ we have
$f\mathbf{1}_{A}+\tilde{h}\mathbf{1}_{A^{c}}\succeq
g\mathbf{1}_{A}+\tilde{h}\mathbf{1}_{A^{c}}$;

(PC)

$\succeq$ is pointwise continuous if, for any uniformly bounded sequence
$(f_{n})_{n}\subseteq\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, such that
$f_{n}(\omega)\rightarrow f(\omega)$ for any $\omega\in\Omega$ then for all
$g\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ such that $g\succ f$ (resp.
$f\succ g$) there exists $N\in\mathbb{N}$ such that $g\succ f_{n}$ (resp.
$f_{n}\succ g$) for every $n>N$.

Assumptions (SM), (ST) and (PC) represent the minimal framework we will be
working with throughout the paper. We recall that, provided ${\mathcal{F}}$
contains at least three distinct non-null events, by Theorem 1.1 any
preference ordering satisfying these assumptions admits an integral
representation.

##  3 Statement of the main results

We are now ready to state the two main results of this paper. The first one
precisely answers to the question posed by [16] as announced in the
introduction and from this perspective it provides an important addendum to
Theorem 1.1.

###### Theorem 3.1

Assume that ${\mathcal{F}}$ contains at least three disjoint non-null events
and $\succeq$ satisfies (SM), (PC) and (ST). Then there exist:

  1. (i)

a probability measure $\mathbb{P}$ on ${\mathcal{F}}$,

  2. (ii)

a ${\mathcal{F}}\otimes\mathbb{B}_{\mathbb{R}}$ measurable function
$\mathbb{u}:\Omega\times\mathbb{R}\to\mathbb{R}$ with
$\mathbb{u}(\omega,\cdot)$ being a continuous and strictly increasing function
on $\mathbb{R}$ for every $\omega\in\Omega$ and
$\mathbb{u}(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ for
every $x\in\mathbb{R}$,

such that given $f,g\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ we have

$f\succeq g\text{ if and only if
}\int_{\Omega}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)\geq\int_{\Omega}\mathbb{u}(\omega,g(\omega))\mathrm{d}\mathbb{P}(\omega)$
(3)

In our second result we apply Theorem 3.1 to obtain an explicit formula for
conditional Chisini means. Given a functional
$T:\mathcal{L}^{\infty}(\Omega,\mathcal{F})\rightarrow\mathbb{R}$ and a random
variable $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, [9] proposes
sufficient conditions to guarantee the existence of a solution
$g\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})$ to the infinite dimensional
system of equations $T(f\mathbf{1}_{A})=T(g\mathbf{1}_{A})$ for any
$A\in{\mathcal{G}}$. The family of solutions is the so-called conditional
Chisini mean, denoted by $\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)$. If
compared with (1), it evidently appears that
$\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)$ is the nonlinear counterpart
of the standard conditional expectation. It is known that the problem of
existence of nonlinear conditional expectation has been addressed also in [7],
yet the provided solution relies on a Brownian framework which in turn demands
for more technical assumptions.
Embedding such a problem in a decision theoretical setup, we could rephrase
the issue as follows: given a preference order $\succeq$ on
$\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ then the certainty equivalent of
$f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ is $a\in\mathbb{R}$ such that
$a\sim f$. Similarly if the agent is provided with additional information,
represented by the $\sigma$-algebra ${\mathcal{G}}\subset{\mathcal{F}}$, then
the conditional certainty equivalent of
$f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ is any
$g\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})$ such that

$f\mathbf{1}_{A}\sim g\mathbf{1}_{A}\quad\forall\;A\in{\mathcal{G}}.$ (4)

Loosely speaking, $g\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})$ can be
considered as a ${\mathcal{G}}$-measurable counterpart of the
${\mathcal{F}}$-measurable function
$f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ (see also [10] and [13] for
applications of the conditional certainty equivalent).
Since this paper deals with state dependent preferences we shall directly
refer to the conditional Chisini mean as the class

$\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)=\left\\{g\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})\mid
f\mathbf{1}_{A}\sim g\mathbf{1}_{A}\;\forall\,A\in{\mathcal{G}}\right\\},$ (5)

keeping in mind that Eq. (5) translates automatically to the framework in [9]
as soon as we characterize preferences $\succeq$ through a representing
functional $T$222$f\succeq g$ if and only if $T(f)\geq T(g)$.

###### Theorem 3.2

Assume that the hypotheses of Theorem 3.1 are satisfied and consider
$(\mathbb{u},\mathbb{P})$ as in Eq. (3). Then, for any given a sub-sigma
algebra ${\mathcal{G}}\subseteq{\mathcal{F}}$,
$\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)\neq\varnothing$ and there
exist

  * 1.

a function $\mathbb{u}_{\mathcal{G}}:\Omega\times\mathbb{R}\to\mathbb{R}$,
with

    1. (i)

$\mathbb{u}_{\mathcal{G}}(\omega,\cdot)$ continuous and strictly increasing on
$\mathbb{R}$ for every $\omega\in\Omega$,

    2. (ii)

$\mathbb{u}_{\mathcal{G}}(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{G}},\mathbb{P})$
for every $x\in\mathbb{R}$,

    3. (iii)

$\mathbb{u}_{\mathcal{G}}(\cdot,x)$ is a version of
$\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,x)\middle|{\mathcal{G}}\right]$
for every $x\in\mathbb{R}$.

  * 2.

a version $g$ of
$\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,f(\cdot))\middle|{\mathcal{G}}\right]$
such that

$\Phi_{\mathcal{G}}(\cdot,g(\cdot))\in\mathfrak{m}\left(f\middle|{\mathcal{G}}\right),$
(6)

where $\Phi_{\mathcal{G}}:\Omega\times\mathbb{R}\rightarrow[-\infty,+\infty]$
defined by

$\Phi_{\mathcal{G}}(\omega,x):=\inf\\{y\in\mathbb{R}\mid\mathbb{u}_{\mathcal{G}}(\omega,y)>x\\}$

is ${\mathcal{G}}\otimes\mathbb{B}_{\mathbb{R}}$ measurable.

We write informally (6) as

$\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)=\mathbb{u}_{\mathcal{G}}^{-1}(\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(f)\middle|{\mathcal{G}}\right]),$

and recall that, under the assumptions of Theorem 3.2,
$\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)$ is unique up to
$\succeq$-null events (see [9], Theorem 3.2).

To clarify the statement of Theorem 3.2, it seems important at this stage to
point out that the use of $\Phi_{\mathcal{G}}$ in (6) conceals some technical
subtleties. To begin with, the dependence of the image of
$\mathbb{u}_{\mathcal{G}}$ on $\omega$ requires cautious handling of its
domain. Additionally, for $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, the
nature of
$g\in\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(f)\middle|{\mathcal{G}}\right]$
implies that it may not be an element of
$\mathcal{L}^{\infty}(\Omega,\mathcal{G})$ itself. While the case of a
finitely generated $\sigma$-algebra is rather simple (as we shall depict in
Example 3.3 below), we shall dedicate a comprehensive treatment to the issue
for an arbitrary $\sigma$-algebra ${\mathcal{G}}$ in the proof postponed in
Section 5.

###### Example 3.3

Consider $\\{A_{1},...,A_{n}\\}$ being a finite partition of $\Omega$ such
that $\mathbb{P}(A_{i})>0$ for every $i\leq n$ and the $\sigma$-algebra
${\mathcal{G}}$ generated by this partition. Then every
${\mathcal{G}}$-measurable function $g$ can be written as a finite linear
combination of indicator functions of the atoms $A_{i}$. In particular, for
$\omega\in A_{i}$, $\mathbb{u}_{\mathcal{G}}(\omega,x)$ coincides with

$u_{i}(x)=\frac{1}{\mathbb{P}(A_{i})}\int_{A_{i}}\mathbb{u}(\omega,x)\mathrm{d}\mathbb{P}(\omega),$

$u_{i}$ being finite since
$\mathbb{u}(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$.
Clearly
$\mathbb{u}_{\mathcal{G}}(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{G}},\mathbb{P})$
for all $x\in\mathbb{R}$,
$\mathbb{u}_{\mathcal{G}}(\omega,x)=\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,x)|{\mathcal{G}}\right](\omega)$
and $x\mapsto\mathbb{u}_{\mathcal{G}}(\omega,x)$ is increasing and continuous
for all $\omega\in\Omega$.
For any fixed $A_{i}$ we notice that the image of
$\mathbb{u}_{\mathcal{G}}(\omega,\cdot)$, namely
$R_{i}:=\mathrm{Im}(\mathbb{u}_{\mathcal{G}}(\omega,\cdot))$ does not depend
on $\omega\in A_{i}$. Given the generalized inverse
$\Phi_{\mathcal{G}}:\Omega\times\mathbb{R}\to\mathbb{R}$ defined in Theorem
3.2, it is crucial to observe that the map
$x\mapsto\Phi_{\mathcal{G}}(\omega,x)=\mathbb{u}_{\mathcal{G}}^{-1}(\omega,x)$
is well defined as a standard inverse function for $\omega\in A_{i}$ and $x\in
R_{i}$ (as a result of strict monotonicity of
$\mathbb{u}_{\mathcal{G}}(\omega,\cdot)$).

