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)}.

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7 January 19

Proof of Optional and Predictable Section

In this post I give a proof of the theorems of optional and predictable section. These are often considered among the more advanced results in stochastic calculus, and many texts on the subject skip their proofs entirely. The approach here makes use of the measurable section theorem but, other than that, is relatively self-contained and will not require any knowledge of advanced topics beyond basic properties of probability measures.

Given a probability space {(\Omega,\mathcal F,{\mathbb P})} we denote the projection map from {\Omega\times{\mathbb R}^+} to {\Omega} by

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

For a set {S\subseteq\Omega\times{\mathbb R}^+} then, by construction, for every {\omega\in\pi_\Omega(S)} there exists a {t\in{\mathbb R}^+} with {(\omega,t)\in S}. Measurable section states that this choice can be made in a measurable way. That is, assuming that the probability space is complete, {\pi_\Omega(S)} is measurable and there is a measurable section {\tau\colon\Omega\rightarrow{\mathbb R}^+} satisfying {\tau\in S}. I use the shorthand {\tau\in S} to mean {(\omega,\tau(\omega))\in S}, and it is convenient to extend the domain of {\tau} to all of {\Omega} by setting {\tau=\infty} outside of {\pi_\Omega(S)}. So, we consider random times taking values in the extended nonnegative real numbers {\bar{\mathbb R}^+={\mathbb R}^+\cup\{\infty\}}. The property that {\tau\in S} whenever {\tau < \infty} can be expressed by stating that the graph of {\tau} is contained in S, where the graph is defined as

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

The optional section theorem is a significant extension of measurable section which is very important to the general theory of stochastic processes. It starts with the concept of stopping times and with the optional sigma-algebra on {\Omega\times{\mathbb R}^+}. Then, it says that if S is optional its section {\tau} can be chosen to be a stopping time. However, there is a slight restriction. It might not be possible to define such {\tau} everywhere on {\pi_\Omega(S)}, but instead only up to a set of positive probability {\epsilon}, where {\epsilon} can be made as small as we like. There is also a corresponding predictable section theorem, which says that if S is in the predictable sigma-algebra, its section {\tau} can be chosen to be a predictable stopping time.

I give precise statements and proofs of optional and predictable section further below, and also prove a much more general section theorem which applies to any collection of random times satisfying a small number of required properties. Optional and predictable section will follow as consequences of this generalised section theorem.

Both the optional and predictable sigma-algebras, as well as the sigma-algebra used in the generalised section theorem, can be generated by collections of stochastic intervals. Any pair of random times {\sigma,\tau\colon\Omega\rightarrow\bar{\mathbb R}^+} defines a stochastic interval,

\displaystyle  [\sigma,\tau)\equiv\left\{(\omega,t)\in\Omega\times{\mathbb R}^+\colon\sigma(\omega)\le t < \tau(\omega)\right\}.

The debut of a set {S\subseteq\Omega\times{\mathbb R}^+} is defined to be the random time

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle D(S)\colon\Omega\rightarrow\bar{\mathbb R}^+,\smallskip\\ &\displaystyle D(S)(\omega)=\inf\left\{t\in{\mathbb R}^+\colon(\omega,t)\in S\right\}. \end{array}

In general, even if S is measurable, its debut need not be, although it can be shown to be measurable in the case that the probability space is complete. For a random time {\tau} and a measurable set {A\subseteq\Omega}, we use {\tau_A} to denote the restriction of {\tau} to A defined by

\displaystyle  \tau_A(\omega)=\begin{cases} \tau(\omega),&{\rm\ if\ }\omega\in A,\\ \infty,&{\rm\ if\ }\omega\not\in A. \end{cases}

We start with the general situation of a collection of random times {\mathcal T} satisfying a few required properties and show that, for sufficiently simple subsets of {\Omega\times{\mathbb R}^+}, the section can be chosen to be almost surely equal to the debut. It is straightforward that the collection of all stopping times defined with respect to some filtration do indeed satisfy the required properties for {\mathcal T}, but I also give a proof of this further below. A nonempty collection {\mathcal A} of subsets of a set X is called an algebra, Boolean algebra or, alternatively, a ring, if it is closed under finite unions, finite intersections, and under taking the complement {A^c=X\setminus A} of sets {A\in\mathcal A}. Recall, also, that {\mathcal A_\delta} represents the countable intersections of A, which is the collection of sets of the form {\bigcap_nA_n} for sequences {A_1,A_2,\ldots} in {\mathcal A}.

