Continuity Form Above for Measurable Sequence of Sets

The aim of this post is to give a direct proof of the theorems of measurable projection and measurable section. These are generally regarded as rather difficult results, and proofs often use ideas from descriptive set theory such as analytic sets. I did previously post a proof along those lines on this blog. However, the results can be obtained in a more direct way, which is the purpose of this post. Here, I present relatively self-contained proofs which do not require knowledge of any advanced topics beyond basic probability theory.

The projection theorem states that if ${(\Omega,\mathcal F,{\mathbb P})}$ is a complete probability space, then the projection of a measurable subset of ${{\mathbb R}\times\Omega}$ onto ${\Omega}$ is measurable. To be precise, the condition is that S is in the product sigma-algebra ${\mathcal B({\mathbb R})\otimes\mathcal F}$ , where ${\mathcal B({\mathbb R})}$ denotes the Borel sets in ${{\mathbb R}}$ , and the projection map is denoted

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\pi_\Omega\colon{\mathbb R}\times\Omega\rightarrow\Omega,\smallskip\\ &\displaystyle\pi_\Omega(t,\omega)=\omega. \end{array}$

Then, measurable projection states that ${\pi_\Omega(S)\in\mathcal{F}}$ . Although it looks like a very basic property of measurable sets, maybe even obvious, measurable projection is a surprisingly difficult result to prove. In fact, the requirement that the probability space is complete is necessary and, if it is dropped, then ${\pi_\Omega(S)}$ need not be measurable. Counterexamples exist for commonly used measurable spaces such as ${\Omega= {\mathbb R}}$ and ${\mathcal F=\mathcal B({\mathbb R})}$ . This suggests that there is something deeper going on here than basic manipulations of measurable sets.

By definition, if ${S\subseteq{\mathbb R}\times\Omega}$ then, for every ${\omega\in\pi_\Omega(S)}$ , there exists a ${t\in{\mathbb R}}$ such that ${(t,\omega)\in S}$ . The measurable section theorem — also known as measurable selection — says that this choice can be made in a measurable way. That is, if S is in ${\mathcal B({\mathbb R})\otimes\mathcal F}$ then there is a measurable section,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\tau\colon\pi_\Omega(S)\rightarrow{\mathbb R},\smallskip\\ &\displaystyle(\tau(\omega),\omega)\in S. \end{array}$

It is convenient to extend ${\tau}$ to the whole of ${\Omega}$ by setting ${\tau=\infty}$ outside of ${\pi_\Omega(S)}$ .

measurable section — Figure 1: A section of a measurable set

The graph of ${\tau}$ is

$\displaystyle [\tau]=\left\{(t,\omega)\in{\mathbb R}\times\Omega\colon t=\tau(\omega)\right\}.$

The condition that ${(\tau(\omega),\omega)\in S}$ whenever ${\tau < \infty}$ can alternatively be expressed by stating that ${[\tau]\subseteq S}$ . This also ensures that ${\{\tau < \infty\}}$ is a subset of ${\pi_\Omega(S)}$ , and ${\tau}$ is a section of S on the whole of ${\pi_\Omega(S)}$ if and only if ${\{\tau < \infty\}=\pi_\Omega(S)}$ .

The results described here can also be used to prove the optional and predictable section theorems which, at first appearances, also seem to be quite basic statements. The section theorems are fundamental to the powerful and interesting theory of optional and predictable projection which is, consequently, generally considered to be a hard part of stochastic calculus. In fact, the projection and section theorems are really not that hard to prove.

Let us consider how one might try and approach a proof of the projection theorem. As with many statements regarding measurable sets, we could try and prove the result first for certain simple sets, and then generalise to measurable sets by use of the monotone class theorem or similar. For example, let ${\mathcal S}$ denote the collection of all ${S\subseteq{\mathbb R}\times\Omega}$ for which ${\pi_\Omega(S)\in\mathcal F}$ . It is straightforward to show that any finite union of sets of the form ${A\times B}$ for ${A\in\mathcal B({\mathbb R})}$ and ${B\in\mathcal F}$ are in ${\mathcal S}$ . If it could be shown that ${\mathcal S}$ is closed under taking limits of increasing and decreasing sequences of sets, then the result would follow from the monotone class theorem. Increasing sequences are easily handled — if ${S_n}$ is a sequence of subsets of ${{\mathbb R}\times\Omega}$ then from the definition of the projection map,