In particular if we now choose any
$f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ and set
$h:=\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,f(\cdot))|{\mathcal{G}}\right]$
then $h(\omega)\in R_{i}$ for $\omega\in A_{i}$ and indeed

$(\mathbb{u}_{\mathcal{G}}\circ\Phi_{\mathcal{G}})(\omega,h(\omega))=h(\omega).$

For $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ denote
$\mathfrak{m}_{i}=u_{i}^{-1}\left(\frac{1}{\mathbb{P}(A_{i})}\int_{A_{i}}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)\right)$
so that for any $\omega\in A_{i}$ we have

$\mathbb{u}_{\mathcal{G}}(\omega,\mathfrak{m}_{i})=\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,f(\cdot))|{\mathcal{G}}\right](\omega).$

We conclude the example noting that the Chisini mean
$\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)$ for the functional
$T(f)=\int_{\Omega}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)$
consists, in the current setup, of a single random variable
$\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)$ which assumes the value

$u_{i}^{-1}\left(\frac{1}{\mathbb{P}(A_{i})}\int_{A_{i}}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)\right)$

on the set $A_{i}$. This can be readily verified following the procedure in
Section 5, step 4.

##  4 Proof of Theorem 3.1

This section is devoted to the proof of Theorem 3.1, which is quite involved
and requires some preliminary groundwork.
In the first part of the proof we lay the basis for the main argument by
rephrasing some known results from [16, 9], obtaining in particular a
functional that represents $\succeq$, which turns out to be additive on
measurable sets. We provide some auxiliary lemmata: Lemma 4.6 proves that it
is possible to construct an integrand function with the desired properties;
this is based on Lemma 4.4 which firstly ensures the suitable measurability by
a construction on rational numbers; Lemma 4.7 states that the integral
representation is well defined and enjoys the desired properties; finally
Lemma 4.8 shows continuity of $\mathbb{u}$. The concluding argument leads to
Proposition 4.1 and illustrates how ultimately Theorem 3.1 follows.

Consider a preference order $\succeq$ on
$\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ satisfying (SM), (PC) and (ST).
Suppose additionally that ${\mathcal{F}}$ contains at least three disjoint
events $A_{1},A_{2},A_{3}$ such that $A_{i}\notin{\mathcal{N}}_{\succeq}$ for
$i=1,2,3$. It is known (see [16], Theorem 11) that $\succeq$ admits a
numerical representation, i.e. there exists a functional
$T:\mathcal{L}^{\infty}(\Omega,\mathcal{F})\to\mathbb{R}$ such that for any
given $f,g\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$

$T(f)\geq T(g)\quad\text{ if and only if }\quad f\succeq g.$

In particular [16], Theorem 11, or [4], Theorem 4, imply that the Assumptions
in [9], Theorem 5.4333Adopting ${\mathcal{F}}$ in place of ${\mathcal{G}}$,
are met by the functional $T$ and we can therefore provide a further
refinement of the representing functional. Specifically one can find a
$V:{\mathcal{F}}\times\mathcal{L}^{\infty}(\Omega,\mathcal{F})\to\mathbb{R},\;(A,f)\mapsto
V_{A}(f)$ such that for
$f_{1},f_{2}\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ and
$A\in{\mathcal{F}}$ we have $f_{2}\mathbf{1}_{A}\succeq f_{1}\mathbf{1}_{A}$
if and only if $V_{A}(f_{1})\leq V_{A}(f_{2})$. Furthermore, $V$ can be taken
to satisfy the following properties (we refer the reader to the aforementioned
literature for further details):

  1. (P1)

for every $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ the map $A\mapsto
V_{A}(f)$ is a signed measure on $(\Omega,{\mathcal{F}})$ with $V_{A}(0)=0$
for every $A\in{\mathcal{F}}$. Moreover, $A\mapsto
V_{A}(\mathbf{1}_{\Omega})=:\mathbb{P}(A)$ is a probability measure on
$(\Omega,{\mathcal{F}})$;

  2. (P2)

for every $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, $A\in{\mathcal{F}}$
we have $V_{A}(f)=V_{\Omega}(f\mathbf{1}_{A})$;

  3. (P3)

the functional $V$ is strictly monotone in that for any
$f,g\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ such that $f\leq g$ then
$V_{A}(f)\leq V_{A}(g)$ for any $A\in{\mathcal{F}}$, and if $A$ is such that
$\mathbb{P}(A)>0$ then $V_{A}(x)<V_{A}(y)$ for any $x,y\in\mathbb{R}$ with
$x<y$;

  4. (P4)

for every $A\in{\mathcal{F}}$ the functional $f\mapsto V_{A}(f)$, defined on
$\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, is pointwise continuous, in that
for any uniformly bounded444Meaning $\sup_{n}\lVert
f_{n}\rVert_{\infty}<+\infty$ sequence
$(f_{n})_{n}\subset\mathcal{L}^{\infty}(\Omega,\mathcal{F})$,
$f_{n}(\omega)\to f(\omega)$ for any $\omega\in\Omega$, then $V_{A}(f_{n})\to
V_{A}(f)$.

The following proposition provides a general representation result which
stretches somewhat beyond the scope of the present work. A similar problem is
in fact addressed within some relevant contributions in the stream of
literature of [2] and [3, Ch. 2], under the name of Nonlinear Superposition
Operator and exploiting the theory of Carathéodory functions. Another
application of these representation results to subjective expected utility is
presented in [15].

###### Proposition 4.1

Let
$V:{\mathcal{F}}\times\mathcal{L}^{\infty}(\Omega,\mathcal{F})\to\mathbb{R}$
be a functional satisfying (P1), (P2), (P3)and (P4). Then there exists a
${\mathcal{F}}\otimes\mathbb{B}_{\mathbb{R}}$ measurable function
$\mathbb{u}:\Omega\times\mathbb{R}\to\mathbb{R}$ with
$\mathbb{u}(\omega,\cdot)$ being a continuous and strictly increasing function
on $\mathbb{R}$ for every $\omega\in\Omega$ and
$\mathbb{u}(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ for
every $x\in\mathbb{R}$, such that

$V_{A}(f)=\int_{A}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)\quad\text{for
any }f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F}),A\in{\mathcal{F}}.$

###### Remark 4.2

We emphasize that Proposition 4.1 provides a more general result than Theorem
3.1. Intuitively speaking, a functional that represents a preference relation
as in [16] always satisfies the assumptions of the proposition, as will be
explained more in detail at the end of this section.

Consider a functional
$V:{\mathcal{F}}\times\mathcal{L}^{\infty}(\Omega,\mathcal{F})\to\mathbb{R}$
satisfying (P1), (P2), (P3) and (P4). For any
$A\in{\mathcal{F}},f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ we denote

$\mu_{f}(A):=V_{\Omega\cap A}(f)=V_{A}(f),$

and observe that, for all $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$,
$\mu_{f}$ is dominated by the probability $\mathbb{P}$ defined as in (P1).

We now construct a special version of the state dependent utility, which
enjoys the desired properties.

###### Definition 4.3

For any $q\in{\mathbb{Q}}$, we let
$u(\cdot,q)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ be a version
of the Radon-Nikodym derivative
$\frac{\mathrm{d}\mu_{q}}{\mathrm{d}\mathbb{P}}$. In particular, we set
$u(\omega,0)=0$ for all $\omega\in\Omega$ by virtue of (P1).

For any $A\in{\mathcal{F}}$ and $q\in\mathbb{Q}$, Definition 4.3 motivates the
following notation

$V_{A}(q)=\int_{A}\mathrm{d}\mu_{q}(\omega)=\int_{\Omega}u(\omega,q)\mathbf{1}_{A}(\omega)\mathrm{d}\mathbb{P}(\omega)=\int_{\Omega}u(\omega,q\mathbf{1}_{A}(\omega))\mathrm{d}\mathbb{P}(\omega)$
(7)

###### Lemma 4.4

Denote by

$A^{\mathbb{Q}}:=\bigl{\\{}\omega\in\Omega:u(\omega,q_{1})<u(\omega,q_{2}),\;\forall\;q_{1}<q_{2}\in{\mathbb{Q}}\bigr{\\}}$

and

$B^{\mathbb{Q}}:=\bigl{\\{}\omega\in\Omega:u(\omega,q)=\inf_{{\mathbb{Q}}\ni\tilde{q}>q}u(\omega,\tilde{q}),\;\forall\;q\in{\mathbb{Q}}\bigl{\\}}.$

Then, $A^{\mathbb{Q}},B^{\mathbb{Q}}\in{\mathcal{F}}$ and
$\mathbb{P}(A^{\mathbb{Q}})=\mathbb{P}(B^{\mathbb{Q}})=1$.

