# Complete Conditional Type Structures††thanks: Preliminary version. I thank
Pierpaolo Battigalli and Emiliano Catonini for comments on a previous draft of
this paper. Financial support from the Italian Ministry of Education, PRIN
2017, Grant Number 2017K8ANN4, is gratefully acknowledged.

Nicodemo De Vito Department of Decision Sciences, Bocconi University, Milan,
Italy. E-mail address<EMAIL_ADDRESS>
(May 2023)

###### Abstract

Hierarchies of conditional beliefs (Battigalli and Siniscalchi 1999) play a
central role for the epistemic analysis of solution concepts in sequential
games. They are practically modelled by type structures, which allow the
analyst to represent the players’ hierarchies without specifying an infinite
sequence of conditional beliefs. Here, we study type structures that satisfy a
“richness” property, called completeness. This property is defined on the type
structure alone, without explicit reference to hierarchies of beliefs or other
type structures. We provide sufficient conditions under which a complete type
structure represents all hierarchies of conditional beliefs. In particular, we
present an extension of the main result in Friedenberg (2010) to type
structures with conditional beliefs.

KEYWORDS: Conditional probability systems, hierarchies of beliefs, type
structures, completeness, terminality.

JEL: C72, D80

##  1 Introduction

Hierarchies of conditional beliefs (Battigalli and Siniscalchi 1999) play a
central role for the epistemic analysis of solution concepts in sequential
games. Conditional beliefs generalize ordinary probabilistic beliefs, in that
every player is endowed with a collection of conditioning events, and forms
conditional beliefs given each hypothesis in a way that updating is satisfied
whenever possible. Such a collection of measures is called conditional
probability system (CPS, hereafter). A player’s first-order conditional
beliefs are described by a CPS over the space of primitive uncertainty; her
second-order conditional beliefs are described by a CPS over the spaces of
primitive uncertainty and of the co-players’ first-order conditional beliefs;
and so on.

Battigalli and Siniscalchi (1999) show that hierarchies of CPSs can be
practically represented by conditional type structures, i.e., compact models
which mimic Harsanyi’s representation of hierarchies of probabilistic beliefs.
Namely, for each agent there is a set of types. Each type is associated with a
CPS over the set of primitive uncertainty and the set of the co-players’
types. Such structure induces a(n infinite) hierarchy of CPSs for each type.

Here, we study conditional type structures that satisfy a “richness” property,
called completeness. This property—which plays a crucial role for epistemic
foundations111See Dekel and Siniscalchi (2015) for a survey. of some solution
concepts—is defined on the type structure alone, without explicit reference to
hierarchies of CPSs or other type structures. Loosely speaking, a type
structure is complete if it induces all possible conditional beliefs about
types.

We ask: When does a complete type structure represent all hierarchies of
conditional beliefs? The main result of the paper (Theorem 1) can be briefly
summarized as follows. Suppose that a (conditional) type structure is
complete. Then:

(i) if the structure is Souslin, then it is finitely terminal, i.e., it
induces all finite order conditional beliefs;

(ii) if the structure is compact and continuous, then it is terminal, i.e., it
induces all hierarchies of conditional beliefs.

Precise definitions are given in the main text. Here we point out that Theorem
1 is an extension of the main result in Friedenberg (2010) to conditional type
structures. Specifically, Friedenberg studies complete type structures with
beliefs represented by ordinary probabilities; her main result shows that (i)
and (ii) are sufficient conditions for finite terminality and terminality,
respectively. Friedenberg (2010, Section 5) leaves open the question whether
her result still holds when beliefs are represented by CPSs: our result
provides an affirmative answer.

To prove the main result of the paper (Theorem 1), we adopt an approach that
is different from the one in Friedenberg (2010). Specifically, we provide a
construction—based on the set-up in Heifetz (1993)—of the canonical space of
hierarchies of CPSs which allows us to characterize the notion of (finite)
terminality in a convenient way (Proposition 2). With this, the crucial step
of the proof relies on Lemma 3, an “extension” result for CPSs whose proof
makes use of a selection argument.

The structure of this paper is as follows. In Section 2 all the relevant
notation and auxiliary mathematical results are introduced. Section 3
introduces basic definitions and main properties of conditional type
structures. Section 4 is devoted to the presentation and the proof of the main
results of the paper. Section 5 discusses some conceptual and technical
issues. All omitted proofs are in Appendix.

##  2 Preliminaries

A measurable space is a pair $\left(X,\Sigma_{X}\right)$, where $X$ is a set
and $\Sigma_{X}$ is a $\sigma$-algebra, the elements of which are called
events. Throughout this paper, when it is clear from the context which
$\sigma$-algebra on $X$ we are considering, we suppress reference to
$\Sigma_{X}$ and simply write $X$ to denote a measurable space. Furthermore,
given a function $f:X\rightarrow Y$ and a family $\mathcal{F}_{Y}$ of subsets
of $Y$, we let $f^{-1}\left(\mathcal{F}_{Y}\right):=\left\\{E\subseteq
X:\exists F\in\mathcal{F}_{Y},E=f^{-1}\left(F\right)\right\\}$. So, if $Y$ is
a measurable space, then $f^{-1}\left(\Sigma_{Y}\right)$ is the
$\sigma$-algebra on $X$ generated by $f$.

We write $\Delta\left(X\right)$ for the set of probability measures on
$\Sigma_{X}$. Fix measurable spaces $X$ and $Y$. Given a measurable function
$f:X\rightarrow Y$, we let
$\mathcal{L}_{f}:\Delta\left(X\right)\rightarrow\Delta\left(Y\right)$ denote
the pushforward-measure map induced by $f$; that is, for each
$\mu\in\Delta\left(X\right)$, $\mathcal{L}_{f}\left(\mu\right)$ is the image
measure of $\mu$ under $f$, and is defined by
$\mathcal{L}_{f}\left(\mu\right)\left(E\right):=\mu\left(f^{-1}\left(E\right)\right)$
for every $E\in\Sigma_{Y}$.

If $X$ is a topological space, we keep using $\Sigma_{X}$ to denote the Borel
$\sigma$-algebra on $X$. All the topological spaces considered in this paper
are assumed to be metrizable. We consider any product, finite or countable, of
metrizable spaces as a metrizable space with the product topology. Moreover,
we endow each subset of a metrizable space with the subspace topology. A
Souslin (resp. Lusin) space is a topological space that is the image of a
complete, separable metric space under a continuous surjection (resp.
bijection). Clearly, a Lusin space is also Souslin. Examples of Souslin (resp.
Lusin) spaces include analytic (resp. Borel) subsets of a complete separable
metric space. In particular, a Polish space (i.e., a topological space which
is homeomorphic to a complete, separable metric space) is a Lusin space.
Furthermore, if $X$ is a Lusin space, then $\left(X,\Sigma_{X}\right)$ is a
standard Borel space, i.e., there is a Polish space $Y$ such that
$\left(X,\Sigma_{X}\right)$ is isomorphic to $\left(Y,\Sigma_{Y}\right)$. If
$X$ is a Souslin space, then $\left(X,\Sigma_{X}\right)$ is an analytic
measurable space, i.e., there is a Polish space $Y$ and an analytic subset
$A\subseteq Y$ such that $\left(X,\Sigma_{X}\right)$ is isomorphic to
$\left(A,\Sigma_{A}\right)$; see Cohn (2013, Chapter 8).

For a metrizable space $X$, the set $\Delta\left(X\right)$ of (Borel)
probability measures is endowed with the topology of weak convergence. With
this topology, $\Delta\left(X\right)$ becomes a metrizable space.

##  3 Conditional probability systems

We represent the players’ beliefs as conditional probability systems (cf.
Rényi 1955). Fix a measurable space $\left(X,\Sigma_{X}\right)$. A family of
conditioning events of $X$ is a non-empty family
$\mathcal{B}\subseteq\Sigma_{X}$ that does not include the empty set. A
possible interpretation is that an individual is uncertain about the
realization of the “state” $x\in X$, and $\mathcal{B}$ represents a family of
observable events or “relevant hypotheses.” If $X$ is a metrizable space, then
each conditioning event $B\in\mathcal{B}$ is a Borel subset of $X$. In this
case, we say that $\mathcal{B}$ is clopen if each element $B\in\mathcal{B}$ is
both closed and open. For instance, $\mathcal{B}$ is clopen if $X$ is a
(finite) set endowed with the discrete topology; if
$\mathcal{B=}\left\\{X\right\\}$, then $\mathcal{B}$ is trivially clopen.

###### Definition 1

Let $\left(X,\Sigma_{X}\right)$ be a measurable space and
$\mathcal{B}\subseteq\Sigma_{X}$ be a family of conditioning events. A
conditional probability system (CPS) on
$\left(X,\Sigma_{X},\mathcal{B}\right)$ is an array of probability measures
$\mu:=\left(\mu\left(\cdot|B\right)\right)_{B\in\mathcal{B}}$ such that:
(i) for all $B\in\mathcal{B}$, $\mu\left(B|B\right)=1$;
(ii) for all $A\in\Sigma_{X}$ and $B,C\in\mathcal{B}$, if $A\subseteq
B\subseteq C$ then
$\mu\left(A|B\right)\mu\left(B|C\right)=\mu\left(A|C\right)$.

Definition 1 says that a CPS $\mu$ is an element of the set
$\Delta\left(X\right)^{\mathcal{B}}$, i.e., $\mu$ is a function from
$\mathcal{B}$ to $\Delta\left(X\right)$.222For every pair of sets $X$ and $Y$,
we let $Y^{X}$ denote the set of functions with domain $X\,$ and codomain $Y$.
We write $\mu\left(\cdot|B\right)$ to stress the interpretation as a
conditional probability given event $B\in\mathcal{B}$. Condition (ii) is the
chain rule of conditional probabilities and it can be written as follows: if
$A\subseteq B\subseteq C$, then

$\mu\left(B|C\right)>0\Rightarrow\mu\left(A|B\right)=\frac{\mu\left(A|C\right)}{\mu\left(B|C\right)}\text{.}$

We let $\Delta^{\mathcal{B}}\left(X\right)$ denote the set of CPSs on
$\left(X,\Sigma_{X},\mathcal{B}\right)$. The following result (whose proof can
be found in Appendix A) records some topological properties of
$\Delta^{\mathcal{B}}\left(X\right)$ when $X$ is a metrizable space and
$\mathcal{B}$ is countable.333Lemma 1 is a generalization of analogous results
(for the case when $X$ is Polish) in Battigalli and Siniscalchi (1999, Lemma
1).

###### Lemma 1

Fix a metrizable space $X$ and a countable family
$\mathcal{B}\subseteq\Sigma_{X}$ of conditioning events.
(i) The space $\Delta^{\mathcal{B}}\left(X\right)$ is metrizable.
(ii) If $X$ is Souslin or Lusin, so is $\Delta^{\mathcal{B}}\left(X\right)$.
(iii) Suppose that $\mathcal{B}$ is clopen. Then
$\Delta^{\mathcal{B}}\left(X\right)$ is compact if and only if $X$ is compact.

Note that if $X$ is a Polish space, then $\Delta^{\mathcal{B}}\left(X\right)$
may fail to be Polish. But, by Lemma 1.(ii),
$\Delta^{\mathcal{B}}\left(X\right)$ is a Lusin space. We can conclude that
$\Delta^{\mathcal{B}}\left(X\right)$ is a Polish space provided that
$\mathcal{B}$ is clopen (cf. Battigalli and Siniscalchi 1999, Lemma 1).

Fix measurable spaces $\left(X,\Sigma_{X}\right)$ and
$\left(Y,\Sigma_{Y}\right)$, and families $\mathcal{B}_{X}\subseteq\Sigma_{X}$
and $\mathcal{B}_{Y}\subseteq\Sigma_{Y}$ of conditioning events. Suppose that
$f:X\rightarrow Y$ is a measurable function such that

$f^{-1}\left(\mathcal{B}_{Y}\right)=\mathcal{B}_{X}\text{.}$

The function
$\overline{\mathcal{L}}_{f}:\Delta^{\mathcal{B}_{X}}\left(X\right)\rightarrow\Delta^{\mathcal{B}_{Y}}\left(Y\right)$
defined by

$\overline{\mathcal{L}}_{f}\left(\mu\right)\left(E|B\right):=\mu\left(f^{-1}\left(E\right)|f^{-1}\left(B\right)\right)\text{,}$

where $E\in\Sigma_{Y}$ and $B\in\mathcal{B}_{Y}$, is the pushforward-CPS map
induced by $f$. Following the terminology in Rao (2005), we call
$\overline{\mathcal{L}}_{f}\left(\mu\right)$ image conditional law of $\mu$
under $f$. Note that, for any $\mu\in\Delta^{\mathcal{B}_{X}}\left(X\right)$,
we can write $\overline{\mathcal{L}}_{f}\left(\mu\right)$ as

$\overline{\mathcal{L}}_{f}\left(\mu\right)=\left(\mathcal{L}_{f}\left(\mu\left(\cdot|f^{-1}\left(B\right)\right)\right)\right)_{B\in\mathcal{B}_{Y}}\text{,}$

where $\mathcal{L}_{f}:\Delta\left(X\right)\rightarrow\Delta\left(Y\right)$ is
the pushforward-measure map induced by $f$.

We record some basic results on the pushforward-CPS map that will be used
extensively throughout the paper (for the proofs, see Appendix A). In
particular, Lemma 2.(i) ensures that $\overline{\mathcal{L}}_{f}$ is well-
defined and justifies the terminology: if
$\mu\in\Delta^{\mathcal{B}_{X}}\left(X\right)$, then
$\overline{\mathcal{L}}_{f}\left(\mu\right)$ is a CPS on
$\left(Y,\Sigma_{Y},\mathcal{B}_{Y}\right)$.

###### Lemma 2

Fix measurable spaces $\left(X,\Sigma_{X}\right)$ and
$\left(Y,\Sigma_{Y}\right)$, and families $\mathcal{B}_{X}\subseteq\Sigma_{X}$
and $\mathcal{B}_{Y}\subseteq\Sigma_{Y}$ of conditioning events. Suppose that
$f:X\rightarrow Y$ is a measurable function such that
$f^{-1}\left(\mathcal{B}_{Y}\right)=\mathcal{B}_{X}$. The following statements
hold.
(i) The map
$\overline{\mathcal{L}}_{f}:\Delta^{\mathcal{B}_{X}}\left(X\right)\rightarrow\Delta^{\mathcal{B}_{Y}}\left(Y\right)$
is well-defined.
(ii) Suppose that $\mathcal{B}_{X}$ and $\mathcal{B}_{Y}$ are countable, $X$
is a metrizable space and $Y$ is a Souslin space. If $f$ is Borel measurable
(resp. continuous), then $\overline{\mathcal{L}}_{f}$ is Borel measurable
(resp. continuous).

A special case of image CPS induced by a function is of particular
interest—namely, the marginalization of a CPS on a product space. Consider
measurable spaces $X$ and $Y$, and denote by $\pi_{X}$ the coordinate
projection from $X\times Y$ onto $X$. Fix a family
$\mathcal{B}\subseteq\Sigma_{X}$ of conditioning events, and define
$\mathcal{B}_{X\times Y}$ as

$\mathcal{B}_{X\times
Y}:=\left(\pi_{X}\right)^{-1}\left(\mathcal{B}\right)=\left\\{C\subseteq
X\times Y:\exists B\in\mathcal{B},C=B\times Y\right\\}\text{,}$ (3.1)

that is, $\mathcal{B}_{X\times Y}$ is the set of all cylinders $B\times Y$
with $B\in\mathcal{B}$. The function
$\overline{\mathcal{L}}_{\pi_{X}}:\Delta^{\mathcal{B}_{X\times
Y}}\left(X\times Y\right)\rightarrow\Delta^{\mathcal{B}}\left(X\right)$
defined by

$\overline{\mathcal{L}}_{\pi_{X}}\left(\mu\right):=\left(\mathcal{L}_{\pi_{X}}\left(\mu\left(\cdot|B\right)\right)\right)_{B\in\mathcal{B}}$

is called marginal-CPS map, and
$\overline{\mathcal{L}}_{\pi_{X}}\left(\mu\right)$ is called the marginal on
$X$ of $\mu\in\Delta^{\mathcal{B}_{X\times Y}}\left(X\times Y\right)$.

##  4 Type structures and hierarchies of conditional beliefs

Throughout, we fix a two-player set $I$;444The assumption of a two-player set
is merely for notational convenience. The analysis can be equivalently carried
out with any finite set $I$ with cardinality greater than two. given a player
$i\in I$, we let $j$ denote the other player in $I$. We assume that both
players share a common measurable space $\left(S,\Sigma_{S}\right)$, called
space of primitive uncertainty. For each $i\in I$, there is a family
$\mathcal{B}_{i}\subseteq\Sigma_{S}$ of conditioning events. One
interpretation (which is borrowed from Battigalli and De Vito 2021) is the
following: $S$ is a product set, viz. $S:=\mathop{\textstyle\prod}_{i\in
I}S_{i}$, and each element $s:=\left(s_{i}\right)_{i\in I}$ is an objective
description of players’ behavior in a game with complete information and
without chance moves—technically, $\left(s_{i}\right)_{i\in I}$ is a strategy
profile. Each player is uncertain about the “true” behavior $s\in S$,
including his own. If the game has sequential moves, then each
$\mathcal{B}_{i}$ is a collection of observable events; that is, each
$B\in\mathcal{B}_{i}$ is the set of strategy profiles inducing an information
set of player $i$. Other interpretations of $S$ and
$\left(\mathcal{B}_{i}\right)_{i\in I}$ are also possible; a more thorough
discussion can be found in Battigalli and Siniscalchi (1999, pp. 191-192). The
results in this paper do not hinge on a specific interpretation.

From now on, we maintain the following technical assumptions on $S$ and
$\left(\mathcal{B}_{i}\right)_{i\in I}$:

  * •

$S$ is a Souslin space, and

  * •

$\mathcal{B}_{i}\subseteq\Sigma_{S}$ is countable for every $i\in I$.

Following Battigalli and Siniscalchi (1999), we adopt the following notational
convention.

Convention 1. Given a product space $X\times Y$ and a family
$\mathcal{B}\subseteq\Sigma_{X}$ of conditioning events of $X$, the family of
conditioning events of $X\times Y$ is $\mathcal{B}_{X\times Y}$ as defined in
(3.1). Accordingly, we let $\Delta^{\mathcal{B}}\left(X\times Y\right)$ denote
the set of CPSs on $\left(X\times Y,\Sigma_{X\times Y},\mathcal{B}_{X\times
Y}\right)$.

###  4.1 Type structures

We use the framework of type structures (or “type spaces”) to model players’
hierarchies of conditional beliefs. We adopt the following definition of type
structure (cf. Battigalli and Siniscalchi 1999).

###### Definition 2

An $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
is a tuple

$\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$

such that, for every $i\in I$,
(i) the type set $T_{i}$ is a metrizable space;
(ii) the belief map
$\beta_{i}:T_{i}\rightarrow\Delta^{\mathcal{B}_{i}}\left(S\times T_{j}\right)$
is Borel measurable.

Each element of $T_{i}$, viz. $t_{i}$, is called (player $i$’s) type.