$\displaystyle \pi_\Omega\left(\bigcup\nolimits_n S_n\right)=\bigcup\nolimits_n\pi_\Omega\left(S_n\right).$

If ${S_n\in\mathcal S}$ for each n, this shows that the union ${\bigcup_nS_n}$ is again in ${\mathcal S}$ . Unfortunately, decreasing sequences are much more problematic. If ${S_n\subseteq S_m}$ for all ${n\ge m}$ then we would like to use something like

$\displaystyle \pi_\Omega\left(\bigcap\nolimits_n S_n\right)=\bigcap\nolimits_n\pi_\Omega\left(S_n\right).$

(1)

However, this identity does not hold in general. For example, consider the decreasing sequence ${S_n=(n,\infty)\times\Omega}$ . Then, ${\pi_\Omega(S_n)=\Omega}$ for all n, but ${\bigcap_nS_n}$ is empty, contradicting (1). There is some interesting history involved here. In a paper published in 1905, Henri Lebesgue claimed that the projection of a Borel subset of ${{\mathbb R}^2}$ onto ${{\mathbb R}}$ is itself measurable. This was based upon mistakenly applying (1). The error was spotted in around 1917 by Mikhail Suslin, who realised that the projection need not be Borel, and lead him to develop the theory of analytic sets.

Actually, there is at least one situation where (1) can be shown to hold. Suppose that for each ${\omega\in\Omega}$ , the slices

$\displaystyle S_n(\omega)\equiv\left\{t\in{\mathbb R}\colon(t,\omega)\in S_n\right\}$

(2)

are compact. For each ${\omega\in\bigcap_n\pi_\Omega(S_n)}$ , the slices ${S_n(\omega)}$ give a decreasing sequence of nonempty compact sets, so has nonempty intersection. So, letting S be the intersection ${\bigcap_nS_n}$ , the slice ${S(\omega)=\bigcap_nS_n(\omega)}$ is nonempty. Hence, ${\omega\in\pi_\Omega(S)}$ , and (1) follows.

The starting point for our proof of the projection and section theorems is to consider certain special subsets of ${{\mathbb R}\times\Omega}$ where the compactness argument, as just described, can be used. The notation ${\mathcal A_\delta}$ is used to represent the collection of countable intersections, ${\bigcap_{n=1}^\infty A_n}$ , of sets ${A_n}$ in ${\mathcal A}$ .

Lemma 1 Let ${(\Omega,\mathcal F)}$ be a measurable space, and ${\mathcal A}$ be the collection of subsets of ${{\mathbb R}\times\Omega}$ which are finite unions ${\bigcup_kC_k\times E_k}$ over compact intervals ${C_k\subseteq{\mathbb R}}$ and ${E_k\in\mathcal F}$ . Then, for any ${S\in\mathcal A_\delta}$ , we have ${\pi_\Omega(S)\in\mathcal F}$ , and the debut

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \tau\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\},\smallskip\\ &\displaystyle \omega\mapsto\inf\left\{t\in{\mathbb R}\colon (t,\omega)\in S\right\}. \end{array}$

is a measurable map with ${[\tau]\subseteq S}$ and ${\{\tau < \infty\}=\pi_\Omega(S)}$ .

Proof: Noting that ${\mathcal F}$ and the collection of compact intervals in ${{\mathbb R}}$ are closed under pairwise intersection, the same is true for ${\mathcal A}$ . Then, for ${S\in\mathcal A_\delta}$ there exists, by definition, ${S_n\in\mathcal A}$ such that ${S=\bigcap_nS_n}$ . Replacing ${S_n}$ by ${\bigcap_{m\le n}S_m}$ if necessary, we may suppose that ${S_n}$ is a decreasing sequence.