###### Proof 1

We start fixing $q_{1}<q_{2}\in{\mathbb{Q}}$ and denoting by
$C=\\{\omega\in\Omega\mid u(\omega,q_{1})\geq u(\omega,q_{2})\\}$. By (7) we
can write

$V_{C}(q_{1})=\int_{C}u(\omega,q_{1})\mathrm{d}\mathbb{P}(\omega)\geq\int_{C}u(\omega,q_{2})\mathrm{d}\mathbb{P}(\omega)=V_{C}(q_{2}),$

which contradicts (P3) unless $\mathbb{P}(C)=0$. We conclude that
$\mathbb{P}(\\{\omega\in\Omega:u(\omega,q_{1})<u(\omega,q_{2})\\})=1$. By
(countable) intersection over the rational numbers we obtain

$A^{\mathbb{Q}}=\bigcap_{\begin{subarray}{c}q_{1},q_{2}\in{\mathbb{Q}},\\\
q_{1}<q_{2}\end{subarray}}\bigl{\\{}\omega\in\Omega:u(\omega,q_{1})<u(\omega,q_{2})\bigr{\\}}$

and clearly $A^{\mathbb{Q}}\in{\mathcal{F}}$, with
$\mathbb{P}(A^{\mathbb{Q}})=1$.
We now turn our attention to $B^{\mathbb{Q}}$. We show that

$\mathbb{P}(\\{\omega\in\Omega:u(\omega,q)\leq\inf_{\tilde{q}>q}u(\omega,\tilde{q}),\;\forall\;q\in{\mathbb{Q}}\\})=1$
(8)

and that

$\mathbb{P}(\\{\omega\in\Omega:u(\omega,q)\geq\inf_{\tilde{q}>q}u(\omega,\tilde{q}),\;\forall\;q\in{\mathbb{Q}}\\})=1,$
(9)

so that (8) and (9) together imply our claim. First of all we show that (8)
holds. Fix $q,\tilde{q}\in{\mathbb{Q}}$ such that $\tilde{q}>q$, the previous
argument shows that the event $\\{\omega\in\Omega:u(\omega,q)\leq
u(\omega,\tilde{q})\\}$ has probability $1$. Notice that for
$q\in{\mathbb{Q}}$ fixed, we have

$\bigcap_{\tilde{q}>q}\\{\omega\in\Omega:u(\omega,q)\leq
u(\omega,\tilde{q})\\}=\\{\omega\in\Omega:u(\omega,q)\leq\inf_{\tilde{q}>q}u(\omega,\tilde{q})\\}.$

We denote by $E_{q}$ the event
$\bigcap_{\tilde{q}>q}\\{\omega\in\Omega:u(\omega,q)\leq
u(\omega,\tilde{q})\\}$. Naturally, the countable intersection is still
measurable and in particular $\mathbb{P}(E_{q})=1$. Finally, since

$\bigcap_{q\in{\mathbb{Q}}}E_{q}=\\{\omega\in\Omega:u(\omega,q)\leq\inf_{\tilde{q}>q}u(\omega,\tilde{q}),\;\forall\;q\in{\mathbb{Q}}\\},$

we find $\mathbb{P}\left(\cap_{q\in{\mathbb{Q}}}E_{q}\right)=1$ and (8) holds.
We proceed proving (9) by contradiction and set

$A=\\{\omega\in\Omega:u(\omega,q)<\inf_{\tilde{q}>q}u(\omega,\tilde{q})\\}.$

Suppose that $\mathbb{P}\left(A\right)>0$, then, for some $n\in\mathbb{N}$,
$A_{n}:=\\{\omega\in\Omega:u(\omega,q)+\frac{1}{n}<\inf_{\tilde{q}>q}u(\omega,\tilde{q})\\}$
is such that $\mathbb{P}(A_{n})>0$. Then it holds that

$\int_{A_{n}}\left(u(\omega,q)+\frac{1}{n}\right)\mathrm{d}\mathbb{P}(\omega)\leq\int_{A_{n}}\inf_{\tilde{q}>q}u(\omega,\tilde{q})\mathrm{d}\mathbb{P}(\omega)\leq\inf_{\tilde{q}>q}\int_{A_{n}}u(\omega,\tilde{q})\mathrm{d}\mathbb{P}(\omega).$
(10)

It follows from (10) that

$V_{A_{n}}(q)+\frac{1}{n}\mathbb{P}(A_{n})\leq\inf_{\tilde{q}>q}V_{A_{n}}(\tilde{q}).$
(11)

Since by (P3) and (P4) $V$ is strictly monotone and pointwise continuous we
have that $\inf_{\tilde{q}>q}V_{A_{n}}(\tilde{q})=V_{A_{n}}(q)$, which in turn
yields the desired contradiction. To conclude, since both (8) and (9) hold
with probability $1$, we have

$\mathbb{P}(B^{\mathbb{Q}})=\mathbb{P}(\\{\omega\in\Omega:u(\omega,q)=\inf_{\tilde{q}>q}u(\omega,\tilde{q}),\;\forall\;q\in{\mathbb{Q}}\\})=1.$

We define the set

$\displaystyle\Theta=A^{\mathbb{Q}}\cap B^{\mathbb{Q}}\in{\mathcal{F}}\text{
with }\mathbb{P}(\Theta)=1,$ (12)

and recall that for any $\omega\in\Theta$ we have
$u(\omega,q_{1})<u(\omega,q_{2})$ for all $q_{1}<q_{2}$,
$q_{1},q_{2}\in{\mathbb{Q}}$ and
$u(\omega,q)=\inf_{\tilde{q}>q}u(\omega,\tilde{q})$ for every
$q\in{\mathbb{Q}}$.

###### Definition 4.5

For any $x\in\mathbb{R}$ and $\omega\in\Omega$ we define

$u^{+}(\omega,x)=\inf_{\begin{subarray}{c}q\in{\mathbb{Q}},\\\ q\geq
x\end{subarray}}\\{u(\omega,q)\mathbf{1}_{\Theta}(\omega)+x\mathbf{1}_{\Omega\setminus\Theta}(\omega)\\}.$

We observe that $u^{+}(\omega,x)$ is ${\mathcal{F}}$-measurable for any
$x\in\mathbb{R}$, as it is the countable pointwise infimum of measurable
functions. Notice that $u^{+}(\omega,0)=0$ for all $\omega\in\Omega$.

###### Lemma 4.6

Let $u^{+}:\Omega\times\mathbb{R}\to\mathbb{R}$ be as in Definition 4.5. Then
the following hold:

  1. (i)

$u^{+}(\omega,q)=u(\omega,q)$ for any $\omega\in\Theta$ (as defined in (12))
and $q\in{\mathbb{Q}}$;

  2. (ii)

if $x<y$, $u^{+}(\omega,x)<u^{+}(\omega,y)$ for every $\omega\in\Omega$;

  3. (iii)

$u^{+}(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ for
every $x\in\mathbb{R}$;

  4. (iv)

for every $\omega\in\Omega$ the function $u^{+}(\omega,\cdot)$ is
RCLL555namely right continuous with left limits on $\mathbb{R}$;

###### Proof 2

Item (i) follows from the definition of $\Theta$. Item (ii) holds trivially on
$\Omega\setminus\Theta$. Take now any $\omega\in\Theta$: provided that $x<y$,
there exist $q_{1},q_{2}\in{\mathbb{Q}}$ such that $x\leq q_{1}<q_{2}\leq y$.
By definition of $\Theta$ we have $u(\omega,q_{1})<u(\omega,q_{2})$ and
moreover by monotonicity of $u$ and definition of $u^{+}$

$u(\omega,q)\leq u^{+}(\omega,y),\quad\text{for every
}q\in{\mathbb{Q}},y\in\mathbb{R}\text{ s.t. }q\leq y.$

Then, we see

$u^{+}(\omega,x)=\inf_{\begin{subarray}{c}q\in{\mathbb{Q}},\\\ q\geq
x\end{subarray}}u(\omega,q)\leq
u(\omega,q_{1})<u(\omega,q_{2})\leq\inf_{\begin{subarray}{c}q\in{\mathbb{Q}},\\\
q\geq y\end{subarray}}u(\omega,q)=u^{+}(\omega,y).$

As to item (iii), fix any $x\in\mathbb{R}$ and consider
$q_{1},q_{2}\in{\mathbb{Q}}$ such that $q_{1}\leq x\leq q_{2}$. Then by
definition and item (i) $u(\omega,q_{1})\leq u^{+}(\omega,x)\leq
u(\omega,q_{2})$ for any $\omega\in\Theta$. Since $\mathbb{P}(\Theta)=1$ and
$u(\cdot,q_{1}),u(\cdot,q_{2})$ are versions of the Radon-Nikodym derivatives
$\frac{d\mu_{q_{1}}}{\mathrm{d}\mathbb{P}},\frac{d\mu_{q_{2}}}{\mathrm{d}\mathbb{P}}\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$,
integrability follows.