Definition 2 says that, for any $i\in I$, $T_{i}$ represents the set of player
$i$’s possible “ways to think.” Each type $t_{i}\in T_{i}$ is associated with
a CPS on the set of primitive uncertainty as well as on the possible “ways to
think” (types) of player $j$. Each conditioning event for
$\beta_{i}\left(t_{i}\right)$ has the form $B\times T_{j}$, with
$B\in\mathcal{B}_{i}$.

Note that in Definition 2, and like in Battigalli and Siniscalchi (1999), it
is implicitly assumed that players have introspective beliefs—that is, beliefs
about their own types. We maintain the implicit assumption that players are
introspective, hence they know their own way to think, and there is common
certainty of this conditional on each relevant hypothesis
$B\in\mathcal{B}_{i}$ ($i\in I$). Such assumption is innocuous: an explicit
formalization of introspective beliefs would not change the substance of our
results (cf. Heifetz and Samet 1998, p. 330).

If $\mathcal{B}_{i}\mathcal{=}\left\\{S\right\\}$ for every player $i\in I$,
then each set $\Delta^{\mathcal{B}_{i}}\left(S\times T_{j}\right)$ can be
naturally identified with $\Delta\left(S\times T_{j}\right)$. In this case,
Definition 2 coincides essentially with the definition in Friedenberg
(2010),555The only difference is that $S$ is assumed to be a Polish space in
Friedenberg (2010). Such difference is immaterial for the remainder of the
analysis. and we say that $\mathcal{T}$ is an ordinary type structure.
Moreover, we will sometimes refer to type structures via Definition 2 as
conditional type structures.

For the statement of our main results, we also need the following definitions.

###### Definition 3

An $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$ is
(i) Souslin (resp. Lusin, compact) if, for every $i\in I$, the type set
$T_{i}$ is a Souslin (resp. Lusin, compact) space;666In Friedenberg (2010),
a(n ordinary) type structure is called analytic if each type set is an
analytic subset of a Polish space—hence a metrizable Souslin (sub)space. We
adopt the definition of Souslin type structure because we can extend our
analysis (as we do in the Supplementary Appendix) without assuming
metrizability of the topological spaces.
(ii) continuous if, for every $i\in I$, the belief map $\beta_{i}$ is
continuous.

Next, we introduce the notion of completeness for a type structure. This
notion is due to Brandenburger (2003).

###### Definition 4

An $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$ is complete if, for every $i\in I$, the belief map $\beta_{i}$ is
surjective.

In words, completeness says that, for every player $i\in I$ and for every
conditional belief $\mu\in\Delta^{\mathcal{B}_{i}}\left(S\times T_{j}\right)$
that player $i$ can hold, there is a type of player $i$ which induces that
belief. Thus, it is a “richness” requirement which may not be satisfied by
some type structures. For instance, suppose that $S$ is not a singleton. Then
a type structure where the type set of some player has finite cardinality is
not complete.

A type structure provides an implicit representation of the hierarchies of
beliefs. To address the question whether a complete type structure represents
all hierarchies of beliefs, we need to formally clarify how type structures
generate a collection of hierarchies of beliefs for each player. This is
illustrated in the following section.

###  4.2 The canonical space of hierarchies

In this section we first offer a construction of the set of all hierarchies of
conditional beliefs satisfying a coherence condition. Loosely speaking,
coherence means that lower-order beliefs are the marginals of higher-order
beliefs. The construction—which is based on the set-up in Heifetz (1993)—shows
that this set of hierarchies identifies in a natural way a type structure,
which we call it “canonical.” Next, we show how each profile of types in a
type structure can be associated with an element of the constructed set of
hierarchies. This part is standard (cf. Heifetz and Samet 1998, 1999).

Our construction of the “canonical” space of hierarchies is formally different
from the one in Battigalli and Siniscalchi (1999), as it relies on a
costruction scheme based on the notion of projective limit. Here, we report
the essential ingredients for the main results of the paper. A detailed
analysis is deferred to Appendix B where we briefly review the background
notions and results on projective limit theory. In Section 6 we will compare
our construction of the canonical space with the construction à la Battigalli
and Siniscalchi (1999).

####  4.2.1 From hierarchies to types

To construct the set of hierarchies of conditional beliefs we define
recursively, for each player, two sequences of sets as well as a sequence of
conditioning events. The first sequence, $\left(\Theta_{i}^{n}\right)_{n\geq
0}$, represents player $i$’s $\left(n+1\right)$-order domain of uncertainty,
for each $n\geq 0$. The second sequence, $\left(H_{i}^{n}\right)_{n\geq 1}$,
consists of player $i$’s $n$-tuples of coherent conditional beliefs over these
space. The notion of coherence, formally defined below, says that, conditional
on any relevant hypothesis, beliefs at different order do not contradict one
another.

Formally, for each player $i\in I$, let

$\displaystyle\Theta_{i}^{0}$ $\displaystyle:$ $\displaystyle=S\text{,}$
$\displaystyle\mathcal{B}_{i}^{0}$ $\displaystyle:$
$\displaystyle=\mathcal{B}_{i}\text{,}$ $\displaystyle H_{i}^{1}$
$\displaystyle:$
$\displaystyle=\Delta^{\mathcal{B}_{i}^{0}}\left(\Theta_{i}^{0}\right)\text{.}$

The set $\Theta_{i}^{0}$ is player $i$’s $1$-order (primitive) domain of
uncertainty, and a first-order belief, viz. $\mu_{i}^{1}$, is an element of
the set $H_{i}^{1}$.

For $n\geq 1$ assume that $\left(\Theta_{i}^{m}\right)_{m=0,...,n-1}$,
$\left(\mathcal{B}_{i}^{m}\right)_{m=0,...,n-1}$ and
$\left(H_{i}^{m}\right)_{m=1,...,n}$ have been defined for each player $i\in
I$. Then, for each $i\in I$, let

$\Theta_{i}^{n}:=\Theta_{i}^{0}\times H_{j}^{n}\text{.}$

That is, $\Theta_{i}^{n}$ is player $i$’s $\left(n+1\right)$-order domain of
uncertainty: it consists of the space of primitive uncertainty and what player
$j\neq i$ believes about the space of primitive uncertainty, what player $j$
believes about what player $i$ believes about the space of primitive
uncertainty,…, and so on, up to level $n$. For each $i\in I$ and $n\geq 1$,
let $\pi_{i}^{n,n+1}:H_{i}^{n+1}\rightarrow H_{i}^{n}$ and
$\rho_{i}^{n-1,n}:\Theta_{i}^{n}\rightarrow\Theta_{i}^{n-1}$ denote the
coordinate projections. By construction, these maps satisfy the following
property:

$\forall i\in I,\forall n\geq
2,\rho_{i}^{n-1,n}=\left(\mathrm{Id}_{\Theta_{i}^{0}},\pi_{j}^{n-1,n}\right)\text{,}$

where $\mathrm{Id}_{\Theta_{i}^{0}}$ is the identity on $\Theta_{i}^{0}$.

To define players’ conditional beliefs on the $\left(n+1\right)$-th order
domain of uncertainty, for each player $i\in I$, let

$\displaystyle\mathcal{B}_{i}^{n}$ $\displaystyle:$
$\displaystyle=\left(\rho_{i}^{n-1,n}\right)^{-1}\left(\mathcal{B}_{i}^{n-1}\right)=\left\\{C\subseteq\Theta_{i}^{n}:\exists
B\in\mathcal{B}_{i}^{n-1},C=\left(\rho_{i}^{n-1,n}\right)^{-1}\left(B\right)\right\\}\text{,}$
$\displaystyle H_{i}^{n+1}$ $\displaystyle:$
$\displaystyle=\left\\{\left(\left(\mu_{i}^{1},...,\mu_{i}^{n}\right),\mu_{i}^{n+1}\right)\in
H_{i}^{n}\times\Delta^{\mathcal{B}_{i}^{n}}\left(\Theta_{i}^{n}\right):\overline{\mathcal{L}}_{\rho_{i}^{n-1,n}}\left(\mu_{i}^{n+1}\right)=\mu_{i}^{n}\right\\}\text{.}$

Specifically, $\mathcal{B}_{i}^{n}$ represents the set of relevant hypotheses
upon which player $i$’s $\left(n+1\right)$-th order conditional beliefs are
defined. That is,
$\mu_{i}^{n+1}\in\Delta^{\mathcal{B}_{i}^{n}}\left(\Theta_{i}^{n}\right)$ is
player $i$’s $\left(n+1\right)$-th order CPS with
$\mu_{i}^{n+1}\left(\cdot|B\right)\in\Delta\left(\Theta_{i}^{n}\right)$,
$B\in\mathcal{B}_{i}^{n}$. Recursively, it can be checked that, for all $n\geq
1$,

$\mathcal{B}_{i}^{n}=\left\\{C\subseteq\Theta_{i}^{n}:\exists
B\in\mathcal{B}_{i},C=B\times H_{j}^{n}\right\\}\text{,}$

i.e., $\mathcal{B}_{i}^{n}$ is a set of cylinders in $\Theta_{i}^{n}$
generated by $\mathcal{B}_{i}$. If $\mathcal{B}_{i}$ is clopen, then every
$B\in\mathcal{B}_{i}^{n}$ is clopen in $\Theta_{i}^{n}$ since each coordinate
projection $\rho_{i}^{n-1,n}$ is a continuous function. By definition of each
$\Theta_{i}^{n}$, we write, according to Convention 1,

$\Delta^{\mathcal{B}_{i}^{n}}\left(\Theta_{i}^{n}\right)=\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right)\text{.}$

The set $H_{i}^{n+1}$ is the set of player $i$’s $\left(n+1\right)$-tuples of
CPSs on $\Theta_{i}^{0}$, $\Theta_{i}^{1}$,…, $\Theta_{i}^{n}$. The condition
on $\mu_{i}^{n+1}$ in the definition of $H_{i}^{n+1}$ is the coherence
condition mentioned above. Given the recursive construction of the sets, CPSs
$\mu_{i}^{n+1}$ and $\mu_{i}^{n}$ both specify a (countable) array of
conditional beliefs on the domain of uncertainty $\Theta_{i}^{n-1}$, and those
beliefs cannot be contradictory. Formally, for all $B\in\mathcal{B}_{i}^{n-1}$
and for every event $E\subseteq\Theta_{i}^{n-1}$,

$\mu_{i}^{n+1}\left(\left(\rho_{i}^{n-1,n}\right)^{-1}\left(E\right)\left|\left(\rho_{i}^{n-1,n}\right)^{-1}\left(B\right)\right.\right)=\mu_{i}^{n}\left(E|B\right)\text{.}$

That is, the conditional belief
$\mu_{i}^{n+1}(\cdot|(\rho_{i}^{n-1,n})^{-1}\left(B\right))$ must assign to
event $\left(\rho_{i}^{n-1,n}\right)^{-1}\left(E\right)$ the same number as
$\mu_{i}^{n}\left(\cdot|B\right)$ assigns to event $E$. Accordingly, we say
that the $(n+1)$-order belief $\mu_{i}^{n+1}$ is a coherent extension of
$\mu_{i}^{n}$.

The proof of the following result can be found in Appendix B.

###### Remark 1

For each $i\in I$ and $n\geq 1$, $H_{i}^{n+1}$ is a closed subset of
$H_{i}^{n}\times\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right)$. So
$H_{i}^{n+1}$ is a Souslin (resp. Lusin) space provided $S$ is a Souslin
(resp. Lusin) space. If $\mathcal{B}_{i}$ is clopen for each $i\in I$, then
$H_{i}^{n+1}$ is compact if and only if $S$ is compact.

In the limit, for each $i\in I$, let

$\displaystyle H_{i}$
$\displaystyle:=\left\\{\left(\mu_{i}^{1},\mu_{i}^{2},...\right)\in\mathop{\textstyle\prod}_{n=0}^{\infty}\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right):\forall
n\geq 1,\left(\mu_{i}^{1},...,\mu_{i}^{n}\right)\in
H_{i}^{n}\right\\}\text{,}$ $\displaystyle\Theta_{i}$ $\displaystyle:=S\times
H_{j}\text{.}$

###### Remark 2

For each $i\in I$, $H_{i}$ is a closed subset of
$\mathop{\textstyle\prod}_{n=0}^{\infty}\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right)$.
So $H_{i}$ is a Souslin (resp. Lusin) space provided $S$ is a Souslin (resp.
Lusin) space. If $\mathcal{B}_{i}$ is clopen for each $i\in I$, then $H_{i}$
is compact if and only if $S$ is compact.

As formally shown in Appendix B (which reports also the proof of Remark 2),
the spaces $H_{i}$ and $\Theta_{i}$ ($i\in I$) are identified with (i.e., they
are homeomorphic to) the non-empty projective limits of the sequences
$\left(H_{n}^{i}\right)_{n\geq 1}$ and $\left(\Theta_{n}^{i}\right)_{n\geq
0}$. The following result corresponds to Proposition 2 in Battigalli and
Siniscalchi (1999), and the proof is in Appendix B.

###### Proposition 1

For each $i\in I$, the spaces $H_{i}$ and
$\Delta^{\mathcal{B}_{i}}\left(S\times H_{j}\right)$ are homeomorphic.

The set $H:=\mathop{\textstyle\prod}_{i\in I}H_{i}$ is the set of all pairs of
collectively coherent hierarchies of conditional beliefs; that is, $H$ is the
set of pairs of coherent hierarchies satisfying common full belief of
coherence.777An event $E$ is fully believed under a CPS
$\left(\mu\left(\cdot|B\right)\right)_{B\in\mathcal{B}}$ if
$\mu\left(E|B\right)=1$ for every $B\in\mathcal{B}$. The notion of “common
full belief of coherence” is made explicit in the alternative, “top-down”
construction of the canonical space à la Battigalli and Siniscalchi (1999), as
we discuss in Section 6. A note on terminology: in Battigalli and Siniscalchi
(1999) the expression “certainty” is used in place of “full belief.”

The homeomorphisms in Proposition 1 are “canonical” in the following sense.
Every coherent hierarchy $\left(\mu_{i}^{1},\mu_{i}^{2},...\right)$ of player
$i$ is associated with a unique CPS $\mu_{i}$ on the space of primitive
uncertainty and the coherent hierarchies of the co-player, i.e., $S\times
H_{j}$. Then, for all $n\geq 0$, the marginal of $\mu_{i}$ on player $i$’s
$\left(n+1\right)$-order domain of uncertainty, viz. $\Theta_{i}^{n}$, is
precisely what it should be, namely $\mu_{i}^{n+1}$. A formal definition of
such homeomorphisms is not needed for the statements and proofs of the results
in this paper—the reader can consult Appendix B, in particular the proof of
Proposition 1 for details. Instead, we will make use of the following
implication of Proposition 1: we can define an
$\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}^{\mathrm{c}}:=\left(S,\left(\mathcal{B}_{i},T_{i}^{\mathrm{c}},\beta_{i}^{\mathrm{c}}\right)_{i\in
I}\right)$ by letting, for each $i\in I$,

$T_{i}^{\mathrm{c}}:=H_{i}\text{,}$

and
$\beta_{i}^{\mathrm{c}}:T_{i}^{\mathrm{c}}\rightarrow\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}^{\mathrm{c}}\right)$ is the “canonical” homeomorphism. Following the
terminology in the literature, we call $\mathcal{T}^{\mathrm{c}}$ the
canonical type structure.

The following properties of $\mathcal{T}^{\mathrm{c}}$ are immediate from
Remark 2 and Proposition 1.

###### Remark 3

Structure $\mathcal{T}^{\mathrm{c}}$ is Souslin, continuous and complete. If
$S$ is Lusin, then $\mathcal{T}^{\mathrm{c}}$ is Lusin. If $\mathcal{B}_{i}$
is clopen for each $i\in I$, then $\mathcal{T}^{\mathrm{c}}$ is a compact type
structure if and only if $S$ is a compact space.

####  4.2.2 From types to hierarchies

The next step is to consider the relationship between the set of hierarchies
constructed in the previous section and any other type structure. In so doing,
we specify how types generate (collectively) coherent hierarchies of
conditional beliefs. As we did in the previous section, given a set $X$, we
let $\mathrm{Id}_{X}$ denote the identity map.

Fix an $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type
structure
$\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$. We construct a natural (Borel) measurable map, called hierarchy
map, which unfolds the higher-order beliefs of each player $i\in I$. This map
assigns to each $t_{i}\in T_{i}$ a hierarchy of beliefs in $H_{i}$.

For each $i\in I$, let $h_{-i}^{0}:\Theta_{i}^{0}\times
T_{j}\rightarrow\Theta_{i}^{0}$ be the projection map (recall that
$\Theta_{i}^{0}:=S$ for each $i\in I$). The “first-order map” for each player
$i$, viz. $h_{i}^{1}:T_{i}\rightarrow H_{i}^{1}$, is defined by

$h_{i}^{1}\left(t_{i}\right):=\overline{\mathcal{L}}_{h_{-i}^{0}}\left(\beta_{i}\left(t_{i}\right)\right)\text{.}$

In words, $h_{i}^{1}\left(t_{i}\right)$ is the marginal on $S$ of CPS
$\beta_{i}\left(t_{i}\right)$. Measurability of each map $h_{i}^{1}$ holds by
Lemma 2 and measurability of belief maps.

With this, for each $i\in I$, let $h_{-i}^{1}:\Theta_{i}^{0}\times
T_{j}\rightarrow\Theta_{i}^{0}\times H_{j}^{1}=\Theta_{i}^{1}$ be the map
defined as $h_{-i}^{1}:=\left(\mathrm{Id}_{S},h_{j}^{1}\right)$. That is, for
each pair $\left(s,t_{j}\right)\in\Theta_{i}^{0}\times T_{j}$, the expression
$h_{-i}^{1}\left(s,t_{j}\right)=\left(s,h_{j}^{1}\left(t_{j}\right)\right)\in\Theta_{i}^{1}$
describes the profile $s$ and the first-order beliefs for type $t_{j}\in
T_{j}$. Standard arguments show that, for each $i\in I$, $h_{-i}^{1}$ is a
measurable map;888Since $\Theta_{i}^{0}$ and $H_{j}^{1}$ are Souslin, hence
separable spaces, the Borel $\sigma$-algebra generated by the product topology
coincides with the product $\sigma$-algebra (Bogachev 2007, Lemma 6.4.2). With
this, Lemma 4.49 in Aliprantis and Border (2006) yields the result.
furthermore, it can be checked that $h_{-i}^{1}$ satisfies

$h_{-i}^{0}=\rho_{i}^{0,1}\circ h_{-i}^{1}\text{ \ and \
}\mathcal{B}_{\Theta_{i}^{0}\times
T_{j}}=\left(h_{-i}^{1}\right)^{-1}\left(\mathcal{B}_{i}^{1}\right)\text{.}$

Recursively, we define the “$\left(n+1\right)$th-orders” maps. For $n\geq 1$,
assume that measurable maps $h_{i}^{n}:T_{i}\rightarrow H_{i}^{n}$ have been
defined for each player $i\in I$. Moreover, for each $i\in I$, assume that
$h_{-i}^{n}:\Theta_{i}^{0}\times T_{j}\rightarrow\Theta_{i}^{0}\times
H_{j}^{n}=\Theta_{i}^{n}$ is the unique measurable function, defined as
$h_{-i}^{n}:=\left(\mathrm{Id}_{S},h_{j}^{n}\right)$, which satisfies

$\mathcal{B}_{\Theta_{i}^{0}\times
T_{j}}=\left(h_{-i}^{n}\right)^{-1}\left(\mathcal{B}_{i}^{n}\right)$ (4.1)

and

$h_{-i}^{n-1}=\rho_{i}^{n-1,n}\circ h_{-i}^{n}\text{.}$ (4.2)

Fix a player $i\in I$. Note that, since (4.1) holds,
$\overline{\mathcal{L}}_{h_{-i}^{n}}:\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}\right)\rightarrow\Delta^{\mathcal{B}_{i}^{n}}\left(\Theta_{i}^{n}\right)$
is a well-defined measurable map by Lemma 2. With this, define
$h_{i}^{n+1}:T_{i}\rightarrow
H_{i}^{n}\times\Delta^{\mathcal{B}_{i}^{n}}\left(\Theta_{i}^{n}\right)$ by

$h_{i}^{n+1}\left(t_{i}\right):=\left(h_{i}^{n}\left(t_{i}\right),\overline{\mathcal{L}}_{h_{-i}^{n}}\left(\beta_{i}\left(t_{i}\right)\right)\right)\text{.}$

Using the same arguments as above, it is easily verified that $h_{i}^{n+1}$ is
a measurable map.