Now, the slices ${S_n(\omega)}$ defined by (2) are finite unions of compact intervals, so are compact. The compactness argument explained above implies that

$\displaystyle \pi_\Omega(S)=\bigcap\nolimits_n\pi_\Omega(S_n).$

(3)

As each ${S_n}$ is a finite union ${\bigcup_kC_k\times E_k}$ for ${E_k\in\mathcal F}$ and nonempty ${C_k}$ , the projection ${\pi_\Omega(S_n)=\bigcup_kE_k}$ is in ${\mathcal F}$ . Then, (3) shows that ${\pi_\Omega(S)}$ is also in ${\mathcal F}$ .

If ${\tau}$ is the debut of S, then ${\tau(\omega)=\inf S(\omega)}$ . This immediately implies ${\{\tau < \infty\}=\pi_\Omega(S)}$ and, as nonempty compact sets contain their infimum, ${[\tau]\subseteq S}$ . For every ${t\in{\mathbb R}}$ , the set ${((-\infty,t]\times\Omega)\cap S}$ is in ${\mathcal A_\delta}$ and,

$\displaystyle \{\tau\le t\}=\pi_\Omega\left(((-\infty,t]\times\Omega)\cap S\right)\in\mathcal F,$

showing that ${\tau}$ is measurable. ⬜

When dealing with more general subsets of ${{\mathbb R}\times\Omega}$ , it will not necessarily be the case that the projection onto ${\Omega}$ is measurable. For that reason, we extend the probability measure to more general subsets of ${\Omega}$ . For a probability space ${(\Omega,\mathcal F,{\mathbb P})}$ , define an outer measure on the power set ${\mathcal P(\Omega)}$ by approximating ${A\subseteq\Omega}$ from above by measurable sets,

$\displaystyle {\mathbb P}^*(A)=\inf\left\{{\mathbb P}(B)\colon B\in\mathcal F,A\subseteq B\right\}.$

(4)

The outer measure has the following basic properties.

Lemma 2 For a probability space ${(\Omega,\mathcal F,{\mathbb P})}$ , the outer measure ${{\mathbb P}^*}$ is increasing and continuous along increasing sequences. That is, ${{\mathbb P}^*(A)\le{\mathbb P}^*(B)}$ for ${A\subseteq B}$ , and ${{\mathbb P}^*(A_n)\rightarrow{\mathbb P}^*(A)}$ for sequences ${A_n\subseteq\Omega}$ increasing to a limit A.

Furthermore, for any ${A\subseteq\Omega}$ , there exists ${B\supseteq A}$ in ${\mathcal F}$ with ${{\mathbb P}(B)={\mathbb P}^*(A)}$ .

Proof: The fact that ${{\mathbb P}^*}$ is increasing is immediate from the definition. Now, let ${A_n\subseteq\Omega}$ be increasing to the limit A. By the definition of ${{\mathbb P}^*(A_n)}$ , there exists ${B_n\supseteq A_n}$ in ${\mathcal F}$ with

$\displaystyle {\mathbb P}(B_n)\le{\mathbb P}^*(A_n)+1/n.$

Replacing ${B_n}$ by ${\bigcap_{m\ge n}B_m}$ if necessary, we may suppose that ${B_n}$ is an increasing sequence. Then, ${B=\bigcup_nB_n\supseteq A}$ is in ${\mathcal F}$ and, by monotone convergence,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb P}^*(A)\le{\mathbb P}(B)&\displaystyle=\lim_n{\mathbb P}(B_n)\smallskip\\ &\displaystyle\le\lim_n({\mathbb P}^*(A_n)+1/n)\smallskip\\ &\displaystyle=\lim_n{\mathbb P}^*(A_n)\le{\mathbb P}^*(A). \end{array}$

So, ${{\mathbb P}^*(A)=\lim_n{\mathbb P}^*(A_n)}$ as required. Incidentally, this also shows that there is a ${B\supseteq A}$ in ${\mathcal F}$ with ${{\mathbb P}(B)={\mathbb P}^*(A)}$ . ⬜