Finally, observe that item (iv) holds trivially for
$\omega\in\Omega\backslash\Theta$. Fix now $\omega\in\Theta$, we wish to show
that for any $\varepsilon>0$ there exists $\delta$ such that
$|u^{+}(\omega,x)-u^{+}(\omega,y)|<\varepsilon$ for any $y\in(x,x+\delta)$.
Fix $\varepsilon>0$ and let $(q_{n})_{n}\subset{\mathbb{Q}}$ be such that
$q_{n}\downarrow_{n}x$. Then there exists $\bar{n}\in\mathbb{N}$ such that,
for all $n>\bar{n}$ we have
$|u(\omega,q_{n})-\inf_{q>x}u(\omega,q)|=u(\omega,q_{n})-\inf_{q>x}u(\omega,q)<\varepsilon$.
Recalling that, from item (i), $u(\omega,q_{n})=u^{+}(\omega,q_{n})$ and
$\inf_{q>x}u(\omega,q)=u^{+}(\omega,x)$ by definition, monotonicity of $u^{+}$
implies that any $y\in(x,q_{\bar{n}})\subset\mathbb{R}$ satisfies
$|u^{+}(\omega,x)-u^{+}(\omega,y)|<\varepsilon$. Existence of the left limit
follows again from monotonicity proved in item (ii).

###### Lemma 4.7

Let $u^{+}:\Omega\times\mathbb{R}\to\mathbb{R}$ be as in Definition 4.5. Then
the map

$f\mapsto
T_{u^{+}}(f):=\int_{\Omega}u^{+}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)$
(13)

is well defined, monotone with respect to the pointwise order in
$\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, pointwise continuous666As
previously mentioned: for any uniformly bounded sequence
$(f_{n})_{n}\subset\mathcal{L}^{\infty}(\Omega,\mathcal{F})$,
$f_{n}(\omega)\to f(\omega)$ for any $\omega\in\Omega$, then
$T_{u^{+}}(f_{n})\to T_{u^{+}}(f)$ and it satisfies
$V_{A}(f)=T_{u^{+}}(f\mathbf{1}_{A})$ for every
$f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ and $A\in{\mathcal{F}}$.

###### Proof 3

Observe that, since $u^{+}(\omega,\cdot)$ is RCLL for every $\omega\in\Omega$,
by Lemma A.1 we have that $T_{u^{+}}$ is well defined on
$\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ and monotone as $f(\omega)\leq
g(\omega)$ implies $u^{+}(\omega,f(\omega))\leq u^{+}(\omega,g(\omega))$. We
now show that $V_{\Omega}(f)=T_{u^{+}}(f)$ for every
$f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, then pointwise continuity of
$T_{u^{+}}$ will follow from (P4). Fix any
$f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, we can find two sequences of
simple functions
$\underline{f_{n}}=\sum_{A\in\underline{\pi}_{n}}\underline{x}^{n}_{A}\mathbf{1}_{A}$,
$\overline{f_{n}}=\sum_{B\in\overline{\pi}_{n}}\overline{x}^{n}_{B}\mathbf{1}_{B}$,
with $\overline{\pi}_{n},\underline{\pi}_{n}\subseteq{\mathcal{F}}$ finite
partitions of $\Omega$, such that
$\underline{f_{n}}\uparrow_{n}f,\overline{f_{n}}\downarrow_{n}f$. Moreover by
standard arguments we can assume for any $n$ that
$\underline{x}^{n}_{A},\overline{x}^{n}_{B}\in\mathbb{Q}$ for any
$A\in\underline{\pi}_{n}$ and $B\in\overline{\pi}_{n}$. We have as a
consequence of (7) and Lemma 4.6

$\displaystyle V_{\Omega}(\underline{f}_{n})$
$\displaystyle=\sum_{A\in\underline{\pi}_{n}}\int_{\Omega}u(\omega,\underline{x}_{n}^{A})\mathbf{1}_{A}\mathrm{d}\mathbb{P}(\omega)=\sum_{A\in\underline{\pi}_{n}}\int_{\Omega}u^{+}(\omega,\underline{x}_{n}^{A})\mathbf{1}_{A}\mathrm{d}\mathbb{P}(\omega)$
$\displaystyle=\int_{\Omega}u^{+}(\omega,\underline{f}_{n}(\omega))\mathrm{d}\mathbb{P}(\omega)\leq\int_{\Omega}u^{+}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)$

and by a similar argument we can get

$\int_{\Omega}u^{+}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)\leq\int_{\Omega}u(\omega,\overline{f}_{n}(\omega))\mathrm{d}\mathbb{P}(\omega)=V_{\Omega}(\overline{f}_{n})$

By (P4) we get the desired equality $V_{\Omega}(f)=T_{u^{+}}(f)$ for every
$f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$. Finally, we see that (P2)
yields $V_{A}(f)=V_{\Omega}(f\mathbf{1}_{A})=T_{u^{+}}(f\mathbf{1}_{A})$,
which concludes the proof.

###### Lemma 4.8

There exists a function $\mathbb{u}:\Omega\times\mathbb{R}\to\mathbb{R}$ such
that $\mathbb{u}(\omega,\cdot)$ is continuous and strictly increasing on
$\mathbb{R}$ for every $\omega\in\Omega$,
$\mathbb{u}(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ for
every $x\in\mathbb{R}$ and

$V_{A}(f)=\int_{A}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)\quad\forall\;f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})\text{
and }A\in{\mathcal{F}}.$ (14)

###### Proof 4

We observe that, by Lemma 4.6, $u^{+}(\omega,\cdot)$ is strictly increasing,
RCLL on $\mathbb{R}$ for all $\omega\in\Omega$ and
$u^{+}(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ for
every $x\in\mathbb{R}$. Moreover, as argued in Lemma 4.7, $T_{u^{+}}$ is
pointwise continuous and in particular continuous from below. Consequently the
assumptions of Corollary A.3 are verified and its application yields the
function $\widehat{u^{+}}$ which we rename $\mathbb{u}:=\widehat{u^{+}}$.
Finally, $\mathbb{u}$ satisfies all the requirements since

$T_{\mathbb{u}}\stackrel{{\scriptstyle\text{Def.}}}{{=}}T_{\widehat{u^{+}}}\stackrel{{\scriptstyle\text{Cor.}\ref{corexistscontversion}}}{{=}}T_{u^{+}}\stackrel{{\scriptstyle\text{Lem.}\ref{claim:pointwise_limit}}}{{=}}V.$

Conclusions. We conclude the section by proving Proposition 4.1 and we show
that Theorem 3.1 follows.

Let
$V:{\mathcal{F}}\times\mathcal{L}^{\infty}(\Omega,\mathcal{F})\to\mathbb{R}$
be a functional satisfying (P1), (P2), (P3) and (P4). Lemma 4.4 and 4.6
provide the construction of a suitable integrand function $u^{+}$ and Lemma
4.7 ensures that the map
$f\mapsto\int_{\Omega}u^{+}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)$ is
well defined. Lemma 4.8 concludes the proof by showing the existence of the
desired version $\mathbb{u}$ with the required properties.

To prove Theorem 3.1, we exploit Proposition 4.1. Consider a preference
$\succeq$ that is (SM), (PC) and (ST) and assume that ${\mathcal{F}}$ contains
at least three disjoint events $A_{1},A_{2},A_{3}$ such that
$A_{i}\notin{\mathcal{N}}_{\succeq}$ for $i=1,2,3$. As argued in the beginning
of Section 4, the preference relation $\succeq$ induces a functional
$V:{\mathcal{F}}\times\mathcal{L}^{\infty}(\Omega,\mathcal{F})\to\mathbb{R}$
that satisfies (P1), (P2), (P3) and (P4) with $V_{\Omega}(\cdot)$ acting as
numerical representation of the preference $\succeq$. Application of
Proposition 4.1 provides a pair $(\mathbb{u},\mathbb{P})$ such that
$V_{\Omega}(f)=\int_{\Omega}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)$
for any $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$.