###### Remark 4

$h_{i}^{n+1}\left(T_{i}\right)\subseteq H_{i}^{n+1}$ for each $i\in I$.

Proof. We show that, for all $t_{i}\in T_{i}$,

$\overline{\mathcal{L}}_{\rho_{i}^{n-1,n}}\left(\overline{\mathcal{L}}_{h_{-i}^{n}}\left(\beta_{i}\left(t_{i}\right)\right)\right)=\overline{\mathcal{L}}_{h_{-i}^{n-1}}\left(\beta_{i}\left(t_{i}\right)\right)\text{.}$

To this end, pick any $t_{i}\in T_{i}$. Then, for all events
$E\subseteq\Theta_{i}^{n-1}$ and all $B\in\mathcal{B}_{i}^{n-1}$,

$\displaystyle\overline{\mathcal{L}}_{\rho_{i}^{n-1,n}}\left(\overline{\mathcal{L}}_{h_{-i}^{n}}\left(\beta_{i}\left(t_{i}\right)\right)\right)\left(E\left|B\right.\right)$
$\displaystyle=$
$\displaystyle\overline{\mathcal{L}}_{h_{-i}^{n}}\left(\beta_{i}\left(t_{i}\right)\right)\left(\left(\rho_{i}^{n-1,n}\right)^{-1}\left(E\right)\left|\left(\rho_{i}^{n-1,n}\right)^{-1}\left(B\right)\right.\right)$
$\displaystyle=$
$\displaystyle\beta_{i}\left(t_{i}\right)\left(\left(\rho_{i}^{n-1,n}\circ
h_{-i}^{n}\right)^{-1}\left(E\right)\left|\left(\rho_{i}^{n-1,n}\circ
h_{-i}^{n}\right)^{-1}\left(B\right)\right.\right)$ $\displaystyle=$
$\displaystyle\overline{\mathcal{L}}_{h_{-i}^{n-1}}\left(\beta_{i}\left(t_{i}\right)\right)\left(E\left|B\right.\right)\text{,}$

where the last equality follows from (4.2). $\blacksquare$

It is easily seen that, for each $t_{i}\in T_{i}$,

$h_{i}^{n+1}\left(t_{i}\right)=\left(\overline{\mathcal{L}}_{h_{-i}^{0}}\left(\beta_{i}\left(t_{i}\right)\right),...,\overline{\mathcal{L}}_{h_{-i}^{n-1}}\left(\beta_{i}\left(t_{i}\right)\right),\overline{\mathcal{L}}_{h_{-i}^{n}}\left(\beta_{i}\left(t_{i}\right)\right)\right)\text{.}$

Finally, for each $i\in I$, the map
$h_{i}:T_{i}\rightarrow\mathop{\textstyle\prod}_{n=0}^{\infty}\Delta^{\mathcal{B}_{i}}\left(S\times
H_{j}^{n}\right)$ is defined by

$h_{i}\left(t_{i}\right):=\left(\overline{\mathcal{L}}_{h_{-i}^{n}}\left(\beta_{i}\left(t_{i}\right)\right)\right)_{n\geq
0}\text{.}$

Thus, $h_{i}\left(t_{i}\right)$ is the hierarchy generated by type $t_{i}\in
T_{i}$. Each type generates a (collectively) coherent hierarchy of beliefs,
i.e., $h_{i}\left(T_{i}\right)\subseteq H_{i}$.

###### Remark 5

For each $i\in I$, the map $h_{i}:T_{i}\rightarrow H_{i}$ is well-defined and
Borel measurable. If $\mathcal{T}$ is continuous, then $h_{i}$ is continuous
for each $i\in I$.

Proof. The induction argument on $n\geq 1$ provided above shows that each map
$h_{i}^{n}$ ($i\in I$) is measurable. Moreover, for every player $i\in I$,
$h_{i}:T_{i}\rightarrow H_{i}$ is a well-defined map because, by Remark 4,
$h_{i}^{n}\left(t_{i}\right)\in H_{i}^{n}$ for all $t_{i}\in T_{i}$ and all
$n\geq 1$. To prove measurability of the maps $h_{i}$ ($i\in I$) notice that,
for every $n\geq 0$,
$\overline{\mathcal{L}}_{h_{-i}^{n}}\circ\beta_{i}:T_{i}\rightarrow\Delta^{\mathcal{B}_{i}}\left(S\times
H_{j}^{n}\right)=\Delta^{\mathcal{B}_{i}^{n}}\left(\Theta_{i}^{n}\right)$ is a
Borel measurable function from a metrizable space to a Souslin space (cf.
Lemma 1 and Remark 1). It follows from Schwartz (1973, Lemma 11, p. 106) that
each map $h_{i}$ is Borel measurable. Finally, if $\mathcal{T}$ is continuous,
then an induction argument on $n\geq 0$ shows—by virtue of Lemma 2—that the
maps $\overline{\mathcal{L}}_{h_{-i}^{n}}\circ\beta_{i}$ ($i\in I$) are
continuous. The conclusion that also the maps $h_{i}$ ($i\in I$) are
continuous follows from Engelking (1989, Proposition 2.3.6). $\blacksquare$

For future reference, we record the following property of the hierarchy map.
The proof is omitted since it can be easily adapted from analogous proofs in
Heifetz and Samet (1998) and Battigalli and Siniscalchi (1999).999The
Supplementary Appendix contains omitted proofs of some results/claims in this
paper.

###### Remark 6

The map $h:=\left(h_{i}\right)_{i\in I}:\mathop{\textstyle\prod}_{i\in
I}T_{i}\rightarrow\mathop{\textstyle\prod}_{i\in I}H_{i}$ is a type morphism
from $\mathcal{T}$ to $\mathcal{T}^{\mathrm{c}}$. That is, for each $i\in I$
and for each $t_{i}\in T_{i}$,

$\beta_{i}^{\mathrm{c}}\left(h_{i}\left(t_{i}\right)\right)=\overline{\mathcal{L}}_{\left(\mathrm{Id}_{S},h_{j}\right)}\left(\beta_{i}\left(t_{i}\right)\right)\text{.}$
(4.3)

##  5 Terminal type structures

Having specified how types generate hierarchies of beliefs, we can formally
introduce the notion of (finite) terminality for conditional type structures.
The following definitions are extensions to conditional type structures of the
definitions put forward by Friedenberg (2010, Section 2) for ordinary type
structures.

###### Definition 5

An $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$ is finitely terminal if, for each
$\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}^{\ast}:=\left(S,\left(\mathcal{B}_{i},T_{i}^{\ast},\beta_{i}^{\ast}\right)_{i\in
I}\right)$, each type $t_{i}^{\ast}\in T_{i}^{\ast}$ and each
$n\in\mathbb{N}$, there is a type $t_{i}\in T_{i}$ such that
$h_{i}^{\ast,n}\left(t_{i}^{\ast}\right)=h_{i}^{n}\left(t_{i}\right)$.

###### Definition 6

An $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$ is terminal if, for each
$\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}^{\ast}:=\left(S,\left(\mathcal{B}_{i},T_{i}^{\ast},\beta_{i}^{\ast}\right)_{i\in
I}\right)$ and each type $t_{i}^{\ast}\in T_{i}^{\ast}$, there is a type
$t_{i}\in T_{i}$ such that
$h_{i}^{\ast}\left(t_{i}^{\ast}\right)=h_{i}\left(t_{i}\right)$.

Definition 5 says that $\mathcal{T}$ is finitely terminal if, for every type
$t_{i}^{\ast}$ that occurs in some structure $\mathcal{T}^{\ast}$ and every
$n\in\mathbb{N}$, there exists a type $t_{i}$ in $\mathcal{T}$ whose hierarchy
agrees with the hierarchy generated by $t_{i}^{\ast}$ up to level $n$.
Definition 6 says that $\mathcal{T}$ is terminal if, for every type
$t_{i}^{\ast}$ that occurs in some structure $\mathcal{T}^{\ast}$, there
exists a type $t_{i}$ in $\mathcal{T}$ which generates the same hierarchy as
$t_{i}^{\ast}$.

The notion of terminality in Definition 6 can be equivalently expressed as
follows: $\mathcal{T}$ is terminal if, for every structure
$\mathcal{T}^{\ast}$, there exists a hierarchy morphism from
$\mathcal{T}^{\ast}$ to $\mathcal{T}$, i.e., a map that preserves the
hierarchies of beliefs. In Section 6 we will compare this notion of
terminality with related notions in the literature. Here we show that (a)
Definition 5 is equivalent to the requirement that a type structure generates
all finite-order beliefs consistent with coherence and common full belief of
coherence; and (b) Definition 6 is equivalent to the requirement that a type
structure generates all collectively coherent hierarchies of beliefs.

The proof of the following remark follows by inspection of Definitions 5 and
6.

###### Remark 7

An $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}$ is finitely terminal if and only if, for each
$\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}^{\ast}$, each player $i\in I$ and each $n\in\mathbb{N}$,

$h_{i}^{\ast,n}\left(T_{i}^{\ast}\right)\subseteq
h_{i}^{n}\left(T_{i}\right)\text{.}$

An $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}$ is terminal if and only if, for each
$\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}^{\ast}$, and for each player $i\in I$,

$h_{i}^{\ast}\left(T_{i}^{\ast}\right)\subseteq
h_{i}\left(T_{i}\right)\text{.}$

The following result establishes the relationship between any (finitely)
terminal type structure and the canonical space of hierarchies.

###### Proposition 2

Fix an $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type
structure
$\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$.
(i) $\mathcal{T}$ is finitely terminal if and only if
$h_{i}^{n}\left(T_{i}\right)=H_{i}^{n}$ for each $i\in I$ and each
$n\in\mathbb{N}$.
(ii) $\mathcal{T}$ is terminal if and only if $h_{i}\left(T_{i}\right)=H_{i}$
for each $i\in I$.

Proof. We show only part (ii) (the proof of part (i) is virtually identical).
Suppose first that $\mathcal{T}$ is terminal. Fix a player $i\in I$. Since
$h_{i}\left(T_{i}\right)\subseteq H_{i}$ holds by Remark 5, we only need to
show that $H_{i}\subseteq h_{i}\left(T_{i}\right)$. Consider structure
$\mathcal{T}^{\mathrm{c}}$. Then

$h_{i}^{\mathrm{c}}\left(H_{i}\right)=H_{i}\subseteq
h_{i}\left(T_{i}\right)\text{,}$

where the first equality holds because
$h_{i}^{\mathrm{c}}:T_{i}^{\mathrm{c}}\rightarrow H_{i}$ is the identity map,
and the inclusion follows from Remark 7.

Suppose now that $\mathcal{T}$ satisfies $h_{i}\left(T_{i}\right)=H_{i}$ for
each $i\in I$. For any structure $\mathcal{T}^{\ast}$, we have
$h_{i}^{\ast}\left(T_{i}^{\ast}\right)\subseteq H_{i}=h_{i}\left(T_{i}\right)$
for each $i\in I$, where the inclusion follows from Remark 5. By Remark 7,
$\mathcal{T}$ is terminal. $\blacksquare$

Proposition 2 provides a characterization of (finite) terminality which turns
out to be useful for the proof of the main result. It is basically a version
of Result 2.1 (and Proposition B1.(ii)) in Friedenberg (2010).

###  5.1 When complete structures are terminal

The main result of this paper is the following theorem.

###### Theorem 1

Fix an $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type
structure
$\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$.
(i) If $\mathcal{T}$ is Souslin and complete, then $\mathcal{T}$ is finitely
terminal.
(ii) If $\mathcal{T}$ is complete, compact and continuous, then $\mathcal{T}$
is terminal.

If $\mathcal{B}_{i}\mathcal{=}\left\\{S\right\\}$ for every $i\in I$, then
Theorem 1 corresponds to Theorem 3.1 in Friedenberg (2010). The proof of
Theorem 1 relies on the following result, whose proof—which makes use of Von
Neumann Selection Theorem—can be found in Appendix C.

###### Lemma 3

Fix Souslin spaces $X$, $Y$ and $Z$, and a countable family
$\mathcal{B}\subseteq\Sigma_{X}$ of conditioning events. Let
$f_{1}:Y\rightarrow Z$ be Borel measurable, and define $f_{2}:X\times
Y\rightarrow X\times Z$ as $f_{2}:=\left(\mathrm{Id}_{X},f_{1}\right)$. Then:
(i) $f_{2}$ is Borel measurable, and the map
$\overline{\mathcal{L}}_{f_{2}}:\Delta^{\mathcal{B}}\left(X\times
Y\right)\rightarrow\Delta^{\mathcal{B}}\left(X\times Z\right)$ is well-
defined;
(ii) if $f_{1}$ is surjective, then $\overline{\mathcal{L}}_{f_{2}}$ is
surjective.

The proof of part (i) of Theorem 1 is by induction on $n\in\mathbb{N}$. The
proof of the base step does not rely on the hypothesis that $\mathcal{T}$ is
Souslin. Lemma 3 is used only in the inductive (and crucial) step. The proof
of part (ii) of Theorem 1 uses the same arguments as in Friedenberg (2010).

Proof of Theorem 1. Part (i): We show that
$h_{i}^{n}\left(T_{i}\right)=H_{i}^{n}$ for each $n\in\mathbb{N}$ and each
$i\in I$. With this, the result follows from Proposition 2.(i). The proof is
by induction on $n\in\mathbb{N}$.

(Base step: $n=1$) Fix an arbitrary type $\bar{t}_{i}\in T_{i}$ for each
player $i\in I$. Next, for each $i\in I$, let $\varphi_{i}:S\rightarrow T_{i}$
be the map such that $\varphi_{i}\left(s\right)=\bar{t}_{i}$ for every $s\in
S$. Each map $\varphi_{i}$ is continuous (so measurable), since it is a
constant function. For each $i\in I$, define the function
$\psi_{i}:S\rightarrow S\times T_{j}$ as
$\psi_{i}:=\left(\mathrm{Id}_{S},\varphi_{j}\right)$. Such function is a
topological embedding, and it satisfies
$h_{-i}^{0}\circ\psi_{i}=\mathrm{Id}_{S}$ and
$\mathcal{B}_{i}=\psi_{i}^{-1}\left(\mathcal{B}_{S\times T_{j}}\right)$. Fix a
player $i\in I$. We show that $h_{i}^{1}\left(T_{i}\right)=H_{i}^{1}$. To this
end, pick any $\mu_{i}^{1}\in H_{i}^{1}$. By Lemma 1.(i),
$\overline{\mathcal{L}}_{\psi_{i}}\left(\mu_{i}^{1}\right)\in\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}\right)$. Moreover, the marginal of
$\overline{\mathcal{L}}_{\psi_{i}}\left(\mu_{i}^{1}\right)$ on $S$ is
$\mu_{i}^{1}$: for all events $E\subseteq S$ and all $B\in\mathcal{B}_{i}$,

$\displaystyle\overline{\mathcal{L}}_{h_{-i}^{0}}\left(\overline{\mathcal{L}}_{\psi_{i}}\left(\mu_{i}^{1}\right)\right)\left(E\left|B\right.\right)$
$\displaystyle=$
$\displaystyle\overline{\mathcal{L}}_{\psi_{i}}\left(\mu_{i}^{1}\right)\left(\left(h_{-i}^{0}\right)^{-1}\left(E\right)\left|\left(h_{-i}^{0}\right)^{-1}\left(B\right)\right.\right)$
$\displaystyle=$
$\displaystyle\mu_{i}^{1}\left(\left(h_{-i}^{0}\circ\psi_{i}\right)^{-1}\left(E\right)\left|\left(h_{-i}^{0}\circ\psi_{i}\right)^{-1}\left(B\right)\right.\right)$
$\displaystyle=$
$\displaystyle\mu_{i}^{1}\left(E\left|B\right.\right)\text{,}$

where the last equality follows from the fact that
$h_{-i}^{0}\circ\psi_{i}=\mathrm{Id}_{S}$. By completeness, there exists
$t_{i}\in T_{i}$ such that
$\beta_{i}\left(t_{i}\right)=\overline{\mathcal{L}}_{\psi_{i}}\left(\mu_{i}^{1}\right)$,
hence
$h_{i}^{1}\left(t_{i}\right):=\overline{\mathcal{L}}_{h_{-i}^{0}}\left(\beta_{i}\left(t_{i}\right)\right)=\mu_{i}^{1}$.

(Inductive step: $n\geq 1$) Suppose that the result is true for $n\geq 1$. We
show that it is also true for $n+1$. Fix a player $i\in I$ and
$\left(\mu_{i}^{1},...,\mu_{i}^{n},\mu_{i}^{n+1}\right)\in H_{i}^{n+1}$. We
need to show the existence of $t_{i}\in T_{i}$ such that
$h_{i}^{n+1}\left(t_{i}\right)=\left(\mu_{i}^{1},...,\mu_{i}^{n},\mu_{i}^{n+1}\right)$.
First note that, by the inductive hypothesis, the function $h_{-i}^{n}:S\times
T_{j}\rightarrow S\times H_{j}^{n}$ defined as
$h_{-i}^{n}:=\left(\mathrm{Id}_{S},h_{j}^{n}\right)$ is a Borel measurable
surjection. By Lemma 3, the induced function
$\overline{\mathcal{L}}_{h_{-i}^{n}}:\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}\right)\rightarrow\Delta^{\mathcal{B}_{i}}\left(S\times H_{j}^{n}\right)$
is surjective as well. With this, we can claim that, since
$\mu_{i}^{n+1}\in\Delta^{\mathcal{B}_{i}^{n}}\left(\Theta_{i}^{n}\right)=\Delta^{\mathcal{B}_{i}}\left(S\times
H_{j}^{n}\right)$, there is $v_{i}\in\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}\right)$ such that
$\overline{\mathcal{L}}_{h_{-i}^{n}}\left(v_{i}\right)=\mu_{i}^{n+1}$.
Completeness yields the existence of a type $t_{i}\in T_{i}$ such that
$\beta_{i}\left(t_{i}\right)=v_{i}$. Hence
$\overline{\mathcal{L}}_{h_{-i}^{n}}\left(\beta_{i}\left(t_{i}\right)\right)=\mu_{i}^{n+1}$.
Now note that, by coherence of $\mu_{i}^{n+1}$ with respect to $\mu_{i}^{m}$
($m\leq n$),

$\mu_{i}^{m}=\overline{\mathcal{L}}_{h_{-i}^{m-1}}\left(\beta_{i}\left(t_{i}\right)\right)$

for all $m\in\mathbb{N}$ such that $1\leq m\leq n$ (this can be verified using
arguments analogous to those in Remark 4). Thus,

$\displaystyle h_{i}^{n+1}\left(t_{i}\right)$ $\displaystyle=$
$\displaystyle\left(\overline{\mathcal{L}}_{h_{-i}^{0}}\left(\beta_{i}\left(t_{i}\right)\right),...,\overline{\mathcal{L}}_{h_{-i}^{n-1}}\left(\beta_{i}\left(t_{i}\right)\right),\overline{\mathcal{L}}_{h_{-i}^{n}}\left(\beta_{i}\left(t_{i}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\left(\mu_{i}^{1},...,\mu_{i}^{n},\mu_{i}^{n+1}\right)\text{,}$

as required. Since $i\in I$ is arbitrary, this concludes the proof of the
inductive step.

