Almost Sure

10 January 19

Proof of the Measurable Projection and Section Theorems

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.

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

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} (5)

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} (6)

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 (6). 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}, write

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

It can be seen that this is a winning strategy. If (5) is satisfied, then it will also hold for the sequence {S^r_n\equiv S_{\theta(r,n)}} over {n\ge1} (for the game {\mathbb G_{A_r}}). As {f^r_n} 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 (6) 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 (5) is satisfied. Setting {S^\prime_1=S_1\cap A_r} and {S^\prime_n=S_n} for {n > 1}, we see that (5) 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 (6). 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\in\mathcal A_\delta} with

\displaystyle  {\mathbb P}(\pi_\Omega(A_n))\ge{\mathbb P}^*(\pi_\Omega(S))-1/n. (7)

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 atisfying {{\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 (7). 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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12 Comments »

  1. Hi the formalization of \tau(\omega) is a little confusing to me. Do you mean something like the following ?
    For \forall S \in \mathcal F \otimes \mathcal B (\mathcal R) we set :
    \tau : \pi_S(\Omega)\in \Omega \to \mathcal B (\mathbb R )
    \omega  \to \tau(\omega), such that (\omega,\tau(\omega) )\in S
    In which case the “measurability” of \tau is not completely clear to me.
    Moreover the graph seems also a little ambiguous as what arrows point to are (\omega, \tau(\omega)) and not \tau(\omega) (which are parts of \mathcal B (\mathbb R) unless mistaken).

    Comment by TheBridge — 16 January 19 @ 12:58 PM | Reply

    • I don’t follow what you are saying. In the second paragraph, \tau is a map from \pi_\Omega(S) (which is a subset of \Omega) to \mathbb R. So, \tau(\omega)\in\mathbb R and (\omega,\tau(\omega))\in\Omega\times\mathbb R. You seem to be suggesting that \pi_\Omega(S) is an element of \Omega, rather than a subset, and that \tau(\omega) is a subset of \mathbb R instead of an element.

      Comment by George Lowther — 16 January 19 @ 11:34 PM | Reply

      • On reflection, maybe you are not suggesting that \pi_\Omega(S) is an element of \Omega and it is just a typo in your comment. However, it still looks like you are suggesting that \tau(\omega) is a subset of \mathbb R, which is not the case.

        Comment by George Lowther — 16 January 19 @ 11:38 PM | Reply

        • As you spotted it’s a typo, I meant \subset intead of \in sorry about that. Coming back to my point, maybe I was confused about this quote :
          “…hat is, if S is in \mathcal F\otimes\mathcal B({\mathbb R}) then there is a measurable section, \tau\colon\pi_\Omega(S)\rightarrow\Omega\times{\mathbb R}
          So shouldn’t you write instead (as I understand your answer to my comment) ?
          \tau\colon\pi_\Omega(S)\rightarrow \mathbb R
          My second point in this regard is pointless and you can delete my other post. I also note that you make clear a “language abuse” shortly after all this when you write :
          “For brevity, the statement (\omega,\tau(\omega))\in S above will also be expressed by writing \tau\in S.” But I think it’s a bit early at this stage in your post to use this convention.
          Last let me correct you on one thing. It’s definitely not you who is happy to see me back (but I fill honored about that nevertheless so thanks), it is me indeed who is happy to see more posts from you on this amazing blog…, I have seen guys on MO forum who do not dare to quote and refer this blog in their papers as it is no “OK to refer a blog, but who can’t find in the literature equivalent theorems claimed and proved in such a clear and self contained manner… cela veut tout dire.

          Comment by TheBridge — 17 January 19 @ 1:21 PM

        • Ah, I fixed the typo which caused your confusion, but it probably occurs elsewhere, so will fix properly later. I’ll also reread through and consider your suggestion regarding the notation when I have some time to properly edit this. Thanks!

          Comment by George Lowther — 17 January 19 @ 1:52 PM

        • Last comment maybe would it be simpler to switch axis in your graph illustrating \tau, as it’s a function of \pi_\Omega (S)\subset \Omega \to \mathbb R and not the other way around.Regards

          Comment by TheBridge — 18 January 19 @ 8:13 AM

        • Rather than changing the graph, maybe it would be better to change the order of the Cartesian products throughout. \mathbb R\times\Omega instead of \Omega\times\mathbb R. That would be consistent with earlier stochastic calculus posts.

          Comment by George Lowther — 18 January 19 @ 9:56 AM

        • Another point is the ambiguity on the notation of sets S_n in the counterexample after equation (2) to illustrate the missing property needed for application of MCT. In some cases it’s a set in \mathbb R (S_n(\omega) and shortly after it is in \Omega \times \mathcal F unless mistaken. Regards

          Comment by TheBridge — 18 January 19 @ 10:36 AM

        • I don’t think there’s ambiguity. S_n is a subset of \Omega\times\mathbb R, whereas S_n(\omega) is a subset of \mathbb R.

          Comment by George Lowther — 18 January 19 @ 10:56 AM

        • I changed the order of all cartesian products. Let me know what you think – if it is better, I’ll update the other posts.

          Comment by George Lowther — 18 January 19 @ 12:04 PM

        • Sorry a few more remarks (I am reading your post very slowly as you can notice;-) ):
          -In the end I think that it would be nice to formalize the notion of “section”, \tau by a fully fledged “definition 1” .

          -Using the S_n in your definition (2) of S_n(\omega) is a bit hard to follow for me as it is easy to forget that it’s only a compact of \mathbb R when \omega is used and a part of \mathbb R \times\Omega when \omega is dropped.

          -You say that a decreasing sequence of compact that’s unless mistaken a theorem from Cantor, could be worth mentioning to be self contained :https://en.wikipedia.org/wiki/Cantor%27s_intersection_theorem

          -The end of the argumentation for the “compact” example could be detailed a little bit more I think, I quote it his part :
          “For each {\omega\in\bigcap_n\pi_\Omega(S_n)}, the slices $latex{S_n(\omega)}$ give a decreasing sequence of nonempty compact sets, so has nonempty intersection. So, letting S be the intersection {\bigcap_n S_n}, the slice {S(\omega)=\bigcap_n S_n(\omega)} is nonempty. Hence, {\omega\in\pi_\Omega(S)}, and (1) follows.”
          So you proved \forall \omega \in \pi_\Omega(S_n) then the slice of the intersection S={\bigcap_n S_n}, namely S(\omega) is nonempty (part of \mathbb R ) in the first part and this is OK for me. But then the fact that this proves that \omega \in \pi_\Omega(S) still need a little more clarification even if it might seem trivial to you. So for your last claim to be true, I think you need to prove the following property :
          For all nonempty S \in \mathcal B(\mathbb R)\otimes\mathcal F and \forall \omega \in \Omega , we have :
          S(\omega) \not \phi \Leftrightarrow \omega \in \pi_\Omega(S).
          Proof :
          \Rightarrow , by definition of a slice if it’s not empty then for t \in S(\omega) then (t,\omega) \in S so that \omega \in \pi_\Omega(S).
          \Leftarrow let’s take a look at the “contrapositive” (i.e. non \Rightarrow), if \omega \not\in \pi_\Omega(S) then there is no t\in \mathbb R such that (t,\omega)\in S and S(\omega)=\phi and we are done. End of proof. Does that seems ok to you ?

          Comment by TheBridge — 22 January 19 @ 1:48 PM

    • And, nice to see you again, TheBridge!

      Comment by George Lowther — 16 January 19 @ 11:40 PM | Reply


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