##  5 Proof of Theorem 3.2

We begin by recalling that we work under the hypotheses of Theorem 3.1. In
analogy to the previous section, it is possible to construct a functional
$V:{\mathcal{F}}\times\mathcal{L}^{\infty}(\Omega,\mathcal{F})\to\mathbb{R}$
that satisfies (P1), (P2), (P3) and (P4). We also point out that properties
(P1-4) are in fact stronger than those required in [9], Theorem 3.2777Indeed,
using now the properties defined in [9], the fact that
$V_{\Omega}:\mathcal{L}^{\infty}(\Omega,\mathcal{F})\to\mathbb{R}$ is
(${\mathcal{G}}$-Mo), (${\mathcal{G}}$-PC) and (${\mathcal{G}}$-NB) follows
trivially from (P3) and (P4), for any sub-$\sigma$-algebra ${\mathcal{G}}$ of
${\mathcal{F}}$. Furthermore, recalling that, by (P1), $V_{A}(0)=0$ we are
able to recover also (${\mathcal{G}}$-QL) and (${\mathcal{G}}$-PS). Notice
that, for
$g_{1},g_{2}\in{\mathcal{L}}^{\infty}(\Omega,{\mathcal{G}}),A\in{\mathcal{G}}$,
it is possible to write
$V_{\Omega}(g_{1}\mathbf{1}_{A}+g_{2}\mathbf{1}_{A^{c}})=V_{A}(g_{1})+V_{A^{c}}(g_{2})$..
Therefore, $\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)\neq\emptyset$ and
for any $g_{1},g_{2}\in\mathfrak{m}\left(f\middle|{\mathcal{G}}\right)$ we
have $\\{g_{1}\neq g_{2}\\}\in{\mathcal{N}}_{\succeq}$.

Step 1: We first show the existence of a state dependent utility on
$\mathcal{L}^{\infty}(\Omega,\mathcal{G})$ for any $\sigma$-algebra
${\mathcal{G}}\subseteq{\mathcal{F}}$. Consider the functional
$V:{\mathcal{F}}\times\mathcal{L}^{\infty}(\Omega,\mathcal{F})\to\mathbb{R}$
defined above. By Proposition 4.1 it follows that $V$ admits a representation
of the kind
$V_{A}(f)=\int_{A}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)$.
Let ${\mathcal{G}}\subseteq{\mathcal{F}}$ be an arbitrary sub-$\sigma$-algebra
on $\Omega$ and denote by
$V^{\mathcal{G}}:{\mathcal{G}}\times\mathcal{L}^{\infty}(\Omega,\mathcal{G})\to\mathbb{R}$
the restriction of the functional $V$ to ${\mathcal{G}}\subseteq{\mathcal{F}}$
and $\mathcal{L}^{\infty}(\Omega,\mathcal{G})$. We stress that, differently
from ${\mathcal{F}}$, the $\sigma$-algebra ${\mathcal{G}}$ need not contain at
least three disjoint events of positive probability. Analogously to $V$,
$V^{\mathcal{G}}$ defines a probability measure $\mathbb{P}_{\mathcal{G}}$ on
${\mathcal{G}}$, which is the restriction of the probability measure
$\mathbb{P}$ induced by $V$ to the sub-$\sigma$-algebra ${\mathcal{G}}$. We
will hence use only $\mathbb{P}$ in the following, to avoid impractical
notations.
Applying Proposition 4.1 to $V^{\mathcal{G}}$ it is possible to construct a
${\mathcal{G}}\otimes\mathbb{B}_{\mathbb{R}}$-measurable function
$\mathbb{u}_{\mathcal{G}}:\Omega\times\mathbb{R}\to\mathbb{R}$ such that

  1. (i)

$\mathbb{u}_{\mathcal{G}}(\omega,\cdot)$ is continuous and strictly increasing
on $\mathbb{R}$ for all $\omega\in\Omega$,

  2. (ii)

$\mathbb{u}_{\mathcal{G}}(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{G}},\mathbb{P})$
for all $x\in\mathbb{R}$,

  3. (iii)

$V_{A}^{\mathcal{G}}(g)=\int_{A}\mathbb{u}_{\mathcal{G}}(\omega,g(\omega))\mathrm{d}\mathbb{P}(\omega),\,\forall
A\in{\mathcal{G}},g\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})$.

Notice that for all $A\in{\mathcal{G}}$ and
$g\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})$

$V_{A}^{\mathcal{G}}(g)=\int_{A}\mathbb{u}_{\mathcal{G}}(\omega,g(\omega))\mathrm{d}\mathbb{P}(\omega)=\int_{A}\mathbb{u}(\omega,g(\omega))\mathrm{d}\mathbb{P}(\omega)=V_{A}(g).$

It is then immediate to verify by definition of conditional expectation that
the random variable $\mathbb{u}_{\mathcal{G}}(\cdot,x)$ is an element of the
class $\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,x)|{\mathcal{G}}\right]$.

To avoid overburdening notation, in the following step 2 we shall write
$\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(f)\middle|{\mathcal{G}}\right]$ in
place of
$\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,f(\cdot))|{\mathcal{G}}\right]$
and $\mathbb{u}_{\mathcal{G}}(g)$ instead of
$\mathbb{u}_{\mathcal{G}}(\cdot,g(\cdot))$. Moreover, we shall often make the
following standard abuse of notation: for any
$f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ we write
$g=\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(f)|{\mathcal{G}}\right]$ to
indicate that
$g\in\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(f)|{\mathcal{G}}\right]$ i.e. $g$
is a version of the conditional expectation
$\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(f)|{\mathcal{G}}\right]$.

Step 2: We show that:

$\mathbb{u}_{\mathcal{G}}(g)\text{ belongs to
}\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(g)\middle|{\mathcal{G}}\right]\quad\forall\,g\in\mathcal{L}^{\infty}(\Omega,\mathcal{G}).$
(15)

Consider indeed a simple function $g=\sum_{A\in\pi}y_{A}\mathbf{1}_{A}$, with
$\pi$ being a finite ${\mathcal{G}}$-measurable partition of $\Omega$. Then we
have

$\displaystyle\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(g)\middle|{\mathcal{G}}\right]$
$\displaystyle=\sum_{A\in\pi}\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(g)\mathbf{1}_{A}\middle|{\mathcal{G}}\right]=\sum_{A\in\pi}\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(y_{A})\mathbf{1}_{A}\middle|{\mathcal{G}}\right]$
$\displaystyle=\sum_{A\in\pi}\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(y_{A})\middle|{\mathcal{G}}\right]\mathbf{1}_{A}=\sum_{A\in\pi}\mathbb{u}_{\mathcal{G}}(y_{A})\mathbf{1}_{A}=\mathbb{u}_{\mathcal{G}}(\sum_{A\in\pi}y_{A}\mathbf{1}_{A}),$

and therefore (15) is verified for ${\mathcal{G}}$-measurable simple
functions. We proceed to show it also holds for general functions
$g\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})$. Take any sequence of
${\mathcal{G}}$-measurable simple functions
$(\underline{g}_{n})_{n}\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})$ such that
$\underline{g}_{n}\uparrow_{n}g$ (again, the existence of such functions is
guaranteed by standard arguments). Monotonicity of
$\mathbb{u}_{\mathcal{G}}(\omega,\cdot)$ ensures that
$(\mathbb{u}_{\mathcal{G}}(\omega,\underline{g}_{n}(\omega)))_{n}$ is an
increasing sequence for all $\omega\in\Omega$. Finally, pointwise continuity
of $\mathbb{u}$ gives

$\mathbb{u}_{\mathcal{G}}(g)=\lim_{n}\mathbb{u}_{\mathcal{G}}(\underline{g}_{n}),$

and monotone convergence theorem yields

$\lim_{n}\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\underline{g}_{n})\middle|{\mathcal{G}}\right]=\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(g)\middle|{\mathcal{G}}\right]$

which provides (15) for general
$g\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})$.