Part (ii): We show that $h_{i}\left(T_{i}\right)=H_{i}$ for each $i\in I$.
With this, Proposition 2.(ii) yields the result. Fix a player $i\in I$. Fix
also an arbitrary $\bar{h}:=\left(\mu_{i}^{1},\mu_{i}^{2},...\right)\in
H_{i}$. We show the existence of $t_{i}\in T_{i}$ such that
$h_{i}\left(t_{i}\right)=\bar{h}$. Since $\mathcal{T}$ is a continuous type
structure, Remark 5 entails that the maps $h_{i}^{n}$ ($n\in\mathbb{N}$) and
$h_{i}$ are continuous. For each $n\in\mathbb{N}$, let

$T_{i}\left[\bar{h}\left|n\right.\right]:=\left\\{t_{i}\in
T_{i}:h_{i}^{n}\left(t_{i}\right)=\left(\mu_{i}^{1},...,\mu_{i}^{n}\right)\right\\}$

be the set of types whose generated hierarchy of beliefs agree with $\bar{h}$
up to level $n$. Note that $T_{i}\left[\bar{h}\left|n\right.\right]$ is the
inverse image under $h_{i}^{n}$ of the singleton
$\left\\{\left(\mu_{i}^{1},...,\mu_{i}^{n}\right)\right\\}\subseteq H^{n}$.
Also note that $\mathcal{T}$ is a Souslin type structure, because each type
space $T_{i}$ is compact metrizable, hence Souslin. Thus, by part (i),
$T_{i}\left[\bar{h}\left|n\right.\right]\neq\emptyset$. Moreover, by
continuity of $h_{i}^{n}$, $T_{i}\left[\bar{h}\left|n\right.\right]$ is a
closed subset of $T_{i}$. Since $T_{i}$ is compact, so is
$T_{i}\left[\bar{h}\left|n\right.\right]$. By inspection of the definition of
$h_{i}^{n}$, it readily follows that
$T_{i}\left[\bar{h}\left|n\right.\right]\subseteq
T_{i}\left[\bar{h}\left|m\right.\right]$ for every $m\leq n$. Hence
$\left(T_{i}\left[\bar{h}\left|n\right.\right]\right)_{n\in\mathbb{N}}$ is
decreasing sequence of non-empty compact sets. By the finite intersection
property,
$\cap_{n\in\mathbb{N}}T_{i}\left[\bar{h}\left|n\right.\right]\neq\emptyset$.
Thus, there exists $t_{i}\in T_{i}$ such that
$h_{i}\left(t_{i}\right)=\bar{h}$, as desired. $\blacksquare$

Some comments on Theorem 1 are in order. First, complete type structures that
are finitely terminal can be easily constructed. A simple example—which uses
the ideas in Brandenburger et al. (2008, proof of Proposition 7.2)—is the
following. For each $i\in I$, let $T_{i}$ be the Baire space, i.e., the (non-
compact) Polish space $\mathbb{N}^{\mathbb{N}}$. Every Souslin space is the
image of $\mathbb{N}^{\mathbb{N}}$ under a continuous function.101010It is
well-known (see, e.g., Cohn 2003, Proposition 8.2.7) that every non-empty
Polish space is the image of $\mathbb{N}^{\mathbb{N}}$ under a continuous
function. Using this result, it is easy to check—by inspection of
definitions—that an analogous conclusion also holds for Souslin spaces (cf.
Cohn 2003, Corollary 8.2.8). Since $\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}\right)$ is a Souslin space by Lemma 1, there exists a continuous
surjection $\beta_{i}:T_{i}\rightarrow\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}\right)$. These maps give us a Souslin and complete type structure
$\mathcal{T}$ that is finitely terminal, but not necessarily terminal. A more
complex example of a(n ordinary) complete, finitely terminal type structure
which is not terminal can be found in Friedenberg and Keisler (2021, Section
6).

Second, we point out that complete, compact and continuous type structures may
not exist. This is so because the structural hypothesis on the families of
conditioning events $\mathcal{B}_{i}$ ($i\in I$) are quite weak—each element
of $\mathcal{B}_{i}$ is a Borel subset of $S$. To elaborate, suppose that
$\mathcal{T}$ is complete and compact. Completeness yields
$\beta_{i}\left(T_{i}\right)=\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}\right)$ for each player $i\in I$. If $\mathcal{T}$ were continuous, then
compactness of $T_{i}$ would imply compactness of
$\Delta^{\mathcal{B}_{i}}\left(S\times T_{j}\right)$ as well, because the
continuous image of a compact set is compact. But, in general,
$\Delta^{\mathcal{B}_{i}}\left(S\times T_{j}\right)$ is not a compact space
even if the underlying space $S\times T_{j}$ is compact (cf. Lemma
1).111111Let $X$ be a compact space. The set
$\Delta^{\mathcal{B}}\left(X\right)$ may fail to be a closed subset of
$\Delta\left(X\right)^{\mathcal{B}}$, which is a compact metrizable space.
Yet, by Lemma 1.(ii), $\Delta^{\mathcal{B}}\left(X\right)$ is a Lusin subspace
of $\Delta\left(X\right)^{\mathcal{B}}$. In particular,
$\Delta^{\mathcal{B}}\left(X\right)$ is a Borel subset of
$\Delta\left(X\right)^{\mathcal{B}}$; see Remark A.1 in Appendix A. In other
words, unless each family $\mathcal{B}_{i}$ ($i\in I$) satisfies some specific
assumptions, there is no guarantee that a complete, compact and continuous
type structure exists—in particular, complete and continuous structures may
fail the compactness requirement.121212Analogously, complete and compact type
structures may fail the continuity requirement. If this is the case, Theorem
1.(ii) still holds, but vacuously because the antecedent of the conditional is
false. An immediate implication of this fact is that the completeness test—as
formalized by completeness, compactness and continuity—cannot be applied.

With this in mind, suppose now that each family $\mathcal{B}_{i}$ ($i\in I$)
is clopen. If $S$ is a compact space, then complete, compact and continuous
type structures do exist. The canonical structure is a prominent example, but
there are also complete, compact and continuous structures which can be
distinct from the canonical one. For instance, one can take each $T_{i}$ to be
the Cantor space $\left\\{0,1\right\\}^{\mathbb{N}}$, a compact metrizable
space. Lemma 1.(iii) yields that each $\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}\right)$ is compact metrizable, so there exists a continuous surjection
$\beta_{i}:T_{i}\rightarrow\Delta^{\mathcal{B}_{i}}\left(S\times T_{j}\right)$
(Aliprantis and Border 2006, Theorem 3.60). The resulting structure
$\mathcal{T}$ is complete, compact and continuous, hence terminal.

Finally, note that if each $\mathcal{B}_{i}$ is clopen, then compactness of
$S$ is a necessary condition for the existence of a complete, compact and
continuous structure $\mathcal{T}$. Indeed, continuity and surjectivity of the
belief maps entail that each set $\Delta^{\mathcal{B}_{i}}\left(S\times
T_{j}\right)$ is compact; by Lemma 1.(iii) and Tychonoff’s theorem, $S$ is a
compact space.

###  5.2 When terminal structures are complete

This section addresses the following question: If $\mathcal{T}$ is a terminal
type structure, is $\mathcal{T}$ complete? As shown by Friedenberg (2010), the
answer is No. Specifically, Section 4.1 in Friedenberg (2010) provides an
example of an ordinary type structure which is terminal but not complete. The
key feature of Friedenberg’s example is that the type structure is redundant,
i.e., there are types associated with the same hierarchy of beliefs.131313For
ease of reference, we briefly report Friedenberg’s example. Suppose that the
two-player set is $I=\left\\{a,b\right\\}$ and $S$ is a singleton, viz.
$S=\left\\{s\right\\}$. Hence, there is a unique hierarchy for each player,
consisting of a sequence of Dirac point masses. Consider the type structure
$\mathcal{T}$ in which
$T_{a}=\left\\{t_{a}^{\prime},t_{a}^{\prime\prime}\right\\}$,
$T_{b}=\left\\{t_{b}\right\\}$,
$\beta_{a}\left(t_{a}^{\prime}\right)\left(\left\\{\left(s,t_{b}\right)\right\\}\right)=\beta_{a}\left(t_{a}^{\prime\prime}\right)\left(\left\\{\left(s,t_{b}\right)\right\\}\right)=1$,
and
$\beta_{b}\left(t_{b}\right)\left(\left\\{\left(s,t_{a}^{\prime}\right)\right\\}\right)=1$.
Structure $\mathcal{T}$ generates the unique belief hierarchy, hence it is
terminal. But $\mathcal{T}$ is not complete because $\beta_{b}$ is not
surjective. It should be noted that types $t_{a}^{\prime}$ and
$t_{a}^{\prime\prime}$ generate the same hierarchy. Proposition 4.1 in
Friedenberg (2010) shows that if an ordinary, terminal type structure
satisfies a certain measurability condition on the type sets, then it is
complete.

That said, we now ask: Is there an analogue of Friedenberg’s result for
conditional type structures? The answer is Yes and the proof is short and
simple. To state and prove the desired result, we need a formal definition of
non-redundancy.

###### Definition 7

An $\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure
$\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$ is non-redundant if, for every $i\in I$, the map
$h_{i}:T_{i}\rightarrow H_{i}$ is injective.

In words, a structure is non-redundant if distinct types generate distinct
hierarchies of beliefs. The canonical structure $\mathcal{T}^{\mathrm{c}}$ is
non-redundant.

###### Theorem 2

Let $\mathcal{T}:=\left(S,\left(\mathcal{B}_{i},T_{i},\beta_{i}\right)_{i\in
I}\right)$ be a Souslin and non-redundant
$\left(S,\left(\mathcal{B}_{i}\right)_{i\in I}\right)$-based type structure.
If $\mathcal{T}$ is terminal, then $\mathcal{T}$ is complete.

Proof. If $\mathcal{T}$ is non-redundant and terminal, then, by Proposition
2.(ii), each map $h_{i}:T_{i}\rightarrow H_{i}$ is a Borel measurable
bijection. Since $T_{i}$ and $H_{i}$ are Souslin spaces (hence,
$\left(T_{i},\Sigma_{T_{i}}\right)$ and $\left(H_{i},\Sigma_{H_{i}}\right)$
are analytic measurable spaces), it follows from Proposition 8.6.2 in Cohn
(2013) (see also Bogachev 2007, Theorem 6.7.3, or Schwartz 1973, Lemma 16, pp.
107-108) that each hierarchy map $h_{i}$ is a Borel isomorphism. Fix a player
$i\in I$. Since $\beta_{i}^{\mathrm{c}}\circ h_{i}$ and
$\overline{\mathcal{L}}_{\left(\mathrm{Id}_{S},h_{j}\right)}$ are Borel
isomorphisms, it follows from (4.3) in Remark 6 that $\beta_{i}$ is a Borel
isomorphism. Thus, $\mathcal{T}$ is complete. $\blacksquare$

We conclude this section with two remarks on Theorem 2. First, the hypothesis
that $\mathcal{T}$ is Souslin is essential for the proof of Theorem 2. Such
hypothesis allows us to say that each map $h_{i}:T_{i}\rightarrow H_{i}$ is a
Borel isomorphism, hence to conclude that also
$\overline{\mathcal{L}}_{\left(\mathrm{Id}_{S},h_{j}\right)}$ is a Borel
isomorphism. Suppose now that $\mathcal{T}$ is non-redundant and terminal, but
not Souslin. In this case, $h_{i}$ is a Borel measurable bijection but not
necessarily an isomorphism. Is this enough to conclude that also
$\overline{\mathcal{L}}_{\left(\mathrm{Id}_{S},h_{j}\right)}$ is a bijection?
The answer is No because
$\overline{\mathcal{L}}_{\left(\mathrm{Id}_{S},h_{j}\right)}$ may fail to be a
surjection! This issue is discussed with an example in Appendix C.

Second, suppose that $S$ is a Polish space as in Friedenberg’s framework. Then
every Souslin, non-redundant and terminal type structure $\mathcal{T}$ has a
“nice” measurability property: for every $i\in I$,
$\left(T_{i},\Sigma_{T_{i}}\right)$ is standard Borel. Indeed, note that
$\mathcal{T}^{\mathrm{c}}$ is a Lusin type structure (Remark 3), hence each
measurable space $\left(H_{i},\Sigma_{H_{i}}\right)$ is standard Borel. Since
each hierarchy map $h_{i}:T_{i}\rightarrow H_{i}$ is a Borel isomorphism, it
readily follows that $\left(T_{i},\Sigma_{T_{i}}\right)$ is standard Borel as
well. In the context of ordinary type structures, Proposition 4.1 in
Friedenberg (2010) states that every non-redundant, terminal structure with
standard Borel type spaces is complete. Theorem 2 shows that this result
remains true also for conditional type structures.

##  6 Discussion

In this section we discuss some conceptual aspects of the paper, and we
compare our work with the closest related literature.

##### On the canonical type structure

The method of the proof of the main result (Theorem 1) takes the canonical
structure constructed in Section 4.2, viz. $\mathcal{T}^{\mathrm{c}}$, as a
natural benchmark for the characterization of terminality of (possibly
different) type structures. The modelling strategy used in Section 4.2 to
obtain $\mathcal{T}^{\mathrm{c}}$ is complementary to the one following the
approch of Brandenburger and Dekel (1993) to coherence, and adopted by
Battigalli and Siniscalchi (1999). This alternative strategy first constructs
the space of all possible (i.e., not necessarily coherent) belief hierarchies,
and then it imposes coherence and common full belief of coherence. Such
construction, unlike the construction of Section 4.2, is more explicit about
what is assumed to be common full belief.

Both approaches are equivalent—they yield two homeomorphic structures. We have
chosen the construction à la Heifetz (1993) because it allows to characterize
the notion of finite terminality in a convenient way. For instance, to
formalize a statement like “structure $\mathcal{T}$ generates all finite order
beliefs,” we simply require that each map $h_{i}^{n}:T_{i}\rightarrow
H_{i}^{n}$ be onto for all $n\in\mathbb{N}$ (cf. Proposition 2.(i)).

In terms of technical assumptions, there are differences between our
construction of $\mathcal{T}^{\mathrm{c}}$ and the one in Battigalli and
Siniscalchi (1999). Actually, our construction is a generalization of theirs
since we allow for weaker conditions on (i) the family of conditioning events,
and (ii) the topological property of the primitive uncertainty space $S$. As
for (i), Battigalli and Siniscalchi (1999) restrict attention to clopen
families of conditioning events, while our approach requires that each
conditioning event be a Borel set. We will discuss in the next subsection why,
from a conceptual standpoint, clopeness of conditioning events could be
restrictive in some applications of interest.

As for (ii), Battigalli and Siniscalchi (1999) consider the case, typically
adopted in the literature, of a Polish space of primitive uncertainty. Here we
considered the more general cases of Souslin and Lusin spaces. There are
reasons to find such generalization useful. The main reason is that the
Lusin/Souslin properties devolve from $X$ to
$\Delta^{\mathcal{B}}\left(X\right)$ (see Lemma 1). An analogous conclusion
does not hold if $X$ is a Polish space—unless clopeness of conditioning events
is assumed. Yet, $\Delta^{\mathcal{B}}\left(X\right)$ has a “nice” topological
property: it is a Lusin space, hence standard Borel as a measurable space.
This is all that matters because the Borel structure on
$\Delta^{\mathcal{B}}\left(X\right)$ generated by a Souslin topology on a set
$X$ coincides with the “natural” measure-theoretic structure on
$\Delta^{\mathcal{B}}\left(X\right)$. Let $\mathcal{A}^{+}$ be the
$\sigma$-algebra on $\Delta^{\mathcal{B}}\left(X\right)$ generated by the sets

$b_{p}\left(E|B\right):=\left\\{v\in\Delta^{\mathcal{B}}\left(X\right):v\left(E|B\right)\geq
p\right\\}\text{,}$

where $E\subseteq X$ is an event, $B\in\mathcal{B}$, and
$p\in\left[0,1\right]$. A statement like “conditional on $B$, player $j$
assigns probability at least $p$ to $E$” corresponds exactly to the set
$b_{p}\left(E|B\right)$. As discussed by Battigalli and Siniscalchi (1999, pp.
192-193), $\mathcal{A}^{+}$ is the conceptually appropriate structure which
allows to formalize complex statements concerning higher-order beliefs.
Appendix A shows that when $X$ is separable and metrizable, the Borel
$\sigma$-algebra on $\Delta^{\mathcal{B}}\left(X\right)$ coincides with
$\mathcal{A}^{+}$.141414An analogous conclusion holds whenever $X$ is not
metrizable but it is a completely regular Souslin space, as shown in the the
Supplementary Appendix.

To sum up, one message of this paper is the following. If we consider the case
of a Polish space of primitive uncertainty and a non-clopen family of
conditioning events, we should not worry if “Polishness” of the higher-order
spaces is lost: the Souslin/Lusin property is always preserved, and it is
well-suited for the formal construction of a canonical structure.

##### On (non-)clopeness of conditioning events

In many game-theoretic applications, the conditioning events of a CPS
correspond—roughly speaking—to information sets; they can be thought of as
representing the possible evidence that a player can obtain in a game. To
illustrate, consider a sequential game with finite sets of strategies,
complete information and imperfect monitoring of past moves. If we
assume—contrary to what we did in Section 4—that each player $i$ is certain
about his own behavior$\backslash$strategy, then his space of primitive
uncertainty is the co-player’s strategy set $S_{j}$. In this environment, a
conditioning event for player $i$’s first-order beliefs is the set
$S_{j}\left(h_{i}\right)$ of co-player’s strategies that do no prevent an
information set $h_{i}$ to be reached;151515If $j$ plays strategy $s_{j}\in
S_{j}\left(h_{i}\right)$, then $h_{i}$ can be reached; whether it is reached
depends on $i$’s play. that is, if $h_{i}$ occurs, then $i$ is informed that
the co-player is playing strategies in $S_{j}\left(h_{i}\right)$. Due to
finiteness of the game, such conditioning events are clopen subsets of $S_{j}$
provided that each (finite) set is given the obvious topology, i.e., the
discrete topology. Any type structure associated with this game has the
property that each conditioning event has the form
$S_{j}\left(h_{i}\right)\times T_{j}$, which is a clopen set.