Lemma 1 Let {(\Omega,\mathcal F,{\mathbb P})} be a probability space and {\mathcal T} be a collection of measurable times {\tau\colon\Omega\rightarrow\bar{\mathbb R}^+} satisfying,

  • the constant function {\tau=0} is in {\mathcal T}.
  • {\sigma\wedge\tau} and {\sigma_{\{\sigma < \tau\}}} are in {\mathcal T}, for all {\sigma,\tau\in\mathcal T}.
  • {\sup_n\mathcal\tau_n\in\mathcal T} for all sequences {\tau_1,\tau_2,\cdots} in {\mathcal T}.

Then, letting {\mathcal A} be the collection of finite unions of stochastic intervals {[\sigma,\tau)} over {\sigma,\tau\in\mathcal T}, we have the following,

  • {\mathcal A} is an algebra on {\Omega\times{\mathbb R}^+}.
  • for all {S\in\mathcal A_\delta}, its debut satisfies

    \displaystyle  [D(S)]\subseteq S,\ \{D(S) < \infty\}=\pi_\Omega(S),

    and there is a {\tau\in\mathcal T} with {[\tau]\subseteq[D(S)]} and {\tau = D(S)} almost surely.

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2 January 19

Proof of Measurable Section

I will give a proof of the measurable section theorem, also known as measurable selection. Given a complete probability space {(\Omega,\mathcal F,{\mathbb P})}, we denote the projection from {\Omega\times{\mathbb R}} by

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

By definition, if {S\subseteq\Omega\times{\mathbb R}} then, for every {\omega\in\pi_\Omega(S)}, there exists a {t\in{\mathbb R}} such that {(\omega,t)\in S}. The measurable section theorem says that this choice can be made in a measurable way. That is, using {\mathcal B({\mathbb R})} to denote the Borel sigma-algebra, if S is in the product sigma-algebra {\mathcal F\otimes\mathcal B({\mathbb R})} then {\pi_\Omega(S)\in\mathcal F} and there is a measurable map

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

We consider measurable functions {\tau\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\}}. The graph of {\tau} is

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

The condition that {(\omega,\tau(\omega))\in S} whenever {\tau < \infty} can then 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 proof of the measurable section theorem will make use of the properties of analytic sets and of the Choquet capacitability theorem, as described in the previous two posts. [Note: I have since posted a more direct proof which does not involve such prerequisites.] Recall that a paving {\mathcal E} on a set X denotes, simply, a collection of subsets of X. The pair {(X,\mathcal E)} is then referred to as a paved space. Given a pair of paved spaces {(X,\mathcal E)} and {(Y,\mathcal F)}, the product paving {\mathcal E\times\mathcal F} denotes the collection of cartesian products {A\times B} for {A\in\mathcal E} and {B\in\mathcal F}, which is a paving on {X\times Y}. The notation {\mathcal E_\delta} is used for the collection of countable intersections of a paving {\mathcal E}.

We start by showing that measurable section holds in a very simple case where, for the section of a set S, its debut will suffice. The debut is the map

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

We use the convention that the infimum of the empty set is {\infty}. It is not clear that {D(S)} is measurable, and we do not rely on this, although measurable projection can be used to show that it is measurable whenever S is in {\mathcal F\otimes\mathcal B({\mathbb R})}.

Lemma 1 Let {(\Omega,\mathcal F)} be a measurable space, {\mathcal K} be the collection of compact intervals in {{\mathbb R}}, and {\mathcal E} be the closure of the paving {\mathcal{F\times K}} under finite unions.

Then, the debut {D(S)} of any {S\in\mathcal E_\delta} is measurable and its graph {[D(S)]} is contained in S.

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