Step 3: we now study the generalized inverse of the state dependent utility
$\mathbb{u}_{\mathcal{G}}$. If on the one hand
$(\omega,x)\mapsto\mathbb{u}_{\mathcal{G}}(\omega,x)$ is
${\mathcal{G}}\otimes\mathbb{B}_{\mathbb{R}}$ measurable (by Proposition 4.1),
we want the generalized inverse $\Phi_{\mathcal{G}}$ to enjoy the same
measurability as $\mathbb{u}_{\mathcal{G}}$. We first extend the domain of
$\mathbb{u}_{\mathcal{G}}:\Omega\times\mathbb{R}\to\mathbb{R}$ to
$\Omega\times\overline{\mathbb{R}}$ by denoting

$\mathbb{u}_{\mathcal{G}}(\omega,+\infty):=\lim_{x\to+\infty}\mathbb{u}_{\mathcal{G}}(\omega,x),\quad\quad\mathbb{u}_{\mathcal{G}}(\omega,-\infty):=\lim_{x\to-\infty}\mathbb{u}_{\mathcal{G}}(\omega,x).$

Monotonicity and continuity of $\mathbb{u}_{\mathcal{G}}(\omega,\cdot)$ for
all $\omega\in\Omega$ ensure that the limits exist. Fixed $\omega\in\Omega$,
we denote with
$\text{Im}_{\omega}(\mathbb{u}_{\mathcal{G}}):=\text{Im}(\mathbb{u}_{\mathcal{G}}(\omega,\cdot))$
the image of $\mathbb{u}_{\mathcal{G}}$, that is the interval
$(\mathbb{u}_{\mathcal{G}}(\omega,-\infty),\mathbb{u}_{\mathcal{G}}(\omega,+\infty))$,
and we stress that it is $\omega$-dependent and open by strict monotonicity.
Let $\Phi_{\mathcal{G}}:\Omega\times\mathbb{R}\to\overline{\mathbb{R}}$ be the
$\omega$-wise right-continuous generalized inverse:

$\Phi_{\mathcal{G}}(\omega,x):=\inf\\{y\in\mathbb{R}\mid\mathbb{u}_{\mathcal{G}}(\omega,y)>x\\}.$

Observe that strict monotonicity and continuity of
$\mathbb{u}_{\mathcal{G}}(\omega,\cdot)$ imply
$\Phi_{\mathcal{G}}(\omega,\cdot)$ is strictly increasing and continuous over
$\text{Im}_{\omega}(\mathbb{u}_{\mathcal{G}})$ for all $\omega\in\Omega$.
Then, in particular,
$\mathbb{u}_{\mathcal{G}}(\omega,\cdot):\mathbb{R}\to\text{Im}_{\omega}(\mathbb{u}_{\mathcal{G}})$
is a bijection and we can write

$\Phi_{\mathcal{G}}(\omega,x)=\begin{cases}+\infty&\quad\text{for
}x\geq\mathbb{u}_{\mathcal{G}}(\omega,+\infty),\\\
\mathbb{u}_{\mathcal{G}}^{-1}(\omega,x)&\quad\text{for
}x\in\text{Im}_{\omega}(\mathbb{u}_{\mathcal{G}}),\\\ -\infty&\quad\text{for
}x\leq\mathbb{u}_{\mathcal{G}}(\omega,-\infty).\end{cases}$ (16)

Notice that, fixed $\omega\in\Omega$, whenever
$x\in\text{Im}_{\omega}(\mathbb{u}_{\mathcal{G}})$ then
$\Phi_{\mathcal{G}}(\omega,x)$ is finite.

We claim that $\Phi_{\mathcal{G}}(\cdot,x):\Omega\to\overline{\mathbb{R}}$ is
${\mathcal{G}}$-measurable for all $x\in\mathbb{R}$. To see this, let
$x\in\mathbb{R}$ be fixed and consider the set
$A=(a,+\infty]\in\mathbb{B}_{\overline{\mathbb{R}}}$ with $a>-\infty$. We show
that $\left\\{\omega\in\Omega:\Phi_{\mathcal{G}}(\omega,x)\in
A\right\\}\in{\mathcal{G}}$. We observe that, by strict monotonicity of
$\mathbb{u}_{\mathcal{G}}$ the latter coincides with
$\left\\{\omega\in\Omega:x>\mathbb{u}_{\mathcal{G}}(\omega,a)\right\\}$ which
is an element of ${\mathcal{G}}$ as $\mathbb{u}_{\mathcal{G}}$ is
${\mathcal{G}}$-measurable.
Addittionally, we wish to emphasize that
$\Phi_{\mathcal{G}}(\omega,\cdot):\mathbb{R}\to\left[-\infty,+\infty\right]$
is continuous for all $\omega\in\Omega$, and therefore
$\Phi_{\mathcal{G}}:\Omega\times\mathbb{R}\to\left[-\infty,+\infty\right]$ is
a Carathéodory function (see [1], Definition 4.50). In particular, we observe
that $\mathbb{R}$ with the usual euclidean topology is a separable metrizable
space and $\overline{\mathbb{R}}=\left[-\infty,+\infty\right]$ with the
standard topology on the extended real numbers is metrizable. Then, by virtue
of [1], Lemma 4.51, $\Phi_{\mathcal{G}}$ is
${\mathcal{G}}\otimes\mathbb{B}_{\mathbb{R}}$ measurable.

###### Remark 5.1

By (16), for any $\omega\in\Omega$ and
$x\in\mathrm{Im}_{\omega}(\mathbb{u}_{\mathcal{G}})$ we have

$\mathbb{u}_{\mathcal{G}}\Big{(}\omega,\Phi_{\mathcal{G}}(\omega,x)\Big{)}=x.$
(17)

Step 4: we conclude proving the desired representation of the conditional
Chisini mean. Chosen $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, by strict
monotonicity of $\mathbb{u}(\omega,\cdot)$ we have that

$\mathbb{u}(\omega,-\left\|f\right\|_{\infty})\leq\mathbb{u}(\omega,f(\omega))\leq\mathbb{u}(\omega,-\left\|f\right\|_{\infty}),\quad\quad\forall\;\omega\in\Omega.$

Taking a version $g$ of
$\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,f)\middle|{\mathcal{G}}\right]$
we can find $A\in{\mathcal{G}}$ such that $\mathbb{P}(A)=1$ and exploting (15)
we may write

$\mathbb{u}_{\mathcal{G}}(\omega,-\left\|f\right\|_{\infty})<g(\omega)<\mathbb{u}_{\mathcal{G}}(\omega,\left\|f\right\|_{\infty})\quad\text{
for all }\omega\in A.$ (18)

Let us define $\tilde{g}=g\mathbf{1}_{A}+0\mathbf{1}_{A^{c}}$ and observe that
now $\tilde{g}(\omega)\in\text{Im}_{\omega}(\mathbb{u}_{\mathcal{G}})$ for all
$\omega\in\Omega$. Moreover, $\tilde{g}$ is still a version of the conditional
expectation of $f$. Recalling that the mapping
$\Phi_{\mathcal{G}}(\omega,\cdot):\mathbb{R}\to\overline{\mathbb{R}}$ is
strictly increasing on
$\text{Im}_{\omega}(\mathbb{u}_{\mathcal{G}})\subseteq\mathbb{R}$ for all
$\omega\in\Omega$, and application to (18) yields

$\Phi_{\mathcal{G}}(\omega,\mathbb{u}_{\mathcal{G}}(\omega,-\left\|f\right\|_{\infty}))<\Phi_{\mathcal{G}}(\omega,\tilde{g}(\omega))<\Phi_{\mathcal{G}}(\omega,\mathbb{u}_{\mathcal{G}}(\omega,\left\|f\right\|_{\infty})).$

Consequently, by (17)
$\Phi_{\mathcal{G}}(\omega,\mathbb{u}_{\mathcal{G}}(\omega,x))=x$ for all
$\omega\in\Omega$ and all $x\in\mathbb{R}$ and it follows that that
$-\left\|f\right\|_{\infty}<\Phi_{\mathcal{G}}(\omega,\tilde{g})<\left\|f\right\|_{\infty}$,
for every $\omega\in\Omega$. Hence
$\Phi_{\mathcal{G}}\left(\cdot,\tilde{g}\right)\in\mathcal{L}^{\infty}(\Omega,\mathcal{G})$.
In particular then we have

$\mathbb{u}_{\mathcal{G}}\Big{(}\omega,\Phi_{\mathcal{G}}\big{(}\omega,\tilde{g}(\omega)\big{)}\Big{)}=\tilde{g}(\omega)\quad\text{
for every }\omega\in\Omega.$ (19)

We prove that $\Phi_{\mathcal{G}}(\omega,\tilde{g}(\omega))$ is the
conditional Chisini mean in that it solves the infinite dimensional system of
equations given in (5), for the preference representing functional
$V_{\Omega}(f)=\int_{\Omega}\mathbb{u}(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)$
defined in Theorem 3.1. Considering an arbitrary set
$A\in{\mathcal{G}}\subseteq{\mathcal{F}}$, we have