Yet, there are situations under which clopeness is not satisfied. Consider
again a sequential game with a finite set of strategy profiles, imperfect
monitoring of past actions, and with incomplete information about payoff
functions. This incomplete-information scenario can be modelled by the
introduction of sets $\Theta_{i}$ of “information-types” for each player $i\in
I$. An information-type of player $i$, viz. $\theta_{i}\in\Theta_{i}$,
describes the private information of $i$ about payoffs: for example,
$\theta_{i}$ may be a preference or a productivity parameter. All uncertainty
and private information about payoffs is captured by the profile
$\left(\theta_{i}\right)_{i\in I}$ with the understanding that each $i$ knows
only his information-type $\theta_{i}$. With this, player $i$’s space of
primitive uncertainty is $\Theta_{j}\times S_{j}$, and a conditioning event
for player $i$’s first-order beliefs is the set $S_{j}\left(h_{i}\right)$ of
profiles $\left(\theta_{j},s_{j}\right)$ consistent with an information set
$h_{i}$.

The requirement that each conditioning event $S_{j}\left(h_{i}\right)$ is
clopen could be restrictive in this case. One concrete example is when
$\Theta_{j}$ is an uncountable set, say $\Theta_{j}=\left[0,1\right]$, and at
the beginning of the game $i$ gets a private signal about the realization of
$\theta_{j}$. Even if the signal is discrete, the set
$S_{j}\left(h_{i}\right)$ may not be clopen. For instance, player $i$’s signal
reveals that $j$’s information type $\theta_{j}$ is greater or equal than a
threshold $\hat{\theta}_{j}\in\left[0,1\right]$.

Other examples of non-clopen conditioning events are discussed by Battigalli
and Tebaldi (2019). These authors extend the epistemic analysis of Battigalli
and Siniscalchi (2002) from finite dynamic games to a special class of
infinite dynamic games—called simple dynamic games—under the clopeness
assumption. Battigalli and Tebaldi (2002, Section 5.6) address the question
whether their results still hold in a more general set-up (type structures)
whereby clopeness is not satisfied. We think that the canonical structure
constructed in this paper may be used to extend their results to a wide class
of infinite dynamic games.

##### Hierarchy vs type morphisms

The notion of terminality used in this paper reflects the following
requirement on a type structure: $\mathcal{T}$ is terminal if there exists a
hierarchy morphism—i.e., a map which preserves the hierarchies of
(conditional) beliefs—from every type structure $\mathcal{T}^{\ast}$ to
$\mathcal{T}$. A distinct notion of terminality, often called universality
(e.g., Heifetz and Samet 1998, Battigalli and Siniscalchi 1999), requires the
existence of a unique type morphism—i.e., a map which preserves the belief
maps—from every type structure $\mathcal{T}^{\ast}$ to $\mathcal{T}$. The
latter notion is stronger than the former: by a straightforward extension of
the proofs in Heifetz and Samet (1998) to conditional type structures, it can
be shown that a type morphism is a hierarchy morphism, but the converse does
not hold. The canonical structure $\mathcal{T}^{\mathrm{c}}$ is
terminal/universal because the hierarchy map is a type morphism (cf. Remark 6)
and, as it can be easily proved by standard techniques, it is also unique.

What is the relationship between terminal structures and the
canonical/universal type structure $\mathcal{T}^{\mathrm{c}}$? Fix a Souslin
and terminal type structure $\mathcal{T}$. In light of the above discussion,
there is a unique type morphism from $\mathcal{T}$ to
$\mathcal{T}^{\mathrm{c}}$. But there is also a hierarchy morphism from
$\mathcal{T}^{\mathrm{c}}$ to $\mathcal{T}$. In particular, such hierarchy
morphism need not be unique because structure $\mathcal{T}$ may be redundant
(cf. Theorem 2). If this is the case, there could be many hierarchy-preserving
maps from $\mathcal{T}^{\mathrm{c}}$ to $\mathcal{T}$ that do not preserve the
beliefs maps, that is, the commutativity property of a type morphism (see
(4.3) in Remark 6) need not be satisfied.

In the context of ordinary type structures, Friedenberg and Meier (2011)
provide a necessary and sufficient condition, called strong measurability,
under which hierarchy and type morphisms coincide: precisely, a hierarchy
morphism from $\mathcal{T}^{\ast}$ to $\mathcal{T}$ is a type morphism if and
only if $\mathcal{T}$ is strongly measurable. They also provide sufficient
conditions on structure $\mathcal{T}$ which guarantee strong measurability.
For instance, if both the space of primitive uncertainty and the type spaces
are standard Borel spaces, then strong measurability is equivalent to non-
redundancy of $\mathcal{T}$. We suspect (but have not proved) that similar
results are also valid for conditional type structures, although their proof
cannot directly extended to our case. This extension requires additional
investigation which is beyond the scope of this paper.

##### Complete type structures and first-order beliefs

As already observed in Section 5.1, a careful reading of the base step of the
proof of Theorem 1.(i) reveals that the hypothesis that $\mathcal{T}$ is a
Souslin structure is not used. Put differently, the proof of the base step
shows that every complete type structure generates all first-order beliefs,
and the result remains true under the assumption that $S$ is a metrizable
space. This observation suggests that, in the topology-free framework à la
Heifetz and Samet (1998), a complete type structure generates all first-order
conditional beliefs. The intuition is correct, and the proof—with obvious
modifications—is similar.161616To provide a topology-free definition of
conditional type structures, the following changes should be made in
Definition 2. Every type set $T_{i}$ is endowed with a $\sigma$-algebra
$\mathcal{A}_{i}$, and every belief map $\beta_{i}$ is
$\mathcal{A}_{i}$-measurable. For any measurable space
$\left(X,\Sigma_{X}\right)$, the set $\Delta^{\mathcal{B}}\left(X\right)$ is
endowed with the $\sigma$-algebra $\mathcal{A}^{+}$ described above. Any
product of measurable spaces is endowed with the product $\sigma$-algebra.
This conclusion is the only possible one if the topological assumptions in
this paper are relaxed. Indeed, Theorem 5 in Friedenberg and Keisler (2021)
shows a powerful negative result: Within Zermelo-Fraenkel set theory with the
axiom of choice, one cannot prove that a complete (ordinary) type structure
generates all second-order beliefs.

##### Completeness for lexicographic type structures

The completeness concept plays a crucial role in the epistemic analysis of
Brandenburger et al. (2008) on iterated admissibility in games. Their analysis
makes use of complete lexicographic type structures, i.e., type structures
where types are mapped to lexicographic probability systems (LPSs). An
overview on LPSs and lexicographic type structures can be found in Dekel and
Siniscalchi (2015, Section 12.8).

Is there an analogue of Theorem 1 (and Theorem 3.1 in Friedenberg 2010) for
lexicographic type structures? The answer is a qualified Yes. Part (i) of
Theorem 1 holds within the lexicographic framework—see Catonini and De Vito
(2018). But the analogue of part (ii) of Theorem 1 holds only vacuously, since
the antecedent of the conditional is never fulfilled: a complete, compact and
continuous type structure does not exist in the lexicographic framework. This
is so because the space of LPSs on a given set is not a compact space
(Catonini and De Vito 2021, Section 3.2). Consequently, a complete and
continuous lexicographic type structure cannot be compact.

Catonini and De Vito (2021, Theorem 5) show the existence of a complete and
continuous lexicographic type structure which is not terminal. Their method of
proof adapts techniques used by Friedenberg and Keisler to establish the
existence of a complete, ordinary type structure that is finitely terminal,
but not terminal (Friedenberg and Keisler 2021, Theorem 1).

##### Non-metrizable spaces

Technically, we have restricted our analysis to metrizable topological spaces
as this simplifies some of the proofs. Actually, all of our results are valid
in a more general setup where each topological space is assumed to be a
perfectly normal Hausdorff space. Such generalization can be found in the
Supplementary Appendix.

## Appendix A. Proofs for Section 3

We first providet the proof of Lemma 1, which relies on three mathematical
facts.

Fact A.1 Fix a metrizable space $X$. Then $\Delta\left(X\right)$ is
metrizable. Furthermore, $\Delta\left(X\right)$ is Souslin (resp. Lusin,
compact) if and only if $X$ is Souslin (resp. Lusin, compact).

Fact A.2 Fix a metrizable space $X$ and a countable family
$\mathcal{B}\subseteq\Sigma_{X}$ of conditioning events. If $\mathcal{B}$ is
clopen, then $\Delta^{\mathcal{B}}\left(X\right)$ is a closed subset of
$\Delta\left(X\right)^{\mathcal{B}}$.

Fact A.3 Suppose $X$ is a metrizable Souslin (resp. Lusin) space. Then for
every countable family $\mathcal{C}$ of Borel subsets of $X$ there exists a
finer Souslin (resp. Lusin), metrizable topology on $X$ generating the same
Borel $\sigma$-algebra and making each $C\in\mathcal{C}$ clopen.

Fact A1 is well-known; see Bogachev (2007, Chapter 8.9). The proof of Fact A.2
can be found in Battigalli and Siniscalchi (1999, Lemma 1): indeed, their
proof for the statement in Fact A.2 remains valid if $X$ is only assumed to be
a metrizable space, not necessarily Polish.171717Specifically, metrizability
of $X$ implies that $\Delta\left(X\right)$ is metrizable. Since $\mathcal{B}$
is countable, the product topology on $\Delta\left(X\right)^{\mathcal{B}}$ is
metrizable as well. The proof in Battigalli and Siniscalchi (1999) relies on
two relevant properties implied by metrizability of the space $X$. First,
every Borel probability measure is both outer regular and inner regular—see
Aliprantis and Border (2006, Theorem 12.5). Second, every metrizable space is
a normal space, and this yields the existence of a Urysohn function satisfying
the required properties for some steps of the proof. Finally, the assumption
that $\mathcal{B}$ is clopen allows to apply the Portmanteau Theorem for
continuity sets (Bogachev 2007, Corollary 8.2.10). The proof of Fact A.3
follows from simple modifications of the results in Srivastava (1998, Section
3.2).181818The results in Srivastava (1998) are stated for Polish spaces. But
the arguments in the proofs can be easily adapted to show that the results
continue to hold with the weaker requirement of Souslin or Lusin topology.
Details can be found in the Supplementary Appendix.

Proof of Lemma 1. Since $\mathcal{B}$ is countable and $\Delta\left(X\right)$
is metrizable (Fact A.1), the product topology on
$\Delta\left(X\right)^{\mathcal{B}}$ is metrizable. By definition of subspace
topology, $\Delta^{\mathcal{B}}\left(X\right)$ is a metrizable space, and this
proves part (i). Next, recall that the class of Souslin (Lusin, compact)
spaces is closed under countable products. With this, Fact A.1 yields that
$\Delta\left(X\right)^{\mathcal{B}}$ is Souslin (resp. Lusin, compact)
provided that $X$ is Souslin (resp. Lusin, compact). Moreover, closed subsets
of a Souslin (resp. Lusin) spaces are Souslin (resp. Lusin) spaces when
endowed with the subspace topology. It follows from Fact A.2 that, if
$\mathcal{B}$ is clopen and $X$ is Souslin (Lusin), so is
$\Delta^{\mathcal{B}}\left(X\right)$. We make use of this result to show part
(ii), that is, $\Delta^{\mathcal{B}}\left(X\right)$ is Souslin (Lusin)
subspace of $\Delta\left(X\right)^{\mathcal{B}}$ even if $\mathcal{B}$ is not
clopen.

By Fact A.3, there exists a metrizable Souslin (Lusin) topology on the Souslin
(Lusin) space $X$ which is finer than the original one, and such that: (a) it
generates the same Borel $\sigma$-algebra, and (b) it makes each element of
$\mathcal{B}$ clopen. Let $X^{\ast}$ denote the topological space endowed with
this topology. Note that $\Delta\left(X\right)=\Delta\left(X^{\ast}\right)$
because the generated Borel $\sigma$-algebra is the same for both spaces. The
topology on $\Delta\left(X^{\ast}\right)$ is finer than the topology on
$\Delta\left(X\right)$. To see this, consider a sequence
$\left(\mu_{n}\right)_{n\in\mathbb{N}}$ in $\Delta\left(X^{\ast}\right)$ that
converges weakly to $\mu$. By definition,
$\mathop{\textstyle\int}fd\mu_{n}\rightarrow\mathop{\textstyle\int}fd\mu$ for
every continuous function $f:X^{\ast}\rightarrow\mathbb{R}$. Since every
continuous function $f:X\rightarrow\mathbb{R}$ is also continuous on
$X^{\ast}$, it follows that $\left(\mu_{n}\right)_{n\in\mathbb{N}}$ converges
weakly to $\mu$ in $\Delta\left(X\right)$.

With this, it follows that the product topology on
$\Delta\left(X^{\ast}\right)^{\mathcal{B}}$ is finer than the product topology
on $\Delta\left(X\right)^{\mathcal{B}}$. This entails that the identity map
from $\Delta\left(X^{\ast}\right)^{\mathcal{B}}$ to
$\Delta\left(X\right)^{\mathcal{B}}$ is a continuous bijection. Thus,
$\Delta^{\mathcal{B}}\left(X\right)$ is the image of
$\Delta^{\mathcal{B}}\left(X^{\ast}\right)$ (a Souslin space) under a
injective continuous map. Since $\Delta^{\mathcal{B}}\left(X^{\ast}\right)$ is
the image of a complete, separable metric space under a continuous map, it
follows that $\Delta^{\mathcal{B}}\left(X\right)$ is a Souslin space. In
particular, if $X$ is Lusin, then $\Delta^{\mathcal{B}}\left(X^{\ast}\right)$
is the image of a complete, separable metric space under a continuous
injective map, hence $\Delta^{\mathcal{B}}\left(X\right)$ is Lusin. This
concludes the proof of part (ii).

The proof of part (iii) is immediate. By Fact A.1 and Tychonoff’s theorem, $X$
is compact if and only if $\Delta\left(X\right)^{\mathcal{B}}$ is compact. If
$\mathcal{B}$ is clopen, then the result follows from Fact A.2. $\blacksquare$

Remark A.1 If $X$ is a Polish space and $\mathcal{B}$ is countable and clopen,
then $\Delta^{\mathcal{B}}\left(X\right)$ is a Polish space (cf. Battigalli
and Siniscalchi 1999). An analogous conclusion does not hold if $\mathcal{B}$
is assumed only countable. Yet, by Lemma 1,
$\Delta^{\mathcal{B}}\left(X\right)$ is a Lusin space. Moreover, it can be
deduced from the above proof that $\Delta^{\mathcal{B}}\left(X\right)$ is a
Borel subset of $\Delta\left(X\right)^{\mathcal{B}}$. To see this, recall that
$X^{\ast}$ is the space obtained by endowing the set $X$ with a Souslin
topology satisfying the same properties as in Fact A.3. Both
$\Delta\left(X^{\ast}\right)^{\mathcal{B}}$ and
$\Delta\left(X\right)^{\mathcal{B}}$ are Souslin spaces, and the identity map
from $\Delta\left(X^{\ast}\right)^{\mathcal{B}}$ to
$\Delta\left(X\right)^{\mathcal{B}}$ is a continuous bijection. Since every
Borel measurable bijection between Souslin spaces is a Borel isomorphism (Cohn
2013, Proposition 8.6.2; Bogachev 2007, Theorem 6.7.3), and
$\Delta^{\mathcal{B}}\left(X^{\ast}\right)$ is a closed (hence Borel) subset
of $\Delta\left(X^{\ast}\right)^{\mathcal{B}}$, the result follows.

Next, we provide the proof of Lemma 2. To this end, we need some additional
definitions and a result.

Fix a metrizable space $X$. Let $\mathcal{F}_{\Delta\left(X\right)}$ be the
$\sigma$-algebra on $\Delta\left(X\right)$ generated by all sets of the form

$\left\\{\mu\in\Delta\left(X\right):\mu\left(E\right)\geq p\right\\}\text{,}$

where $E\in\Sigma_{X}$ and $p\in\mathbb{Q\cap}\left[0,1\right]$. We let
$\Sigma_{\Delta\left(X\right)}$ denote the Borel $\sigma$-algebra on
$\Delta\left(X\right)$ generated by the topology of weak convergence. With
this, fix a countable family $\mathcal{B}\subseteq\Sigma_{X}$ of conditioning
events, and, for each $B\in\mathcal{B}$, let
$e_{B}:\Delta\left(X\right)^{\mathcal{B}}\rightarrow\Delta\left(X\right)$
denote the coordinate projection. We use the following notation for three
$\sigma$-algebras on the set $\Delta\left(X\right)^{\mathcal{B}}$:

  * •

$\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}}}$ is the $\sigma$-algebra
generated by the cylinder algebra
$\cup_{B\in\mathcal{B}}e_{B}^{-1}\left(\mathcal{F}_{\Delta\left(X\right)}\right)$;

  * •

$\mathcal{G}_{\Delta\left(X\right)^{\mathcal{B}}}$ is the $\sigma$-algebra
generated by the cylinder algebra
$\cup_{B\in\mathcal{B}}e_{B}^{-1}\left(\Sigma_{\Delta\left(X\right)}\right)$;

  * •

$\Sigma_{\Delta\left(X\right)^{\mathcal{B}}}$ is the Borel $\sigma$-algebra
generated by the product topology on $\Delta\left(X\right)^{\mathcal{B}}$.

Fact A.4 Fix a metrizable space $X$ and a countable family
$\mathcal{B}\subseteq\Sigma_{X}$ of conditioning events. The following
statements hold.
(i) $\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}}}$ is generated by all
sets of the form

$\left\\{\mu\in\Delta\left(X\right)^{\mathcal{B}}:\mu\left(E|B\right)\geq
p\right\\}\text{,}$

where $E\in\Sigma_{Y}$, $B\in B$ and $p\in Q\cap\left[0,1\right]$.
(ii)
$\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}}}\subseteq\mathcal{G}_{\Delta\left(X\right)^{\mathcal{B}}}\subseteq\Sigma_{\Delta\left(X\right)^{\mathcal{B}}}$.
(iii) If $X$ is separable, then
$\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}}}=\mathcal{G}_{\Delta\left(X\right)^{\mathcal{B}}}=\Sigma_{\Delta\left(X\right)^{\mathcal{B}}}$.