I now move on the the main component of the proof of the projection and section theorems. This will allow us to approximate measurable subsets of ${{\mathbb R}\times\Omega}$ from below by sets in ${\mathcal A_\delta}$ , as defined in lemma 1 above. While the statement of theorem 3 is simple enough, the proof can get a bit tricky. The method used here is elementary and, although the argument is a bit intricate, no advanced mathematics is required. The definition of ${\mathcal{\bar A}}$ means that it is the minimal collection of subsets of X which contains ${\mathcal A}$ and is closed under taking limits of increasing and decreasing sequences. I refer to the result as the `capacitability theorem' as it is a version of Choquet's capacitability theorem although, here, we do not involve the concept of analytic sets. A set ${A\subseteq X}$ can be called capacitable if, for each ${K < I(A)}$ , there exists a decreasing sequence ${A_n\in\mathcal A}$ with ${K < I(A_n)}$ and ${\bigcap_nA_n\subseteq A}$ . So, theorem 3 is saying that all sets in ${\mathcal{\bar A}}$ are capacitable.

Theorem 3 (Capacitability Theorem) Let X be a set, ${\mathcal A\subseteq\mathcal P(X)}$ be closed under pairwise intersections, and ${I\colon\mathcal P(X)\rightarrow{\mathbb R}^+}$ be increasing and continuous along increasing sequences. Denote the closure of ${\mathcal A}$ under limits of increasing and of decreasing sequences by ${\mathcal{\bar A}}$ .

Then, for any ${A\in\mathcal{\bar A}}$ and ${K\in{\mathbb R}}$ with ${I(A) > K}$ , there exists a decreasing sequence ${A_n\in\mathcal A}$ with ${\bigcap_nA_n\subseteq A}$ and ${I(A_n) > K}$ for all n.

Proof: Fixing ${K\in{\mathbb R}}$ , let ${\mathcal C}$ denote the collection of all ${A\subseteq X}$ with ${I(A) > K}$ . The assumptions on I mean that for any ${A\in\mathcal C}$ then every ${B\supseteq A}$ is in ${\mathcal C}$ and, for any sequence ${A_n\subseteq X}$ increasing to A, then ${A_n\in\mathcal C}$ for large n.

The proof of the theorem amounts to finding a collection ${\mathcal B\subseteq\mathcal P(X)}$ containing ${\mathcal A}$ and closed under taking limits of increasing and decreasing sequences, such that, for every ${A\in\mathcal B\cap\mathcal C}$ , we can construct a decreasing sequence ${A_n\in\mathcal A\cap\mathcal C}$ with ${\bigcap_nA_n\subseteq A}$ . In that case, every ${A\in\mathcal{\bar A}}$ will also be in ${\mathcal B}$ , and the claimed result will follow.

The main difficulty in the proof is to describe a collection ${\mathcal B}$ with the required properties. One way of doing this is as follows, and can be described in terms of a game. For ${A\in\mathcal C}$ , consider the following infinite game played between two players, who take turns choosing sets from ${\mathcal C}$ . Starting with ${T_0=A}$ , at rounds ${n=1,2,\ldots}$ , the players make the following moves.

Player 1 chooses an ${S_n\subseteq T_{n-1}}$ in ${\mathcal C}$ .
Player 2 chooses a ${T_n\subseteq S_n}$ in ${\mathcal C}$ .

At each round, both players can, at least, make a valid move. For example, player 1 can set ${S_n=T_{n-1}}$ and player 2 can set ${T_n=S_n}$ . We say that player 2 wins the game if, once completed, she is able to find a sequence ${A_n\supseteq T_n}$ in ${\mathcal A}$ with ${\bigcap_nA_n\subseteq A}$ .

For any ${A\subseteq X}$ , denote the game described above by ${\mathbb G_A}$ . A strategy (for player 2) is just a sequence of functions ${f_n\colon\mathcal C^n\rightarrow\mathcal C}$ satisfying

$\displaystyle f_n(S_1,\ldots,S_n) \subseteq S_n.$

(5)

The idea is that ${f_n(S_1,\ldots,S_n)}$ represents player 2's choice for ${T_n}$ at round n, given that player 1 has chosen ${S_1,\ldots,S_n}$ so far. It is a winning strategy if, for any sequence ${S_1,S_2,\ldots\in\mathcal C}$ satisfying

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle S_1\subseteq A,\smallskip\\ &\displaystyle S_{n+1}\subseteq f_n(S_1,\ldots,S_n) \end{array}$

(6)

for each ${n\ge 1}$ , then there exists a sequence ${A_n\in\mathcal A}$ with

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle f_n(S_1,\ldots,S_n)\subseteq A_n,\smallskip\\ &\displaystyle\bigcap_{n=1}^\infty A_n\subseteq A. \end{array}$

(7)

We note that, combining (5) and (6) shows that ${S_n}$ must be a decreasing sequence of subsets of A.