$\displaystyle
V_{\Omega}\big{(}\Phi_{\mathcal{G}}(\cdot,\tilde{g})\mathbf{1}_{A}\big{)}$
$\displaystyle=\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}\big{(}\cdot,\Phi_{\mathcal{G}}\left(\cdot,\tilde{g}\right)\mathbf{1}_{A}\big{)}\right]=\mathbb{E}_{\mathbb{P}}\left[\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}\big{(}\cdot,\Phi_{\mathcal{G}}\left(\cdot,\tilde{g}\right)\big{)}\mathbf{1}_{A}\middle|{\mathcal{G}}\right]\right]$
$\displaystyle=\mathbb{E}_{\mathbb{P}}\left[\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}\big{(}\cdot,\Phi_{\mathcal{G}}\left(\cdot,\tilde{g}\right)\big{)}\middle|{\mathcal{G}}\right]\mathbf{1}_{A}\right]\stackrel{{\scriptstyle\eqref{almosttakeout}}}{{=}}\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}_{\mathcal{G}}\big{(}\cdot,\Phi_{\mathcal{G}}\left(\cdot,\tilde{g}\right)\big{)}\mathbf{1}_{A}\right]$
$\displaystyle\stackrel{{\scriptstyle\eqref{inverseGmeas}}}{{=}}\mathbb{E}_{\mathbb{P}}\left[\tilde{g}\mathbf{1}_{A}\right]=\mathbb{E}_{\mathbb{P}}\left[\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,f)\mathbf{1}_{A}\middle|{\mathcal{G}}\right]\right]=\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,f)\mathbf{1}_{A}\right]$
$\displaystyle=\mathbb{E}_{\mathbb{P}}\left[\mathbb{u}(\cdot,f\mathbf{1}_{A})\right]=V_{\Omega}(f\mathbf{1}_{A}),$

observing that since $\mathbb{u}(\omega,0)=0$ for all $\omega\in\Omega$ by
construction (whence also $\Phi_{\mathcal{G}}(\omega,0)=0$ for all
$\omega\in\Omega$) we have

$\begin{cases}\mathbb{u}(\omega,f(\omega))\mathbf{1}_{A}(\omega)=\mathbb{u}\big{(}\omega,f(\omega)\mathbf{1}_{A}(\omega)\big{)}\\\
\Phi_{\mathcal{G}}\left(\omega,\tilde{g}(\omega)\right)\mathbf{1}_{A}(\omega)=\Phi_{\mathcal{G}}\big{(}\omega,\tilde{g}(\omega)\mathbf{1}_{A}(\omega)\big{)}\end{cases}\quad\text{
for every }\omega\in\Omega.$

##  Appendix A Appendix: auxiliary results

###### Lemma A.1

Let $u:\Omega\times\mathbb{R}\to\mathbb{R}$ and assume that for every
$\omega\in\Omega$ the function $x\mapsto u(\omega,x)$ is right continuous with
left limit (RCLL), nondecreasing and $u(\omega,0)=0$. Suppose additionally
that $u(\omega,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ for
every $x\in\mathbb{R}$. Then:

  1. (i)

$u$ is ${\mathcal{F}}\otimes\mathbb{B}_{\mathbb{R}}-\mathbb{B}_{\mathbb{R}}$
measurable;

  2. (ii)

for every $f\in\mathcal{L}^{\infty}(\Omega,\mathcal{F})$, $\omega\mapsto
u(\omega,f(\omega))\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$

###### Proof 5

We prove Item (i) in the spirit of [12] (Proposition 1.1.13 and Remark 1.1.14)
as follows. Let $m>0$. For $n\geq 1$, $k=0,1,...,2\cdot 2^{n}-1$, we define

$u^{(n)}(\omega,s):=u\left(\omega,\frac{\left(k+1\right)m}{2^{n}}-m\right)\;\;\;\;\;\text{
for }\;\;\;k\frac{m}{2^{n}}-m\leq s<\frac{\left(k+1\right)m}{2^{n}}-m,$

and fix

$\begin{cases}u^{(n)}(\omega,s):=u\left(\omega,-m\right),\;\;\;\;\;&\text{ for
}s<-m,\\\ u^{(n)}(\omega,s):=u\left(\omega,m\right),\;\;\;\;\;&\text{ for
}s\geq m.\end{cases}$

The newly constructed process is piecewise constant over intervals of length
$\frac{m}{2^{n}}$ and right continuous. The map $(\omega,s)\mapsto
u^{(n)}(\omega,s)$ is
${\mathcal{F}}\otimes\mathbb{B}_{\mathbb{R}}$-measurable. By right continuity
of $u$ we have that $u^{(n)}(\omega,s)\to u(\omega,s)$ for every couple
$(\omega,s)\in\Omega\times[-m,m]$ as $n\to\infty$. Therefore we obtain that
the map $(\omega,s)\mapsto u(\omega,s)$ is itself
${\mathcal{F}}\otimes\mathbb{B}_{[-m,m]}$-measurable as pointwise limit of a
sequence of measurable functions. Finally, by letting $m\to\infty$ we have
that $(\omega,s)\mapsto u(\omega,s)$ is
${\mathcal{F}}\otimes\mathbb{B}_{\mathbb{R}}$-measurable. By the latter
argument we have that $\omega\mapsto u(\omega,x)$ is
${\mathcal{F}}\otimes\mathbb{B}_{\mathbb{R}}-\mathbb{B}_{\mathbb{R}}$
measurable. Moreover, for any ${\mathcal{F}}-\mathbb{B}_{\mathbb{R}}$
measurable and bounded $f$, the map $\varphi:\omega\mapsto(\omega,f(\omega))$
is ${\mathcal{F}}-{\mathcal{F}}\otimes\mathbb{B}_{\mathbb{R}}$ measurable (see
[1], Lemma 4.49). Hence, the composition $(u\circ\varphi):\omega\mapsto
u(\omega,f(\omega))$ is ${\mathcal{F}}-\mathbb{B}_{\mathbb{R}}$ measurable. To
conclude, since $u$ is nondecreasing,
${\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})\ni u(\omega,-\lVert
f\rVert_{\infty})\leq u(\omega,f(\omega))\leq u(\omega,\lVert
f\rVert_{\infty})\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ thus
concluding the proof of Item (ii).

###### Proposition A.2

Let $u:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ satisfying

  1. (i)

for every $\omega\in\Omega$ $u(\omega,\cdot)$ is RCLL, nondecreasing and
$u(\omega,0)=0$;

  2. (ii)

$u(\cdot,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ for every
$x\in\mathbb{R}$;

  3. (iii)

the functional
$T_{u}:\mathcal{L}^{\infty}(\Omega,\mathcal{F})\rightarrow\mathbb{R}$ defined
by $T_{u}(f):=\int_{\Omega}u(\omega,f(\omega))\mathrm{d}\mathbb{P}(\omega)$ is
continuous from below 888For any
$(f_{n})_{n}\subset\mathcal{L}^{\infty}(\Omega,{\mathcal{F}})$ such that
$f_{n}(\omega)\uparrow_{n}f(\omega)$ for any $\omega\in\Omega$ we have
$T_{u}(f_{n})\rightarrow T_{u}(f)$.

Then $u$ is continuous in the following sense:

$A_{\text{cont}}:=\\{\omega\in\Omega\mid x\mapsto u(\omega,x)\text{ is
continuous}\\}\in{\mathcal{F}}$

and $\mathbb{P}(A_{\text{cont}})=1$.

###### Proof 6

In the present proof we shall rely on few concepts related to the theory of
stochastic processes and in particular on stopping times. We refer the reader
to [12], page 6, for an exhaustive treatment of the notions involved.

Lemma A.1 guarantees that
$T_{u}:\mathcal{L}^{\infty}(\Omega,{\mathcal{F}})\to\mathbb{R}$ is well
defined and finite valued. Fix $M>0$ and define $X_{t}(\omega):=u(\omega,t-M)$
for $t\geq 0$, omitting the dependence on $M$ for simplicity. Take also the
filtration ${\mathcal{F}}_{t}:={\mathcal{F}},t\geq 0$, which is trivially
right continuous by definition:
$\bigcap_{s>t}{\mathcal{F}}_{s}={\mathcal{F}}={\mathcal{F}}_{t}$. Then
$(X_{t})_{t}$ is an RCLL process, adapted to $({\mathcal{F}}_{t})_{t}$. Set
$\Delta X_{0}=0$ and for $t>0$ set $\Delta X_{t}:=X_{t}-\sup_{0\leq
s<t}X_{s}=X_{t}-\lim_{s\uparrow t}X_{s}$.