Proof. Part (i) follows by inspection of the definitions of
$\mathcal{F}_{\Delta\left(X\right)}$ and
$\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}}}$. To show part (ii), first
note that, by Theorem 2.2 in Gaudard and Handwin (1989),
$\mathcal{F}_{\Delta\left(X\right)}\subseteq\Sigma_{\Delta\left(X\right)}$.
This entails that
$\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}}}\subseteq\mathcal{G}_{\Delta\left(X\right)^{\mathcal{B}}}$.
Moreover, by Lemma 6.4.1 in Bogachev (2007),
$\mathcal{G}_{\Delta\left(X\right)^{\mathcal{B}}}\subseteq\Sigma_{\Delta\left(X\right)^{\mathcal{B}}}$.
As for part (iii), assume that $X$ is separable. By Theorem 2.3 in Gaudard and
Handwin (1989),
$\mathcal{F}_{\Delta\left(X\right)}=\Sigma_{\Delta\left(X\right)}$. This
yields
$\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}}}=\mathcal{G}_{\Delta\left(X\right)^{\mathcal{B}}}$.
Furthermore, separability of $X$ implies separability of
$\Delta\left(X\right)$ (Bogachev 2007, Theorem 8.9.4). Hence, by Lemma 6.4.2
in Bogachev (2007),
$\mathcal{G}_{\Delta\left(X\right)^{\mathcal{B}}}=\Sigma_{\Delta\left(X\right)^{\mathcal{B}}}$.
$\blacksquare$

Proof of Lemma 2. We begin with part (i) of the lemma. Fix some
$\mu\in\Delta^{\mathcal{B}_{X}}\left(X\right)$. We show that
$\overline{\mathcal{L}}_{f}\left(\mu\right)$ is a CPS on
$\left(Y,\Sigma_{Y},\mathcal{B}_{Y}\right)$. Pick any $B\in\mathcal{B}_{Y}$.
Since $f^{-1}\left(\mathcal{B}_{Y}\right)=\mathcal{B}_{X}$, we have
$f^{-1}\left(B\right)\in\mathcal{B}_{X}$ and

$\overline{\mathcal{L}}_{f}\left(\mu\right)\left(B|B\right)=\mu\left(f^{-1}\left(B\right)|f^{-1}\left(B\right)\right)=1\text{,}$

because $\mu$ is a CPS. Hence Condition (i) in Definition 1 holds.To show that
the chain rule is satisfied, consider any $A\in\Sigma_{Y}$ and
$B,C\in\mathcal{B}_{Y}$ such that $A\subseteq B\subseteq C$. We have
$f^{-1}\left(B\right)$, $f^{-1}\left(C\right)\in\mathcal{B}_{X}$. Furthermore,
$f^{-1}\left(A\right)\in\Sigma_{X}$ and $f^{-1}\left(A\right)\subseteq
f^{-1}\left(B\right)\subseteq f^{-1}\left(C\right)$. Using the fact that $\mu$
is a CPS, we get

$\displaystyle\left(\overline{\mathcal{L}}_{f}\left(\mu\right)\left(A|B\right)\right)\left(\overline{\mathcal{L}}_{f}\left(\mu\right)\left(B|C\right)\right)$
$\displaystyle=$
$\displaystyle\mu\left(f^{-1}\left(A\right)|f^{-1}\left(B\right)\right)\mu\left(f^{-1}\left(B\right)|f^{-1}\left(C\right)\right)$
$\displaystyle=$
$\displaystyle\mu\left(f^{-1}\left(A\right)|f^{-1}\left(C\right)\right)$
$\displaystyle=$
$\displaystyle\overline{\mathcal{L}}_{f}\left(\mu\right)\left(A|C\right)\text{,}$

as required.

Now we turn to part (ii) of the lemma. Suppose that $f$ is Borel measurable.
To show that $\overline{\mathcal{L}}_{f}$ is Borel measurable, first note the
following facts. Consider the set $\Delta\left(Y\right)^{\mathcal{B}_{Y}}$.
Since every Souslin space is separable, it follows from Fact A.4.(iii) that
$\mathcal{F}_{\Delta\left(Y\right)^{\mathcal{B}_{Y}}}=\Sigma_{\Delta\left(Y\right)^{\mathcal{B}_{Y}}}$.
Hence, by Fact A.4.(i), the Borel $\sigma$-algebra on
$\Delta^{\mathcal{B}_{Y}}\left(Y\right)$ is the relative $\sigma$-algebra of
$\mathcal{F}_{\Delta\left(Y\right)^{\mathcal{B}_{Y}}}$ on
$\Delta^{\mathcal{B}_{Y}}\left(Y\right)$, and it is generated by all sets of
the form

$b_{p}\left(E|B\right):=\left\\{v\in\Delta^{\mathcal{B}_{Y}}\left(Y\right):v\left(E|B\right)\geq
p\right\\}\text{,}$

where $E\in\Sigma_{Y}$, $B\in\mathcal{B}_{Y}$ and
$p\in\mathbb{Q\cap}\left[0,1\right]$. We now claim that
$\overline{\mathcal{L}}_{f}^{-1}\left(b_{p}\left(E|B\right)\right)$ belongs to
the relative $\sigma$-algebra of
$\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}_{X}}}$ on
$\Delta^{\mathcal{B}_{X}}\left(X\right)$. Since
$\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}_{X}}}\subseteq\Sigma_{\Delta\left(X\right)^{\mathcal{B}_{X}}}$
by Fact A.4.(ii), this will entail that
$\overline{\mathcal{L}}_{f}^{-1}\left(b_{p}\left(E|B\right)\right)$ is a Borel
subset of $\Delta^{\mathcal{B}_{X}}\left(X\right)$ (i.e., it belongs to the
relative $\sigma$-algebra of $\Sigma_{\Delta\left(X\right)^{\mathcal{B}_{X}}}$
on $\Delta^{\mathcal{B}_{X}}\left(X\right)$). Note that

$\displaystyle\overline{\mathcal{L}}_{f}^{-1}\left(b_{p}\left(E|B\right)\right)$
$\displaystyle=$
$\displaystyle\left\\{\mu\in\Delta^{\mathcal{B}_{X}}\left(X\right):\overline{\mathcal{L}}_{f}\left(\mu\right)\in
b_{p}\left(E|B\right)\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\mu\in\Delta^{\mathcal{B}_{X}}\left(X\right):\mu\left(f^{-1}\left(E\right)|f^{-1}\left(B\right)\right)\geq
p\right\\}\text{.}$

Since $f$ is measurable and
$f^{-1}\left(\mathcal{B}_{Y}\right)=\mathcal{B}_{X}$, we have
$f^{-1}\left(E\right)\in\Sigma_{X}$ and
$f^{-1}\left(B\right)\in\mathcal{B}_{X}$. The claim follows from the
definition of relative $\sigma$-algebra of
$\mathcal{F}_{\Delta\left(X\right)^{\mathcal{B}_{X}}}$ on
$\Delta^{\mathcal{B}_{X}}\left(X\right)$. Finally, as $b_{p}\left(E|B\right)$
is arbitrary and it belongs to the system of generators of the Borel
$\sigma$-algebra on $\Delta^{\mathcal{B}_{Y}}\left(Y\right)$, standard results
(e.g., Aliprantis and Border 2006, Corollary 4.24) yield measurability of
$\overline{\mathcal{L}}_{f}$.

Suppose now that $f$ is continuous. Consider any sequence
$\left(\mu_{n}\right)_{n\in\mathbb{N}}$ in
$\Delta^{\mathcal{B}_{X}}\left(X\right)$ converging to $\mu$. It must be shown
that $\overline{\mathcal{L}}_{f}\left(\mu_{n}\right)$ converges to
$\overline{\mathcal{L}}_{f}\left(\mu\right)$ as $n\rightarrow\infty$. That is,
$\mathcal{L}_{f}\left(\mu_{n}\left(.|f^{-1}\left(C\right)\right)\right)$
converges to
$\mathcal{L}_{f}\left(\mu\left(.|f^{-1}\left(C\right)\right)\right)$ for every
$C\in\mathcal{B}_{Y}$. This readily follows from the fact that the map
$\mathcal{L}_{f}:\Delta\left(X\right)\rightarrow\Delta\left(Y\right)$ is
continuous (Bogachev 2007, Theorem 8.4.1.(i)). $\blacksquare$

## Appendix B. Proofs for Section 4

We first provide the proofs of Remark 1 and Remark 2.

Proof of Remark 1. The proof is by induction on $n\in\mathbb{N}$.

($n=1$): Since $\Theta_{i}^{0}:=S$ and
$H_{i}^{1}:=\Delta^{\mathcal{B}_{i}}\left(S\right)$, the result follows from
Lemma 1.(i).

($n>1$): Suppose that, for each $i\in I$, $H_{i}^{n}$ is a closed subset of
$H_{i}^{n-1}\times\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$, so
Souslin (resp. Lusin) in the subspace topology. Rewrite $H_{i}^{n}$ as
$H_{i}^{n}=H_{i}^{n-1}\times\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$,
where $\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$ is the
subset of $\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$ consisting
of all $\mu_{i}^{n}$ which are coherent extensions of
$\left(\mu_{i}^{1},...,\mu_{i}^{n-1}\right)\in H_{i}^{n-1}$. Thus, if
$H_{i}^{n}$ is a closed subset of
$H_{i}^{n-1}\times\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$, this
amounts to say that
$\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$ is a closed subset
of $\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$. By this, it
follows that $\Theta_{i}^{n}:=\Theta_{i}^{0}\times H_{j}^{n}$ is Souslin
(resp. Lusin). The proof of the inductive step is complete if we show that
$\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right)$ is a closed subset
of $\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right)$. Consider a sequence
$\left(\mu_{i,m}^{n+1}\right)_{m\in\mathbb{N}}$ in
$\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right)$ converging to
$\mu_{i}^{n+1}$. We need to show that
$\mu_{i}^{n+1}\in\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right)$,
that is, $\mu_{i}^{n+1}$ satisfies
$\overline{\mathcal{L}}_{\rho_{i}^{n-1,n}}\left(\mu_{i}^{n+1}\right)\in\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$.
Since $\overline{\mathcal{L}}_{\rho_{i}^{n-1,n}}$ is continuous (by continuity
of $\rho_{i}^{n-1,n}$ and Lemma 2),
$\left(\overline{\mathcal{L}}_{\rho_{i}^{n-1,n}}\left(\mu_{i,m}^{n+1}\right)\right)_{m\in\mathbb{N}}$
converges to
$\overline{\mathcal{L}}_{\rho_{i}^{n-1,n}}\left(\mu_{i}^{n+1}\right)$. Since
$\overline{\mathcal{L}}_{\rho_{i}^{n-1,n}}\left(\mu_{i,m}^{n+1}\right)\in\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$
for every $m\in\mathbb{N}$ and
$\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$ is closed by the
inductive hypothesis, we have
$\overline{\mathcal{L}}_{\rho_{i}^{n-1,n}}\left(\mu_{i}^{n+1}\right)\in\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n-1}\right)$.
Thus
$\mu_{i}^{n+1}\in\Delta_{C}^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right)$.

Lastly, if each $\mathcal{B}_{i}$ is clopen, then, by Lemma 1.(iii),
$H_{i}^{1}:=\Delta^{\mathcal{B}_{i}}\left(S\right)$ is compact if and only if
$S$ is compact. The final claim of Remark 1 follows by induction and Lemma
1.(iii). $\blacksquare$

Proof of Remark 2. Note that

$H_{i}=\mathop{\textstyle\bigcap}_{n\in\mathbb{N}}\left\\{\left(\mu_{i}^{1},\mu_{i}^{2},...\right)\in\mathop{\textstyle\prod}_{n=0}^{\infty}\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}^{n}\right):\left(\mu_{i}^{1},...,\mu_{i}^{n}\right)\in
H_{i}^{n}\right\\}\text{,}$

and since each $H_{i}^{n}$ is closed by Remark 1, $H_{i}$ is a countable
intersection of closed cylinders in $H_{i}$, hence closed. $\blacksquare$

As claimed in the main text, $H_{i}$ and $\Theta_{i}$ ($i\in I$) can be
identified as non-empty projective limits of the sequences
$\left(H_{n}^{i}\right)_{n\geq 1}$ and $\left(\Theta_{n}^{i}\right)_{n\geq
0}$, respectively. We now make this claim precise. To this end, we first
review some notions from the theory of projective (or inverse) systems that
are necessary for the proof of Proposition 1. For a more thorough treatment
see Engelking (1989) or Rao (1981). Here we restrict attention to projective
systems of spaces indexed by $\mathbb{N}$, the set of natural numbers.

A projective sequence is a sequence
$\left(Y_{n},f_{m,n}\right)_{m,n\in\mathbb{N}}$ of spaces $Y_{n}$ and
functions $f_{m,n}:Y_{n}\rightarrow Y_{m}$ such that:

  * •

for all $n\in\mathbb{N}$, $Y_{n}$ is non-empty topological space;

  * •

$f_{m,n}$ is continuous for all $m,n\in\mathbb{N}$ such that $m\leq n$;

  * •

$f_{m,p}=f_{m,n}\circ f_{n,p}$ for all $m,n,p\in\mathbb{N}$ such that $m\leq
n\leq p$, and $f_{n,n}=\mathrm{Id}_{Y_{n}}$ for each $n\in\mathbb{N}$.

Each space $Y_{n}$ ($n\in\mathbb{N}$) is called coordinate (or factor) space
and the functions $f_{m,n}$ are called bonding maps. The projective limit of
$\left(Y_{n},f_{m,n}\right)_{m,n\in\mathbb{N}}$ is defined as follows:

$\underleftarrow{\lim}Y_{n}:=\left\\{\left(y_{n}\right)_{n\in\mathbb{N}}\in\mathop{\textstyle\prod}_{n\in\mathbb{N}}Y_{n}:\forall
m,n\in\mathbb{N},\text{ }m\leq
n,y_{m}=f_{m,n}\left(y_{n}\right)\right\\}\text{.}$

For each $m\in\mathbb{N}$, the function
$\overline{f}_{m}:\underleftarrow{\lim}Y_{n}\rightarrow Y_{m}$ is defined as
the restriction of the projection map
$\mathrm{P}$ro$\mathrm{j}_{Y_{m}}:\mathop{\textstyle\prod}_{n\in\mathbb{N}}Y_{n}\rightarrow
Y_{m}$ to $\underleftarrow{\lim}Y_{n}$. For all $m,n\in\mathbb{N}$ such that
$m\leq n$, the functions $\overline{f}_{n}$ and $\overline{f}_{m}$ satisfy the
equality $\overline{f}_{m}=f_{m,n}\circ\overline{f}_{n}$.

The set $\underleftarrow{\lim}Y_{n}$ inherits the subspace topology as a
subset of the product $\prod_{n\in\mathbb{N}}Y_{n}$. It is known (see
Engelking 1989, Proposition 2.5.1) that if
$\left(Y_{n},f_{m,n}\right)_{m,n\in\mathbb{N}}$ is a projective sequence of
Hausdorff spaces $Y_{n}$, then $\underleftarrow{\lim}Y_{n}$ is a closed subset
of the Cartesian product $\prod_{n\in\mathbb{N}}Y_{n}$. Moreover,
$\underleftarrow{\lim}Y_{n}$ is a metrizable space if each $Y_{n}$ is a
metrizable space (Engelking 1989, Corollary 4.2.5).

Lemma B.1 Let $\left(Y_{n},f_{m,n}\right)_{m,n\in\mathbb{N}}$ be a projective
sequence of Hausdorff spaces $Y_{n}$ and surjective bonding maps $f_{m,n}$.
The following statements hold.
(i) $Y:=\underleftarrow{\lim}Y_{n}$ is non-empty, and, for each
$n\in\mathbb{N}$, $\overline{f}_{n}:Y\rightarrow Y_{n}$ is continuous and
surjective.
(ii) Suppose that each $Y_{n}$ is a Souslin space. Then $Y$ is a Souslin space
and the Borel $\sigma$-algebra $\Sigma_{Y}$ is generated by the algebra
$\mathcal{A}_{Y}:=\cup_{n\in\mathbb{N}}\overline{f}_{n}^{-1}\left(\Sigma_{n}\right)$.

Proof. For part (i), see Engelking (1989, Exercises 2.5.A and 2.5.B). To show
part (ii), suppose that each $Y_{n}$ is a Souslin space. Then
$\prod_{n\in\mathbb{N}}Y_{n}$ is a Souslin space, and since
$Y:=\underleftarrow{\lim}Y_{n}$ is a closed subset of
$\prod_{n\in\mathbb{N}}Y_{n}$, it follows that $Y$ is Souslin in the subspace
topology (see Schwartz 1973, Part I, Chapter II). Hence $\Sigma_{Y}$ is the
relative $\sigma$-algebra induced by the Borel $\sigma$-algebra generated by
the product topology on $\prod_{n\in\mathbb{N}}Y_{n}$ (Bogachev 2007, Lemma
6.2.4). By Lemma 6.4.2 and Lemma 6.6.4 in Bogachev (2007), the Borel
$\sigma$-algebra generated by the product topology equals the Borel
$\sigma$-algebra generated by the cylinder algebra
$\cup_{n\in\mathbb{N}}\mathrm{Proj}_{Y_{n}}^{-1}\left(\Sigma_{n}\right)$. The
conclusion follows from the fact that each $\overline{f}_{n}$ is the
restriction to $Y$ of the projection map $\mathrm{P}$ro$\mathrm{j}_{Y_{n}}$.
$\blacksquare$

Consider the sequences $\left(H_{i}^{n}\right)_{n\geq 1}$ and
$\left(\Theta_{i}^{n}\right)_{n\geq 0}$, respectively. Recall that, for each
$i\in I$ and $n\geq 1$, we defined the projections
$\pi_{i}^{n,n+1}:H_{i}^{n+1}\rightarrow H_{i}^{n}$ and
$\rho_{i}^{n-1,n}:\Theta_{i}^{n}\rightarrow\Theta_{i}^{n-1}$ which satisfy, by
construction,
$\rho_{i}^{n-1,n}=\left(\mathrm{Id}_{\Theta_{i}^{0}},\pi_{j}^{n-1,n}\right)$
for each $n\geq 2$ (cf. Mertens et al. 2015, Theorem III.1.1.(3)).

Clearly, $\left(H_{i}^{n},\pi_{i}^{n,n+1}\right)_{n\in\mathbb{N}}$ and
$\left(\Theta_{i}^{n-1},\rho_{i}^{n-1,n}\right)_{n\in\mathbb{N}}$ are
projective sequences of Souslin spaces. Proposition B.1 below, which is an
analogue of Theorem 6 in Heifetz (1993), ensures that each bonding map
$\pi_{i}^{n,n+1}$ is a surjective open map—hence, it is a well-defined
projection. Moreover, since each set $H_{i}^{1}$ is non-empty, it will follow
that for each $n\in\mathbb{N}$ the set $H_{i}^{n+1}$ is non-empty as well. The
proof is similar to Heifetz’s proof and thus omitted.191919The proof can be
found in the Supplementary Appendix. Differently from Theorem 6 of Heifetz
(1993), the proof additionally shows that each bonding map is an open
function, i.e., the image of any open subset of the domain is an open subset
of the codomain.

Proposition B.1 Consider the projective sequences
$\left(H_{i}^{n},\pi_{i}^{n,n+1}\right)_{n\in\mathbb{N}}$ and
$\left(\Theta_{i}^{n-1},\rho_{i}^{n-1,n}\right)_{n\in\mathbb{N}}$. Then, for
each $i\in I$ and each $n\in\mathbb{N}$, the bonding maps
$\pi_{i}^{n,n+1}:H_{i}^{n+1}\rightarrow H_{i}^{n}$ and
$\rho_{i}^{n-1,n}:\Theta_{i}^{n}\rightarrow\Theta_{i}^{n-1}$ are surjective
and open.