Now, let ${\mathcal B}$ be the collection of ${A\subseteq X}$ for which the game ${\mathbb G_A}$ has a winning strategy. The case with ${A\in\mathcal A}$ is easy. Any strategy is a winning strategy simply by taking ${A_n=A}$ in (7). For ${f_n}$ we may as well take ${f_n(S_1,\ldots,S_n)=S_n}$ , which is a valid strategy.

Now, consider a sequence ${A_k\in\mathcal B}$ and let ${\{f^k_n\}_{n=1,2,\ldots}}$ be winning strategies for ${\mathbb G_{A_k}}$ . Construct a winning strategy for ${\mathbb G_A}$ , with ${A=\bigcap_kA_k}$ , as follows. Choose a bijection ${\theta\colon{\mathbb N}^2\rightarrow{\mathbb N}}$ such that ${\theta(r,s)}$ is increasing in s. For example, take ${\theta(r,s)=(2r-1)2^{s-1}}$ . Then for ${n=\theta(r,s)}$ and ${S_1,\ldots,S_n\in\mathcal C}$ , define

$\displaystyle f_n(S_1,\ldots,S_n)=f^r_s(S_{\theta(r,1)},\ldots,S_{\theta(r,s)})\subseteq S_n.$

It can be seen that this is a winning strategy. If (6) is satisfied then, writing ${S^r_s\equiv S_{\theta(r,s)}}$ , we use the fact that the sequence ${S_n}$ is decreasing and ${\theta(r,s+1)\ge\theta(r,s)+1}$ to write

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{ll} \displaystyle S^r_s\subseteq A\subseteq A_r,\smallskip\\ \displaystyle S^r_{s+1}\subseteq S_{\theta(r,s)+1}&\displaystyle\subseteq f_n(S_1,\ldots,S_n)\smallskip\\ &\displaystyle=f^r_s(S^r_1,\ldots,S^r_s) \end{array}$

for any ${n=\theta(r,s)}$ . So, (6) is also satisfied for the sequence ${S^r_1,S^r_2,\ldots}$ (for the strategy ${f^r_\cdot}$ and game ${\mathbb G_{A_r}}$ ). As ${f^r_\cdot}$ is a winning strategy for ${\mathbb G_{A_r}}$ , there exists ${A_{rs}\supseteq f^r_s(S^r_1,\ldots,S^r_s)}$ in ${\mathcal A}$ satisfying ${\bigcap_sA_{rs}\subseteq A_r}$ . In particular, writing ${B_{\theta(r,s)}=A_{rs}}$ gives

$\displaystyle \bigcap_nB_n=\bigcap_r\bigcap_s A_{rs}\subseteq\bigcap_rA_r=A$

so (7) is satisfied, and ${A\in\mathcal B}$ .

If ${A_k}$ is increasing, construct a winning strategy for ${A=\bigcup_kA_k}$ as follows. For any ${S_1,\ldots,S_n\in\mathcal C}$ with ${S_1\subseteq A}$ , the sequence ${S_1\cap A_k}$ increases to ${S_1}$ . Hence, there is a minimum r such that ${S_1\cap A_r\in\mathcal C}$ . Set,

$\displaystyle f_n(S_1,\ldots,S_n)=f^r_n(S_1\cap A_r,S_2,\ldots,S_n).$

For ${S_1\not\subseteq A}$ then we do not really care, so can just take ${f_n(S_1,\ldots,S_n)=S_n}$ . This clearly gives a valid strategy. To see that it is a winning strategy, suppose that (6) is satisfied. Setting ${S^\prime_1=S_1\cap A_r}$ and ${S^\prime_n=S_n}$ for ${n > 1}$ , we see that (6) is also satisfied with ${S^\prime_n}$ in place of ${S_n}$ and ${f^r_n}$ in place of ${f_n}$ . So, as ${f^r_n}$ is a winning strategy for the game ${\mathbb G_{A_r}}$ , there exists a sequence ${B_n\in\mathcal A}$ with

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle f_n(S_1,\ldots,S_n)=f^r_n(S^\prime_1,\ldots,S^\prime_n)\subseteq B_n,\smallskip\\ &\displaystyle\bigcap_{n=1}^\infty B_n\subseteq A_r\subseteq A. \end{array}$

So, ${\{f_n\}}$ is a winning strategy for ${\mathbb G_{A}}$ and, hence, ${A\in\mathcal B}$ .