Following the idea in [14], we define $\tau^{\varepsilon}$ as the first jump
time for the size $\varepsilon>0$, namely

$\tau^{\varepsilon}:=\inf\\{t\geq 0\mid\Delta
X_{t}>\varepsilon\\}\stackrel{{\scriptstyle(\star)}}{{=}}\inf\\{t>0\mid
X_{t}-\sup_{0\leq s<t}X_{s}>\varepsilon\\},$

where equality $(\star)$ follows from posing $\Delta X_{0}=0$. Looking back at
$u(\omega,\cdot)$, $\tau^{\varepsilon}(\omega)$ is the first time
$u(\omega,\cdot)$ jumps, in the interval $(-M,+\infty)$, with a jump size
strictly greater than $\varepsilon$, with the usual convention that such time
is set to be $+\infty$ if no such jump actually occurs. By [14],
$\tau^{\varepsilon}:\Omega\rightarrow[0,+\infty]$ is a
$({\mathcal{F}}_{t})_{t}$-stopping time, hence in particular it is
${\mathcal{F}}$-measurable by definition of such a filtration. Setting

$A_{cont}^{M}:=\\{\omega\in\Omega\mid[0,+\infty)\ni t\mapsto
X_{t}(\omega)\text{ is continuous}\\},$

we observe that

$\Omega\setminus
A_{cont}^{M}=\bigcup_{n\in\mathbb{N}}\\{\tau^{\frac{1}{n}}<+\infty\\},$

hence the measurability of $A^{M}_{cont}$ follows. Suppose now that
$\mathbb{P}(A^{M}_{cont})<1$, then for some $n$ we would have
$\mathbb{P}(\tau^{\frac{1}{n}}<+\infty)>0$ and by
$\\{\tau^{\frac{1}{n}}<+\infty\\}=\bigcup_{N\in\mathbb{N}}\\{\tau^{\frac{1}{n}}<N\\}$
we could find $N$ big enough such that $\mathbb{P}(\tau^{\frac{1}{n}}\leq
N)>0$.
Set, for a fixed pair $n,N$ satisfying $\mathbb{P}(\tau^{\frac{1}{n}}\leq
N)>0$, $B:=\\{\tau^{\frac{1}{n}}\leq N\\}$ and
$Z:=\tau^{\frac{1}{n}}\mathbf{1}_{B}\in\mathcal{L}^{\infty}(\Omega,{\mathcal{F}})$.
For any fixed $\omega\in B$ we know that
$\tau^{\frac{1}{n}}(\omega)=\inf\\{t\geq 0\mid\Delta
X_{t}(\omega)>1/n\\}=\lim_{k\to\infty}t_{k}(\omega)$ for some
$\omega$-dependent, non-increasing sequence
$t_{k}(\omega)\downarrow_{k}\tau^{\frac{1}{n}}(\omega)$ which satisfies
$\Delta X_{t_{k}(\omega)}(\omega)>1/n$ for every $k$. This means that, for any
$k$,

$\frac{1}{n}\leq X_{t_{k}(\omega)}(\omega)-\sup_{0\leq
s<t_{k}(\omega)}X_{s}(\omega)\leq X_{t_{k}(\omega)}-\sup_{0\leq
s<\tau^{\frac{1}{n}}(\omega)}X_{s}(\omega)$

as $\sup_{0\leq s<t_{k}(\omega)}X_{s}(\omega)\geq\sup_{0\leq
s<\tau^{\frac{1}{n}}(\omega)}X_{s}(\omega)$. Letting $k\to\infty$ this implies
by right continuity of $(X_{t})_{t}$

$\frac{1}{n}\leq X_{\tau^{\frac{1}{n}}(\omega)}(\omega)-\sup_{0\leq
s<\tau^{\frac{1}{n}}(\omega)}X_{s}(\omega).$ (20)

Take now $s_{k}:=(Z-\frac{1}{k}\mathbf{1}_{B})^{+}$ for $k\in\mathbb{N}$.
$s_{k}$ is ${\mathcal{F}}-\mathbb{B}_{[0,+\infty)}$ measurable and bounded.
Observe that since $(X_{t})_{t}$ is RCLL, by [12] Remark 1.1.14 we have
$(t,\omega)\mapsto X_{t}(\omega)$ is
$\mathbb{B}_{[0,+\infty)}\otimes{\mathcal{F}}-\mathbb{B}_{[0,+\infty)}$
measurable, and $\omega\mapsto(\omega,Y(\omega))$ is
${\mathcal{F}}-{\mathcal{F}}\otimes\mathbb{B}_{[0,+\infty)}$ measurable for
every ${\mathcal{F}}-\mathbb{B}_{[0,+\infty)}$ measurable bounded
$Y:\Omega\rightarrow[0,+\infty)$. Hence $\omega\mapsto
X_{s_{k}(\omega)}(\omega)$ is ${\mathcal{F}}-\mathbb{B}_{[0,+\infty)}$
measurable. By continuity from below, argued at the beginning of this proof,

$T_{u}\big{(}(s_{k}-M)\mathbf{1}_{B}\big{)}\rightarrow_{k}T_{u}\big{(}(Z-M)\mathbf{1}_{B}\big{)}.$
(21)

At the same time, $s_{k}(\omega)=Z(\omega)=0$ for every
$\omega\in\Omega\setminus B$ and using (20)
$X_{s_{k}(\omega)}(\omega)\leq\sup_{0\leq
s<\tau^{\frac{1}{n}}(\omega)}X_{s}(\omega)\leq-\frac{1}{n}+X_{\tau^{\frac{1}{n}}(\omega)}(\omega)$,
for any $\omega\in B$. Recalling that $u(\omega,0)=0$ for every
$\omega\in\Omega$, we can then write

$\displaystyle T_{u}\big{(}(s_{k}-M)\mathbf{1}_{B}\big{)}$
$\displaystyle=\int_{B}u(\omega,s_{k}(\omega)-M)\mathrm{d}\mathbb{P}(\omega)=\int_{B}X_{s_{k}(\omega)}(\omega)\mathrm{d}\mathbb{P}(\omega)$
$\displaystyle\stackrel{{\scriptstyle\eqref{ineqstopping}}}{{\leq}}-\frac{1}{n}{\mathbb{P}}(B)+\int_{B}X_{\tau^{\frac{1}{n}}(\omega)}(\omega)\mathrm{d}\mathbb{P}(\omega)$
$\displaystyle=-\frac{1}{n}\mathbb{P}(B)+\int_{\Omega}u(\omega,Z(\omega)-M)\mathrm{d}\mathbb{P}(\omega)$

so that $T_{u}((s_{k}-M)\mathbf{1}_{B})+\frac{1}{n}\mathbb{P}(B)\leq
T_{u}((Z-M)\mathbf{1}_{B})$ reaching a contradiction with (21), as
$\frac{1}{n}\mathbb{P}(B)>0$ does not depend on $k$. Now, we conclude that
${\mathbb{P}}(A^{M}_{cont})=1$ for every arbitrarily fixed $M>0$. Since
$A_{cont}=\bigcup_{M\in\mathbb{N}}A^{M}_{cont}$ we conclude that both
$A_{cont}\in{\mathcal{F}}$ and ${\mathbb{P}}(A)=1$.

###### Corollary A.3

Under the same assumptions of Proposition A.2, there exists a function
$\widehat{u}$ such that $\widehat{u}(\omega,\cdot)$ is nondecreasing and
continuous for every $\omega\in\Omega$, $E=\\{\omega\in\Omega\mid
u(\omega,x)=\widehat{u}(\omega,x)\,\forall x\in\mathbb{R}\\}\in{\mathcal{F}}$
and ${\mathbb{P}}(E)=1$. In particular, $T_{u}(f)=T_{\widehat{u}}(f),\,\forall
f\in\mathcal{L}^{\infty}(\Omega,{\mathcal{F}})$. Finally, if $u(\omega,\cdot)$
is strictly increasing on $\mathbb{R}$ for every $\omega\in\Omega$,
$\widehat{u}(\omega,\cdot)$ can be taken strictly increasing on $\mathbb{R}$
for every $\omega\in\Omega$.

###### Proof 7

As a consequence of Proposition A.2, $A_{cont}\in{\mathcal{F}}$. Set
$A:=A_{cont}$ and define now
$\widehat{u}(\omega,x):=u(\omega,x)\mathbf{1}_{A}(\omega)+x\mathbf{1}_{\Omega\setminus
A}$. Then clearly $\widehat{u}(\omega,\cdot)$ is continuous and nondecreasing
either $\omega\in A$ or $\omega\in\Omega\setminus A$. Further for any fixed
$x\in\mathbb{R}$ the map $\omega\mapsto\widehat{u}(\omega,x)$ is
${\mathcal{F}}$ measurable. As
$u(\omega,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$ implies
$\widehat{u}(\omega,x)\in{\mathcal{L}}^{1}(\Omega,{\mathcal{F}},\mathbb{P})$
for every $x\in\mathbb{R}$ we can invoke Lemma A.1 to obtain that
$\widehat{u}$ is
${\mathcal{F}}\otimes\mathbb{B}_{\mathbb{R}}-\mathbb{B}_{\mathbb{R}}$
measurable, $T_{\widehat{u}}$ is well defined in
${\mathcal{L}}^{\infty}(\Omega,{\mathcal{F}})$ and $T_{u}\equiv
T_{\widehat{u}}$.

## Ethics declarations

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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