Let $\underleftarrow{\lim}H_{i}^{n}$ and
$\underleftarrow{\lim}\Theta_{i}^{n-1}$ be the projective limits of
$\left(H_{i}^{n},\pi_{i}^{n,n+1}\right)_{n\in\mathbb{N}}$ and
$\left(\Theta_{i}^{n-1},\rho_{i}^{n-1,n}\right)_{n\in\mathbb{N}}$. By
Proposition B.1 and Lemma B.1, $\underleftarrow{\lim}H_{i}^{n}$ and
$\underleftarrow{\lim}\Theta_{i}^{n-1}$ are non-empty Souslin spaces.
Furthermore, let
$\overline{\pi}_{i}^{n}:\underleftarrow{\lim}H_{i}^{n}\rightarrow H_{i}^{n}$
and
$\overline{\rho}_{i}^{n}:\underleftarrow{\lim}\Theta_{i}^{n-1}\rightarrow\Theta_{i}^{n}$
denote the corresponding projections. By Lemma B.1, both
$\overline{\pi}_{i}^{n}$ and $\overline{\rho}_{i}^{n}$ are continuous and
surjective. Moreover, they are also open functions (Engelking 1989, Exercise
2.7.19.(b)).

Next, refer back to the spaces $H_{i}$ and $\Theta_{i}$. For each $i\in I$ and
each $n\in\mathbb{N}$, we let $\pi_{i}^{n}:H_{i}\rightarrow H_{i}^{n}$ denote
the projection map. For all $m,n\in\mathbb{N}$ such that $m\leq n$, the
functions $\pi_{i}^{m}$ and $\pi_{i}^{n}$ satisfy the equality
$\pi_{i}^{m}=\pi_{i}^{m,n}\circ\pi_{i}^{n}$. Furthermore, for each $i\in I$
and each $n\geq 0$, we let $\rho_{i}^{n}:\Theta_{i}\rightarrow\Theta_{i}^{n}$
denote the projection map, which satisfies

$\rho_{i}^{n}=\left(\mathrm{Id}_{\Theta_{i}^{0}},\pi_{j}^{n}\right)$

for all $n\geq 1$. It should be noted that the sets $H_{i}$ and $\Theta_{i}$
are formally distinct from $\underleftarrow{\lim}H_{i}^{n}$ and
$\underleftarrow{\lim}\Theta_{i}^{n-1}$. In particular, the elements of
$H_{i}$ are sequences of CPSs, while—according to the definition of projective
limit given here—the elements of $\underleftarrow{\lim}H_{i}^{n}$ are
sequences of finite sequences of CPSs. Yet, we show that $H_{i}$ and
$\Theta_{i}$ are naturally identified as $\underleftarrow{\lim}H_{i}^{n}$ and
$\underleftarrow{\lim}\Theta_{i}^{n-1}$. Specifically, Proposition B.2 below
shows the existence of homeomorphisms
$\gamma_{i}:H_{i}\rightarrow\underleftarrow{\lim}H_{i}^{n}$ and
$\varphi_{i}:\Theta_{i}\rightarrow\underleftarrow{\lim}\Theta_{i}^{n}$ that
preserve the projections: for each $n\in\mathbb{N}$,

$\pi_{i}^{n}=\overline{\pi}_{i}^{n}\circ\gamma_{i}\text{ \ and \
}\rho_{i}^{n-1}=\overline{\rho}_{i}^{n-1}\circ\varphi_{i}\text{.}$

Proposition B.2 For each $i\in I$, $H_{i}$ and $\Theta_{i}$ are homeomorphic
to $\underleftarrow{\lim}H_{i}^{n}$ and
$\underleftarrow{\lim}\Theta_{i}^{n-1}$, respectively.

Proof.202020The proof uses standard techniques in the theory of projective
systems; see, e.g., Rao (1981, pp. 117-118) and Engelking (1989, Examples
2.5.3-2.5.4). Fix a player $i\in I$. Recall that (by definition of
$\underleftarrow{\lim}H_{i}^{n}$) for all $m,n\in\mathbb{N}$ such that $m\leq
n$, the functions $\overline{\pi}_{i}^{m}$ and $\overline{\pi}_{i}^{n}$
satisfy the equality
$\overline{\pi}_{i}^{m}=\pi_{i}^{m,n}\circ\overline{\pi}_{i}^{n}$. We first
construct a homeomorphism
$\gamma_{i}:H_{i}\rightarrow\underleftarrow{\lim}H_{i}^{n}$ such that
$\pi_{i}^{n}=\overline{\pi}_{i}^{n}\circ\gamma_{i}$ for each $n\in\mathbb{N}$.
Let $\gamma_{i}:H_{i}\rightarrow\underleftarrow{\lim}H_{i}^{n}$ be the
function defined by
$\gamma_{i}\left(\cdot\right):=\left(\pi_{i}^{1}\left(\cdot\right),\pi_{i}^{2}\left(\cdot\right),...\right)$.
The map $\gamma_{i}$ is injective: to see this, consider any
$h^{\prime},h^{\prime\prime}\in H_{i}$ such that
$\gamma_{i}\left(h^{\prime}\right)=\gamma_{i}\left(h^{\prime\prime}\right)$.
Then $h^{\prime}=h^{\prime\prime}$ because all the coordinates of $h^{\prime}$
are equal to all those of $h^{\prime\prime}$. To show that $\gamma_{i}$ is
surjective, pick any
$\left(\left(\mu_{i}^{1}\right),\left(\mu_{i}^{1},\mu_{i}^{2}\right),...\right)\in\underleftarrow{\lim}H_{i}^{n}$.
The element $\bar{h}:=\left(\mu_{i}^{1},\mu_{i}^{2},...\right)\in H_{i}$ is
such that

$\gamma_{i}\left(\bar{h}\right)=\left(\pi_{i}^{1}\left(\bar{h}\right),\pi_{i}^{2}\left(\bar{h}\right),...\right)=\left(\left(\mu_{i}^{1}\right),\left(\mu_{i}^{1},\mu_{i}^{2}\right),...\right)\text{,}$

as desired. Thus, $\gamma_{i}:H_{i}\rightarrow\underleftarrow{\lim}H_{i}^{n}$
is a bijection. Since each projection $\pi_{i}^{n}\left(\cdot\right)$
($n\in\mathbb{N}$) is a continuous function, so is $\gamma_{i}$ by Engelking
(1989, Proposition 2.3.6). The inverse of $\gamma_{i}$ is also continuous:
consider a sequence
$\left(\hat{h}_{m}\right)_{m\in\mathbb{N}}:=\left(\left(\mu_{i,m}^{1}\right),\left(\mu_{i,m}^{1},\mu_{i,m}^{2}\right),...\right)_{m\in\mathbb{N}}$
in $\underleftarrow{\lim}H_{i}^{n}$ converging to
$\hat{h}:=\left(\left(\mu_{i}^{1}\right),\left(\mu_{i}^{1},\mu_{i}^{2}\right),...\right)$.
By definition of product topology (i.e., the topology of pointwise
convergence), it is the case that, for all $n\in\mathbb{N}$,
$\mu_{i,m}^{n}\rightarrow\mu_{i}^{n}$ as $m\rightarrow\infty$. Hence
$\gamma_{i}^{-1}\left(\hat{h}_{m}\right)=\left(\mu_{i,m}^{1},\mu_{i,m}^{2},...\right)$
converges to
$\gamma_{i}^{-1}\left(\hat{h}\right)=\left(\mu_{i}^{1},\mu_{i}^{2},...\right)$.

To show the result for the spaces $\Theta_{i}$ and
$\underleftarrow{\lim}\Theta_{i}^{n-1}$, let
$\mathbb{N}_{0}:=\mathbb{N}\cup\left\\{0\right\\}$ be the set of natural
numbers including $0$. Denote by
$\theta_{i}:=\left(\theta_{i}^{0},\theta_{i}^{1},...\right)$ each element of
the product set $\mathop{\textstyle\prod}_{n\in\mathbb{N}_{0}}\Theta_{i}^{n}$.
Note that, by definition,

$\underleftarrow{\lim}\Theta_{i}^{n-1}=\left\\{\theta_{i}\in\mathop{\textstyle\prod}_{n\in\mathbb{N}_{0}}\Theta_{i}^{n}:\begin{tabular}[]{l}1)
$\rho_{i}^{0,1}\left(\theta_{i}^{1}\right)=\theta_{i}^{0}$,\\\ 2) $\forall
n\in\mathbb{N},\rho_{i}^{n-1,n}\left(\theta_{i}^{n}\right)=\theta_{i}^{n-1}$\end{tabular}\right\\}\text{,}$

where each $\theta_{i}^{n}$ ($n\in\mathbb{N}$) can be written as
$\theta_{i}^{n}=\left(s,\left(\mu_{j}^{m}\right)_{m=1}^{n}\right)$. That is,
each element $\theta_{i}\in\underleftarrow{\lim}\Theta_{i}^{n-1}$ has the form
$\theta_{i}=\left(\left(s\right),\left(s,\left(\mu_{j}^{1}\right)\right),\left(s,\left(\mu_{j}^{1},\mu_{j}^{2}\right)\right),...\right)$.
Therefore
$\underleftarrow{\lim}\Theta_{i}^{n}=$D$\left(S^{\mathbb{N}_{0}}\right)\times\underleftarrow{\lim}H_{j}^{n}$,
where D$\left(S^{\mathbb{N}_{0}}\right)$ is the set of all constant sequences
on $S$ (i.e., D$\left(S^{\mathbb{N}_{0}}\right)$ is the “diagonal” of
$S^{\mathbb{N}_{0}}$). The map $\psi_{i}:S\rightarrow S^{\mathbb{N}_{0}}$
defined by

$\psi_{i}\left(s\right)=\left(s,s,...\right)$

is a homeomorphism from $S$ onto D$\left(S^{\mathbb{N}_{0}}\right)$.
Consequently, the map
$\varphi_{i}:=\left(\psi_{i},\gamma_{j}\right):\Theta_{i}\rightarrow\underleftarrow{\lim}\Theta_{i}^{n}$
is a homeomorphism such that
$\rho_{i}^{n-1}=\overline{\rho}_{i}^{n-1}\circ\varphi_{i}$ for every
$n\in\mathbb{N}$. $\blacksquare$

The next step is to provide the proof of Proposition 1. To this end, we need
the following definition and result.

Definition B.1 We call
$\left(\left(Y_{n},\Sigma_{Y_{n}},\mathcal{B}_{n}\right),f_{m,n},\mu_{n}\right)_{m,n\in\mathbb{N}}$
a projective sequence of CPSs if:
(i) $\left(Y_{n},f_{m,n}\right)_{m,n\in\mathbb{N}}$ is a projective sequence
of Hausdorff spaces $Y_{n}$ and surjective bonding maps $f_{m,n}$;
(ii) for all $n\in\mathbb{N}$, the family
$\mathcal{B}_{n}\subseteq\Sigma_{Y_{n}}$ of conditioning events satisfies the
following property:
$\mathcal{B}_{n}:=f_{m,n}^{-1}\left(\mathcal{B}_{m}\right)$ for all
$m,n\in\mathbb{N}$ such that $m\leq n$;
(iii) $\mu_{n}$ is a CPS on
$\left(Y_{n},\Sigma_{Y_{n}},\mathcal{B}_{n}\right)$ for all $n\in\mathbb{N}$;
(iv) $\mu_{m}=\overline{\mathcal{L}}_{f_{m,n}}\left(\mu_{n}\right)$ for all
$m,n\in\mathbb{N}$ such that $m\leq n$.

We call
$\left(\left(Y,\Sigma_{Y},\mathcal{B}_{Y}\right),\overline{f}_{n},\mu\right)_{n\in\mathbb{N}}$
the CPS projective limit of the projective sequence of CPSs
$\left(\left(Y_{n},\Sigma_{Y_{n}},\mathcal{B}_{n}\right),f_{m,n},\mu_{n}\right)_{m,n\in\mathbb{N}}$
if:
(v) $Y=\underleftarrow{\lim}Y_{n}$, and $\overline{f}_{m}:Y\rightarrow Y_{m}$
satisfies $\overline{f}_{m}=f_{m,n}\circ\overline{f}_{n}$ for all
$m,n\in\mathbb{N}$ such that $m\leq n$;
(vi) $\mathcal{B}_{Y}\subseteq\Sigma_{Y}$ is such that
$B_{Y}=\overline{f}_{n}^{-1}\left(\mathcal{B}_{n}\right)$ for all
$n\in\mathbb{N}$, so that
$B_{Y}=\overline{f}_{1}^{-1}\left(\mathcal{B}_{1}\right)$;
(vii) $\mu$ is a CPS (called limit CPS) on
$\left(Y,\Sigma_{Y},\mathcal{B}_{Y}\right)$ such that, for all
$n\in\mathbb{N}$,

$\overline{\mathcal{L}}_{\overline{f}_{n}}\left(\mu\right)=\mu_{n}\text{{.}}$

Note that a CPS projective limit may fail to exist, that is, there could be no
limit CPS $\mu$ which “extends” the sequence
$\left(\mu_{n}\right)_{n\in\mathbb{N}}$. The following result (a version of
Kolmogorov’s extension theorem for CPSs) gives a sufficient condition for the
existence and uniqueness of a limit CPS.

Theorem B.1 Let
$\left(\left(Y_{n},\Sigma_{Y_{n}},\mathcal{B}_{n}\right),f_{m,n},\mu_{n}\right)_{m,n\in\mathbb{N}}$
be a projective sequence of CPSs such that, for each $n\in\mathbb{N}$, $Y_{n}$
is a Souslin space. Then the CPS projective limit
$\left(\left(Y,\Sigma_{Y},\mathcal{B}_{Y}\right),\overline{f}_{n},\mu\right)_{n\in\mathbb{N}}$
exists and $\mu$ is the unique limit CPS.

Proof. Each CPS $\mu_{n}$ on
$\left(Y_{n},\Sigma_{Y_{n}},\mathcal{B}_{n}\right)$ is an array

$\mu_{n}=\left(\mu_{n}\left(\cdot|B\right)\right)_{B\in\mathcal{B}_{n}}$

such that, for all $B\in\mathcal{B}_{n}$, $\mu_{n}\left(\cdot|B\right)$ is a
Radon probability measure on $\left(Y_{n},\Sigma_{Y_{n}}\right)$, i.e., for
each $E\in\Sigma_{Y_{n}}$ and each $\varepsilon>0$, there exists a compact set
$K\subseteq E$ such that
$\mu\left(E|B_{n}\right)-\mu\left(K|B\right)\leq\varepsilon$. This follows
from the fact that every Borel probability measure on a Souslin space is Radon
(Bogachev 2007, Theorem 7.4.3). With this, we can apply Prokhorov’s Theorem
for Radon probability measures (Schwartz 1974, Theorem 21 and Corollary, p.
81) to claim the existence of a unique array
$\left(\mu\left(\cdot|B\right)\right)_{B\in\mathcal{B}_{Y}}$ of (Radon)
probability measures on the Borel $\sigma$-algebra $\Sigma_{Y}$ of the Souslin
space $Y$ (see Lemma B1.(ii)) such that

$\mu\left(\overline{f}_{n}^{-1}\left(\cdot\right)\left|\overline{f}_{n}^{-1}\left(B_{n}\right)\right.\right)=\mu_{n}\left(\cdot|B_{n}\right)$

for all $n\in\mathbb{N}$ and $B_{n}\in\mathcal{B}_{n}$. It remains to check
that the array $\left(\mu\left(\cdot|B\right)\right)_{B\in\mathcal{B}_{Y}}$ is
a CPS. Condition (i) of Definition 1 clearly holds. To show that Condition
(ii) of Definition 1 is satisfied, pick any $B,C\in\mathcal{B}_{Y}$ such that
$B\subseteq C$. We show that the chain rule is satisfied for every
$A\in\Sigma_{Y}$ such that $A\subseteq B$. To this end, we first consider the
case when $A$ belongs to the algebra
$\mathcal{A}_{Y}:=\cup_{n\in\mathbb{N}}\overline{f}_{n}^{-1}\left(\Sigma_{n}\right)$,
which generates the Borel $\sigma$-algebra $\Sigma_{Y}$ (Lemma B1.(ii)). If
$A\in\mathcal{A}_{Y}$, then there exist $n\in\mathbb{N}$ and
$A_{n}\in\Sigma_{Y_{n}}$ such that
$A=\overline{f}_{n}^{-1}\left(A_{n}\right)$. Since
$\mathcal{B}_{Y}=\overline{f}_{n}^{-1}\left(\mathcal{B}_{n}\right)$, there are
$B_{n},C_{n}\in\mathcal{B}_{n}$ such that
$B=\overline{f}_{n}^{-1}\left(B_{n}\right)$ and
$C=\overline{f}_{n}^{-1}\left(C_{n}\right)$. By surjectivity of
$\overline{f}_{n}$ (Lemma B.1.(i)), $A_{n}\subseteq B_{n}\subseteq C_{n}$.
Using the fact that $\mu_{n}$ is a CPS, we obtain

$\displaystyle\mu\left(A|B\right)\mu\left(B|C\right)$ $\displaystyle=$
$\displaystyle\mu\left(\overline{f}_{n}^{-1}\left(A_{n}\right)|\overline{f}_{n}^{-1}\left(B_{n}\right)\right)\mu\left(\overline{f}_{n}^{-1}\left(B_{n}\right)|\overline{f}_{n}^{-1}\left(C_{n}\right)\right)$
$\displaystyle=$
$\displaystyle\mu_{n}\left(A_{n}|B_{n}\right)\mu_{n}\left(B_{n}|C_{n}\right)=\mu_{n}\left(A_{n}|C_{n}\right)$
$\displaystyle=$
$\displaystyle\mu\left(\overline{f}_{n}^{-1}\left(A_{n}\right)|\overline{f}_{n}^{-1}\left(C_{n}\right)\right)=\mu\left(A|C\right)\text{.}$

Hence, the chain rule holds whenever $A\in\mathcal{A}_{Y}$. Next, let
$\mathcal{M}\left(B,C\right)\subseteq\Sigma_{Y}$ be the collection of all
Borel subsets of $Y$ such that Condition (ii) of Definition 1 holds for those
fixed $B,C\in\mathcal{B}_{Y}$. By $\sigma$-additivity of probability measures,
$\mathcal{M}\left(B,C\right)$ is a monotone class. As seen,
$\mathcal{A}_{Y}\subseteq\mathcal{M}\left(B,C\right)$. It follows that
$\Sigma_{Y}\subseteq\mathcal{M}\left(B,C\right)$ because, by the Monotone
Class Theorem, the smallest monotone class containing $\mathcal{A}_{Y}$ is
$\Sigma_{Y}$. Thus, $\mathcal{M}\left(B,C\right)=\Sigma_{Y}$. This shows that
the chain rule is satisfied, and concludes the proof. $\blacksquare$

We now prove Proposition 1.