We have shown that ${\mathcal B}$ contains ${\mathcal A}$ and is closed under taking limits of increasing and decreasing sequences and, so, contains ${\mathcal{\bar A}}$ . Finally, for any ${A\in\mathcal{\bar A}\cap\mathcal C}$ , let ${\{f_n\}_{n=1,2,\ldots}}$ be a winning strategy for ${\mathbb G_A}$ and define a sequence ${S_n\in\mathcal C}$ by ${S_1=A}$ and

$\displaystyle S_{n+1}=f_n(S_1,\ldots,S_n)$

for all ${n\ge1}$ . As ${\{f_n\}}$ is a winning strategy, there exists a sequence ${A_n\in\mathcal A}$ satisfying (7). Replacing ${A_n}$ by ${\bigcap_{m\le n}A_n}$ if required, we can suppose that the sequence is decreasing. Finally, as ${S_{n+1}\subseteq A_n}$ , we have ${A_n\in\mathcal C}$ as required. ⬜

The argument above is along similar lines to the `rabotages de Sierpinski' used by Dellacherie, Ensembles aléatoires II (1969). Although the description of the collection ${\mathcal B}$ in terms of winning strategies of the games ${\mathbb G_A}$ may not seem like an obvious approach, it is really quite natural. As a first attempt to prove the result, we could try defining ${\mathcal B}$ to be the collection of sets for which the conclusion of the theorem holds. That is, the sets A for which there is a decreasing sequence ${A_n\in\mathcal A\cap\mathcal C}$ with ${\bigcap_nA_n\subseteq A}$ . We would then have to show that ${\mathcal B}$ is closed under taking limits of increasing and decreasing sequences. While increasing sequences are easy to deal with, decreasing ones are problematic. Suppose that ${A_n}$ decreases to A and that, for each n, there is a decreasing sequence ${\{A_{nk}\}_{k=1,2,\ldots}\in\mathcal A\cap\mathcal C}$ with ${\bigcap_kA_{nk}\subseteq A_n}$ . To construct a sequence of sets ${B_n\in\mathcal A\cap\mathcal C}$ we could try to do the following. Reorder the doubly-indexed sequence ${A_{nk}}$ into a singly-indexed one, ${A_{n_1k_1},A_{n_2k_2},\ldots}$ and set ${B_r=A_{n_rk_r}}$ . Then, it is clear that ${B_r\in\mathcal A\cap\mathcal C}$ and ${\bigcap_rB_r\subseteq A}$ . However, ${B_r}$ is not decreasing. We could try and ensure that it is decreasing by setting

$\displaystyle B_r=A_{n_1k_1}\cap\cdots\cap A_{n_rk_r}.$

Unfortunately, it is no longer necessarily true that ${B_r}$ is in ${\mathcal C}$ . When we take intersections ${A_{n_rk_r}\cap A_{n_sk_s}}$ we need no longer be in ${\mathcal C}$ . The easiest way around this, it seems, is to allow the choice of ${A_{n_rk_r}}$ to depend on the previous choices of ${A_{n_sk_s}}$ . That is, the choice of ${A_{n_rk_r}}$ should depend on ${B_{r-1}}$ so as to enforce the condition that ${B_r=B_{r-1}\cap A_{n_rk_r}}$ is in ${\mathcal C}$ . This leads, essentially, to the requirement of winning strategies for the games ${\mathbb G_{A_n}}$ as described in the proof of theorem 3.

We use theorem 3 to show that measurable subsets of ${\Omega\times{\mathbb R}}$ can be approximated from below by ${\mathcal A_\delta}$ .