Proof of Proposition 1. Fix $i\in I$. By Proposition B.2 and Theorem B.1, to
every $\left(\mu_{i}^{1},\mu_{i}^{2},...\right)\in H_{i}$ there corresponds a
unique $\mu_{i}\in\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}\right)$ such that
$\overline{\mathcal{L}}_{\rho_{i}^{n-1}}\left(\mu_{i}\right)=\mu_{i}^{n}$ for
all $n\in\mathbb{N}$. Hence, there exists a bijection between $H_{i}$ and
$\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}\right)$. Let
$g_{i}:\Delta^{\mathcal{B}_{i}}\left(\Theta_{i}\right)\rightarrow H_{i}$ be
the function defined by

$g_{i}\left(\mu_{i}\right):=\left(\overline{\mathcal{L}}_{\rho_{i}^{n-1}}\left(\mu_{i}\right)\right)_{n\in\mathbb{N}}\text{.}$

Note that, for all $n\in\mathbb{N}$,

$\left(\overline{\mathcal{L}}_{\rho_{i}^{m-1}}\left(\mu_{i}\right)\right)_{m\leq
n}\in H_{i}^{n}\text{,}$

so that $g_{i}$ is a well-defined bijection. Since each coordinate projection
is a continuous function, it follows from Lemma 2 that
$\overline{\mathcal{L}}_{\rho_{i}^{n-1}}$ is continuous for all
$n\in\mathbb{N}$. With this, continuity of $g_{i}$ follows from Engelking
(1989, Proposition 2.3.6). We now show that the inverse of $g_{i}$, denoted by
$f_{i}$, is also continuous. Consider a sequence in $H_{i}$, viz.
$\left(h_{m}\right)_{m\in\mathbb{N}}:=\left(\left(\mu_{i,m}^{1},\mu_{i,m}^{2},...\right)\right)_{m\in\mathbb{N}}$,
converging to $h:=\left(\mu_{i}^{1},\mu_{i}^{2},...\right)\in H_{i}$. That is,
for all $n\in\mathbb{N}$, it is the case that
$\mu_{i,m}^{n}\rightarrow\mu_{i}^{n}$ as $m\rightarrow+\infty$. For
convenience, let $\mu_{i,m}:=f_{i}\left(h_{m}\right)$ for all $m\in\mathbb{N}$
and $\mu_{i}:=f_{i}\left(h\right)$. We need to prove that
$\mu_{i,m}\rightarrow\mu_{i}$ as $m\rightarrow+\infty$. We will prove the
following claim:

$\forall
B\in\mathcal{B}_{\Theta_{i}},\mu_{i,m}\left(\cdot\left|B\right.\right)\rightarrow\mu_{i}\left(\cdot\left|B\right.\right)\text{.}$

Fix some $B\in\mathcal{B}_{\Theta_{i}}$, so that, for all $n\geq 0$,
$B=\left(\rho_{i}^{n}\right)^{-1}\left(B_{n}\right)$ with
$B_{n}\in\mathcal{B}_{i}^{n}$. We show that
$\lim\inf_{m}\mu_{i,m}\left(O\left|B\right.\right)\geq\mu_{i}\left(O\left|B\right.\right)$
for every open set $O\subseteq\Theta_{i}$; with this, the Portmanteau Theorem
(Aliprantis and Border 2006, Theorem 15.3) entails that
$\mu_{i,m}\left(\cdot\left|B\right.\right)$ converges weakly to
$\mu_{i}\left(\cdot\left|B\right.\right)$ as $m\rightarrow+\infty$.

Towards this end, we first record some facts that will be used in the proof.
By Engelking (1989, Proposition 2.5.5), a base for the topology on
$\Theta_{i}$ is the family $\mathcal{C}$ of sets of the form
$\left(\rho_{i}^{n}\right)^{-1}\left(O_{n}\right)$, where $O_{n}$ is a basic
open subset of $\Theta_{i}^{n}$, for some $n\geq 0$. This base can be taken
countable, because each $\Theta_{i}^{n}$ is a metrizable Souslin space, hence
second countable.

Consider now any open set $U\subseteq\Theta_{i}$. Since every open set is the
union of elements of the (countable) basis, there exists a sequence
$\left(G_{k}\right)_{k\geq 0}$ of sets such that $G_{k}\in\mathcal{C}$ for all
$k\geq 0$ and $\cup_{k\geq 0}G_{k}=U$. We derive from this sequence a new
sequence satisfying some desirable properties.212121This argument is standard,
and it can be generalized to nets whenever the coordinate spaces are not
metrizable (cf. Heifetz 1991, p. 337). Define, for every $n\geq 0$, the
following sets:

$\displaystyle V_{n}$ $\displaystyle:$
$\displaystyle=\mathop{\textstyle\bigcup}_{k\geq
0}\left\\{U_{k}\subseteq\Theta_{i}^{n}:U_{k}\text{ is open and
}G_{k}=\left(\rho_{i}^{n}\right)^{-1}\left(U_{k}\right)\right\\}\text{,}$
$\displaystyle V_{n}^{\ast}$ $\displaystyle:$
$\displaystyle=\left(\rho_{i}^{n}\right)^{-1}\left(V_{n}\right)\text{.}$

It can be checked that $\left(V_{n}^{\ast}\right)_{n\geq 0}$ is an increasing
sequence such that $V_{n}^{\ast}\in\mathcal{C}$ for all $n\geq 0$ and
$\cup_{n\geq 0}V_{n}^{\ast}=U$.

We now prove the claim. Fix any open set $O\subseteq\Theta_{i}$. By the above
argument, there is an increasing sequence
$\left(G_{k}\right)_{k\in\mathbb{N\cup}\left\\{0\right\\}}$ of open subsets of
$\Theta_{i}$ such that: (a) $\cup_{k\geq 0}G_{k}=O$, and (b) for each $G_{k}$,
there exist $\ell_{k}\geq 0$ and an open set
$U_{\ell_{k}}\subseteq\Theta_{i}^{\ell_{k}}$ such that
$G_{k}=\left(\rho_{i}^{\ell_{k}}\right)^{-1}\left(U_{\ell_{k}}\right)$. Thus,
for each $k,m\in\mathbb{N\cup}\left\\{0\right\\}$, we have

$\mu_{i,m}\left(G_{k}\left|B\right.\right)=\mu_{i,m}\left(\left(\rho_{i}^{\ell_{k}}\right)^{-1}\left(U_{\ell_{k}}\right)\left|\left(\rho_{i}^{\ell_{k}}\right)^{-1}\left(B_{\ell_{k}}\right)\right.\right)=\mu_{i,m}^{\ell_{k}}\left(\left(U_{\ell_{k}}\right)\left|B_{\ell_{k}}\right.\right)\text{,}$
(6.1)

and, for each $k\in\mathbb{N\cup}\left\\{0\right\\}$,

$\mu_{i}\left(G_{k}\left|B\right.\right)=\mu_{i}\left(\left(\rho_{i}^{\ell_{k}}\right)^{-1}\left(U_{\ell_{k}}\right)\left|\left(\rho_{i}^{\ell_{k}}\right)^{-1}\left(B_{\ell_{k}}\right)\right.\right)=\mu_{i}^{\ell_{k}}\left(\left(U_{\ell_{k}}\right)\left|B_{\ell_{k}}\right.\right)\text{.}$
(6.2)

Since $\mu_{i,m}^{n}\rightarrow\mu_{i}^{n}$ for all $n\in\mathbb{N}$, it
follows that $\mu_{i,m}^{\ell_{k}}\left(\cdot\left|B_{\ell_{k}}\right.\right)$
converges weakly to
$\mu_{i}^{\ell_{k}}\left(\cdot\left|B_{\ell_{k}}\right.\right)$. Hence, by the
Portmanteau Theorem,

$\lim\inf_{m}\mu_{i,m}^{\ell_{k}}\left(\left(U_{\ell_{k}}\right)\left|B_{\ell_{k}}\right.\right)\geq\mu_{i}^{\ell_{k}}\left(\left(U_{\ell_{k}}\right)\left|B_{\ell_{k}}\right.\right)\text{.}$

By (6.1) and (6.2), it follows that, for each
$k,m\in\mathbb{N\cup}\left\\{0\right\\}$,

$\lim\inf_{m}\mu_{i,m}\left(G_{k}\left|B\right.\right)\geq\mu_{i}\left(G_{k}\left|B\right.\right)\text{.}$
(6.3)

The left expression in (6.3) is non-decreasing in $k\geq 0$ and bounded above
by $\lim\inf_{m}\mu_{i,m}\left(O\left|B\right.\right)$, while the right one
converges to $\mu_{i}\left(O\left|B\right.\right)$ as $k\rightarrow+\infty$.
Thus, $\mu_{i,m}\left(O\left|B\right.\right)$ converges weakly to
$\mu_{i}\left(O\left|B\right.\right)$. Since $O\subseteq\Theta_{i}$ and
$B\in\mathcal{B}_{\Theta_{i}}$ are arbitrary, the conclusion follows.
$\blacksquare$

## Appendix C. Proofs for Section 5

Here we prove Lemma 3. To this end, we begin by reviewing the required
mathematical background and results, then we briefly discuss the underlying
idea of the proof, and finally we move on to the formal analysis.

Fix topological spaces $X$ and $Y$. Given a Borel measure
$\mu\in\Delta\left(X\right)$, a function $f:X\rightarrow Y$ is
$\mu$-measurable if $f^{-1}\left(E\right)\in\Sigma_{X}^{\mu}$ for each
$E\in\Sigma_{Y}$, where $\Sigma_{X}^{\mu}$ stands for the completion of the
Borel $\sigma$-algebra $\Sigma_{X}$ with respect to $\mu$. Sets in
$\Sigma_{X}^{\mu}\backslash\Sigma_{X}$ are called $\mu$-null. A function
$f:X\rightarrow Y$ is called universally measurable if it is $\mu$-measurable
for every $\mu\in\Delta\left(X\right)$, or—equivalently—if
$f^{-1}\left(E\right)\in\Sigma_{X}^{\ast}$ for every $E\in\Sigma_{Y}$, where
$\Sigma_{X}^{\ast}:=\cap_{\mu\in\Delta\left(X\right)}\Sigma_{X}^{\mu}$. Sets
in $\Sigma_{X}^{\ast}$ are called universally measurable sets, and
$\Sigma_{X}^{\ast}$ is called the universally measurable $\sigma$-algebra.
Clearly, $\Sigma_{X}\subseteq\Sigma_{X}^{\ast}$.

Fact C.1 Fix Souslin spaces $X_{1}$, $X_{2}$, $Y_{1}$, and $Y_{2}$. Fix
functions $f_{1}:X_{1}\rightarrow Y_{1}$ and $f_{2}:X_{2}\rightarrow Y_{2}$,
and define $f:X_{1}\times X_{2}\rightarrow Y_{1}\times Y_{2}$ as
$f:=\left(f_{1},f_{2}\right)$. If $f_{1}$ and $f_{2}$ are universally
measurable, so is $f$.

Fact C.1 follows from Lemma 9.1 in Friedenberg and Meier (2017) because
Souslin spaces are separable.

To understand the idea of the proof of Lemma 3, fix topological spaces $X$ and
$Y$, and a Borel measurable function $f:X\rightarrow Y$. Consider the
following problem pertaining to the pushforward-measure map induced by $f$.

Problem C.1 If $f:X\rightarrow Y$ is surjective, is the induced function
$\mathcal{L}_{f}:\Delta\left(X\right)\rightarrow\Delta\left(Y\right)$
surjective?

Problem C.1 can be restated as follows: Given $\nu\in\Delta\left(Y\right)$,
does there exist a solution $\mu\in\Delta\left(X\right)$ to the “equation”
$\mathcal{L}_{f}\left(\mu\right)=\nu$? In general the answer is negative, as
simple examples show. In particular, there could be no solution even if $f$ is
a bijection—cf. the discussion below Theorem 2 in Section 5.2 of the main
text. The following example illustrates this issue.

Example C.1 Suppose that $X$ is the unit interval $\left[0,1\right]$ endowed
with the discrete topology, and $Y$ is the unit interval $\left[0,1\right]$
endowed with the Euclidean topology. Let $f:X\rightarrow Y$ be the identity
map. Then $f$ is a continuous bijection, but it is neither a homeomorphism nor
a Borel isomorphism. The induced function
$\mathcal{L}_{f}:\Delta\left(X\right)\rightarrow\Delta\left(Y\right)$ is not
surjective because $\Delta\left(X\right)$ does not contain the Lebesgue
measure, which is an element of $\Delta\left(Y\right)$.

To provide a positive answer, one has to impose topological restrictions on
the relevant spaces. Problem C.1 can be reframed as an instance of “measurable
selection problem.” Recall that if a map $f:X\rightarrow Y$ is surjective, it
admits (by the Axiom of Choice) a right inverse, i.e., an injective function
$g:Y\rightarrow X$ such that $f\circ g=\mathrm{Id}_{Y}$. Put differently, $g$
is a selection of the correspondence $f^{-1}:Y\rightarrow 2^{X}$. The
following result, which is a version of Von Neumann Selection Theorem, gives
sufficient conditions for the existence of a right inverse $g:Y\rightarrow X$
satisfying a “nice” measurability condition. (For a proof, see Bogachev 2007,
Theorem 9.1.3 or Schwartz, 1973, Theorem 13, pp. 127-128.)

Theorem C.1 Fix Souslin spaces $X$ and $Y$. If $f:X\rightarrow Y$ is a
surjective Borel measurable function, then there exists a universally
measurable function $g:Y\rightarrow X$ which is a right inverse of $f$.

Given a Borel measurable function $f:X\rightarrow Y$ between Souslin spaces
and a probability measure $\nu\in\Delta\left(Y\right)$, a solution
$\mu\in\Delta\left(X\right)$ which solves the “equation”
$\mathcal{L}_{f}\left(\mu\right)=\nu$ can be found by means of the following
proof scheme:

1) Invoke Theorem C.1 to assert the existence of a universally measurable
function $g:Y\rightarrow X$ which is a right inverse of $f$.

2) Extend $\nu$ to $\left(Y,\Sigma_{Y}^{\ast}\right)$: such extension, denoted
by $\bar{\nu}$, exists and it is unique (Corbae et al. 2009, Corollary 7.9.6).

3) Use $g$ to push $\bar{\nu}$ forward: this gives a measure
$\mathcal{L}_{g}\left(\bar{\nu}\right)$ on $\left(X,\Sigma_{X}\right)$ such
that $\mathcal{L}_{f}\left(\mathcal{L}_{g}\left(\bar{\nu}\right)\right)=\nu$.

A solution to Problem C.1 was first given by Varadarajan (1963, p. 194), and
the proof follows the above scheme.222222It is worth noting that the function
$f$ in Example C.1 does not have a(n universally) measurable right inverse. We
adopt an analogous strategy to prove Lemma 3.

Proof of Lemma 3. To show part (i), first note that$\ X$, $Y$ and $Z$ are
Souslin spaces, hence thery are separable. It follows from standard results
(e.g., Bogachev 2007, Lemma 6.4.2) that the Borel $\sigma$-algebras generated
by the product topology on $X\times Y$ and $X\times Z$ coincide with the
product $\sigma$-algebras. The conclusion that
$f_{2}:=\left(\mathrm{Id}_{X},f_{1}\right):X\times Y\rightarrow X\times Z$ is
measurable follows from Aliprantis and Border (2006, Lemma 4.49). Next note
that every $C\in\mathcal{B}_{X\times Y}$ (resp. $C\in\mathcal{B}_{X\times Z}$)
is such that $C=B\times Y$ (resp. $C=B\times Z$) for some $B\in\mathcal{B}$.
Hence, it is the case that $\mathcal{B}_{X\times
Y}=\left(f_{2}\right)^{-1}\left(\mathcal{B}_{X\times Z}\right)$. By Lemma
1.(i), the map
$\overline{\mathcal{L}}_{f_{2}}:\Delta^{\mathcal{B}}\left(X\times
Y\right)\rightarrow\Delta^{\mathcal{B}}\left(X\times Z\right)$ is well-
defined.

To show part (ii), let $v$ be a CPS on $\left(X\times Z,\Sigma_{X\times
Z},\mathcal{B}_{X\times Z}\right)$. We have to find
$\mu\in\Delta^{\mathcal{B}}\left(X\times Y\right)$ such that
$\overline{\mathcal{L}}_{f_{2}}\left(\mu\right)=v$. To this end, first note
that, since $f_{1}:Y\rightarrow Z$ is surjective,
$f_{2}:=\left(\mathrm{Id}_{X},f_{1}\right):X\times Y\rightarrow X\times Z$ is
a Borel measurable surjection. By Theorem C.1, there exists a universally
measurable right inverse of $f_{1}$, that is, an injective function
$g_{1}:Z\rightarrow Y$ such that $f_{1}\circ g_{1}=\mathrm{Id}_{Z}$ and
$g_{1}^{-1}\left(E\right)\in\Sigma_{Z}^{\ast}$ for every $E\in\Sigma_{Y}$.

Let $g_{2}:X\times Z\rightarrow X\times Y$ be the function defined as
$g_{2}:=\left(\mathrm{Id}_{X},g_{1}\right)$. Clearly, $g_{2}$ satisfies
$f_{2}\circ g_{2}=\mathrm{Id}_{X\times Z}$, and, by Fact C.1, $g_{2}$ is a
universally measurable right inverse of $f_{2}$. Furthermore, $g_{2}$ is such
that $\mathcal{B}_{X\times Z}=g_{2}^{-1}\left(\mathcal{B}_{X\times Y}\right)$.
Hence, by Lemma 2.(i), the map $\overline{\mathcal{L}}_{g_{2}}$ from the set
of CPSs on $\left(X\times Z,\Sigma_{X\times Z}^{\ast},\mathcal{B}_{X\times
Z}\right)$ to the set of CPSs on $\left(X\times Y,\Sigma_{X\times
Y},\mathcal{B}_{X\times Y}\right)$ is well-defined.

Next, for each $B\in\mathcal{B}_{X\times Z}$, let
$\bar{v}\left(\cdot|B\right)$ denote the unique extension of
$v\left(\cdot|B\right)$ to $\Sigma_{X\times Z}^{\ast}$. We claim that the
array
$\bar{v}:=\left(\bar{v}\left(\cdot|B\right)\right)_{B\in\mathcal{B}_{X\times
Z}}$ is a CPS on $\left(X\times Z,\Sigma_{X\times
Z}^{\ast},\mathcal{B}_{X\times Z}\right)$. To see this, it is enough to verify
that the chain rule holds. Pick any $A\in\Sigma_{X\times Z}^{\ast}$ and
$B,C\in\mathcal{B}_{X\times Z}$ such that $A\subseteq B\subseteq C$. Notice
that, since $B\in\Sigma_{X\times Z}$, it is the case that
$\bar{v}\left(B|C\right)=v\left(B|C\right)$. Suppose first that
$A\in\Sigma_{X\times Z}$. Then the chain rule is satisfied because
$\bar{v}\left(A|C\right)=v\left(A|C\right)$ and
$\bar{v}\left(A|B\right)=v\left(A|B\right)$. If instead
$A\notin\Sigma_{X\times Z}$, then the chain rule trivially holds as $0=0$: we
have $\bar{v}\left(A|B\right)=\bar{v}\left(A|C\right)=0$ because $A$ is a set
that is both $v\left(\cdot|B\right)$-null and $v\left(\cdot|C\right)$-null.

By Lemma 2.(i), the image conditional law
$\mu:=\overline{\mathcal{L}}_{g_{2}}\left(\bar{v}\right)$ of $\bar{v}$ under