Corollary 4 Let ${(\Omega,\mathcal F,{\mathbb P})}$ be a probability space and ${\mathcal A}$ be the collection of subsets of ${{\mathbb R}\times\Omega}$ given in lemma 1. Then, for any ${S\in\mathcal B({\mathbb R})\otimes\mathcal F}$ and ${\epsilon > 0}$ , there exists ${A\subseteq S}$ in ${\mathcal A_\delta}$ satisfying

$\displaystyle {\mathbb P}\left(\pi_\Omega(A)\right)\ge{\mathbb P}^*\left(\pi_\Omega(S)\right)-\epsilon.$

Proof: Setting ${X={\mathbb R}\times\Omega}$ , define

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle I\colon\mathcal P(X)\rightarrow{\mathbb R}^+\smallskip\\ &\displaystyle A\mapsto{\mathbb P}^*(\pi_\Omega(A)). \end{array}$

This is clearly increasing. Also, if ${A_n\subseteq X}$ is increasing to a limit A then ${\pi_\Omega(A_n)}$ increases to ${\pi_\Omega(A)}$ . Lemma 2 implies that ${I(A_n)\rightarrow I(A)}$ , and I is continuous along increasing sequences.

As the complement of a compact interval in ${{\mathbb R}}$ is a countable union of compact intervals, the complement of any ${A\in\mathcal A}$ is a countable union of ${\mathcal A}$ . The monotone class theorem then says that the closure of ${\mathcal A}$ under limits of increasing and decreasing sequences is the entire sigma-algebra generated by ${\mathcal A}$ . Hence,

$\displaystyle \mathcal{\bar A}=\mathcal B({\mathbb R})\otimes\mathcal F.$

We apply theorem 3. For ${S\in\mathcal B({\mathbb R})\otimes\mathcal F}$ and ${\epsilon > 0}$ , setting ${K=I(S)-\epsilon}$ , there exists a decreasing sequence ${A_n\in\mathcal A}$ with ${\bigcap_nA_n\subseteq S}$ and ${I(A_n) > K}$ . Take ${A=\bigcap_nA_n}$ which is in ${\mathcal A_\delta}$ . As in the proof of lemma 1, ${\pi_\Omega(A_n)\in\mathcal F}$ decreases to ${\pi_\Omega(A)}$ . By monotone convergence,

$\displaystyle {\mathbb P}(\pi_\Omega(A))=\lim_n{\mathbb P}(\pi_\Omega(A_n))\ge K$

as required. ⬜

Combining this result with the statement, in lemma 1, of measurable projection for sets in ${\mathcal A_\delta}$ gives the measurable projection theorem.

Theorem 5 (Measurable Projection) Let ${(\Omega,\mathcal F,{\mathbb P})}$ be a complete probability space, and ${S\in\mathcal B({\mathbb R})\otimes\mathcal F}$ . Then, ${\pi_\Omega(S)\in\mathcal F}$ .

Proof: By corollary 4, for each positive integer n, there is an ${A_n\subseteq S}$ in ${\in\mathcal A_\delta}$ with

$\displaystyle {\mathbb P}(\pi_\Omega(A_n))\ge{\mathbb P}^*(\pi_\Omega(S))-1/n.$

(8)

We know from lemma 1 that ${\pi_\Omega(A_n)}$ are measurable, so ${A\equiv\bigcup_n\pi_\Omega(A_n)}$ is in ${\mathcal F}$ , is contained in ${\pi_\Omega(S)}$ , and satisfies ${{\mathbb P}(A)={\mathbb P}^*(\pi_\Omega(S))}$ . Lemma 2 states that there is a ${B\supseteq\pi_\Omega(S)}$ in ${\mathcal F}$ and satisfying ${{\mathbb P}(B)={\mathbb P}^*(\pi_\Omega(S))}$ .

We have constructed sets ${A\subseteq\pi_\Omega(S)\subseteq B}$ in ${\mathcal F}$ and satisfying ${{\mathbb P}(A)={\mathbb P}(B)}$ . By definition, this means that ${\pi_\Omega(S)}$ is in the completion of ${\mathcal F}$ and, if the probability space is complete, it is in ${\mathcal F}$ . ⬜

In a similar way, corollary 4 combined with the statement of measurable section for sets in ${\mathcal A_\delta}$ , given by lemma 1, gives the measurable section theorem.

Theorem 6 (Measurable Section) Let ${(\Omega,\mathcal F,{\mathbb P})}$ be a probability space and ${S\in\mathcal B({\mathbb R})\otimes\mathcal F}$ . Then, there exists a measurable ${\tau\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\}}$ , such that ${[\tau]\subseteq S}$ and ${\pi_\Omega(S)\setminus\{\tau < \infty\}}$ is ${{\mathbb P}}$ -null.

Proof: As in the proof of theorem 5, there is a sequence ${A_n\subseteq S}$ in ${\mathcal A_\delta}$ satisfying (8). Replacing ${A_n}$ by ${\bigcup_{m\le n}A_m}$ if necessary, we suppose that the sequence ${A_n}$ is increasing. Let ${\tau_n}$ be the debut of ${A_n}$ , Lemma 1 states that this is measurable and ${[\tau_n]\subseteq A_n}$ . Define a random time ${\tau}$ by,

$\displaystyle \tau(\omega)=\begin{cases} \tau_n(\omega),&{\rm\ for\ }\omega\in\pi_\Omega(A_n)\setminus\pi_\Omega(A_{n-1})\\ \infty,&{\rm\ for\ }\omega\in\Omega\setminus\bigcup_n\pi_\Omega(A_n) \end{cases}$

(I am using ${A_0=\emptyset}$ ). This is measurable with graph ${[\tau]}$ contained in S and,

$\displaystyle {\mathbb P}(\tau < \infty)={\mathbb P}\left(\bigcup\nolimits_n\pi_\Omega(A_n)\right)\ge{\mathbb P}^*(\pi_\Omega(S))$

By lemma 2, there exists ${B\in\mathcal F}$ containing ${\pi_\Omega(S)}$ with ${{\mathbb P}(B)={\mathbb P}^*(\pi_\Omega(S))}$ . So, ${B\setminus\{\tau < \infty\}}$ has zero probability and contains ${\pi_\Omega(S)\setminus\{\tau < \infty\}}$ , which is ${{\mathbb P}}$ -null as required. ⬜

Finally, we state the theorem for complete probability spaces, in which case the section is defined on all of ${\pi_\Omega(S)}$ , and not just up to a ${{\mathbb P}}$ -null set.

Theorem 7 (Measurable Section) Let ${(\Omega,\mathcal F,{\mathbb P})}$ be a complete probability space and ${S\in\mathcal B({\mathbb R})\otimes\mathcal F}$ . Then, there exists a measurable ${\tau\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\}}$ , such that ${[\tau]\subseteq S}$ and ${\{\tau < \infty\}=\pi_\Omega(S)}$ .

Proof: By theorem 6 there exists a measurable map ${\tau_0\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\}}$ such that ${[\tau_0]\subseteq S}$ and ${\pi_\Omega(S)\setminus\{\tau_0 < \infty\}}$ is ${{\mathbb P}}$ -null. Define ${\tau\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\}}$ by

$\displaystyle \displaystyle\tau(\omega)=\begin{cases} \tau_0(\omega),&{\rm\ if\ }\tau_0(\omega) < \infty,\\ \infty,&{\rm\ if\ }\omega\not\in\pi_\Omega(S),\\ t\in S(\omega),&{\rm\ if\ }\omega\in\pi_\Omega(S)\setminus\{\tau_0 < \infty\}. \end{cases}$

Here, ${S(\omega)}$ represents the slice of S defined as in (2). We do not care about which t is chosen in the third case but, as ${S(\omega)}$ is nonempty on ${\pi_\Omega(S)}$ , a choice does exist. By construction, ${[\tau]\subseteq S}$ , ${\{\tau < \infty\}=\pi_\Omega(S)}$ , and ${\tau=\tau_0}$ almost surely. As ${\tau_0}$ is measurable, completeness of the probability space implies that ${\tau}$ is also measurable. ⬜

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Source: https://almostsuremath.com/2019/01/10/proof-of-the-measurable-projection-and-section-theorems/

Continuity Form Above for Measurable Sequence of Sets

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