Almost Sure

6 October 19

The Monotone Class Theorem

Filed under: Probability Theory — George Lowther @ 11:00 AM
Tags: , , , ,

The monotone class theorem, and closely related {\pi}-system lemma, are simple but fundamental theorems in measure theory, and form an essential step in the proofs of many results. General measurable sets are difficult to describe explicitly so, when proving results in measure theory, it is often necessary to start by considering much simpler sets. The monotone class theorem is then used to extend to arbitrary measurable sets. For example, when proving a result about Borel subsets of {{\mathbb R}}, we may start by considering compact intervals and then apply the monotone class theorem. I include this post on the monotone class theorem for reference. (more…)

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24 February 19

Properties of the Dual Projections

In the previous post I introduced the definitions of the dual optional and predictable projections, firstly for processes of integrable variation and, then, generalised to processes which are only required to be locally (or prelocally) of integrable variation. We did not look at the properties of these dual projections beyond the fact that they exist and are uniquely defined, which are significant and important statements in their own right.

To recap, recall that an IV process, A, is right-continuous and such that its variation

\displaystyle  V_t\equiv \lvert A_0\rvert+\int_0^t\,\lvert dA\rvert (1)

is integrable at time {t=\infty}, so that {{\mathbb E}[V_\infty] < \infty}. The dual optional projection is defined for processes which are prelocally IV. That is, A has a dual optional projection {A^{\rm o}} if it is right-continuous and its variation process is prelocally integrable, so that there exist a sequence {\tau_n} of stopping times increasing to infinity with {1_{\{\tau_n > 0\}}V_{\tau_n-}} integrable. More generally, A is a raw FV process if it is right-continuous with almost-surely finite variation over finite time intervals, so {V_t < \infty} (a.s.) for all {t\in{\mathbb R}^+}. Then, if a jointly measurable process {\xi} is A-integrable on finite time intervals, we use

\displaystyle  \xi\cdot A_t\equiv\xi_0A_0+\int_0^t\xi\,dA

to denote the integral of {\xi} with respect to A over the interval {[0,t]}, which takes into account the value of {\xi} at time 0 (unlike the integral {\int_0^t\xi\,dA} which, implicitly, is defined on the interval {(0,t]}). In what follows, whenever we state that {\xi\cdot A} has any properties, such as being IV or prelocally IV, we are also including the statement that {\xi} is A-integrable so that {\xi\cdot A} is a well-defined process. Also, whenever we state that a process has a dual optional projection, then we are also implicitly stating that it is prelocally IV.

From theorem 3 of the previous post, the dual optional projection {A^{\rm o}} is the unique prelocally IV process satisfying

\displaystyle  {\mathbb E}[\xi\cdot A^{\rm o}_\infty]={\mathbb E}[{}^{\rm o}\xi\cdot A_\infty]

for all measurable processes {\xi} with optional projection {{}^{\rm o}\xi} such that {\xi\cdot A^{\rm o}} and {{}^{\rm o}\xi\cdot A} are IV. Equivalently, {A^{\rm o}} is the unique optional FV process such that

\displaystyle  {\mathbb E}[\xi\cdot A^{\rm o}_\infty]={\mathbb E}[\xi\cdot A_\infty]

for all optional {\xi} such that {\xi\cdot A} is IV, in which case {\xi\cdot A^{\rm o}} is also IV so that the expectations in this identity are well-defined.

I now look at the elementary properties of dual optional projections, as well as the corresponding properties of dual predictable projections. The most important property is that, according to the definition just stated, the dual projection exists and is uniquely defined. By comparison, the properties considered in this post are elementary and relatively easy to prove. So, I will simply state a theorem consisting of a list of all the properties under consideration, and will then run through their proofs. Starting with the dual optional projection, the main properties are listed below as Theorem 1.

Note that the first three statements are saying that the dual projection is indeed a linear projection from the prelocally IV processes onto the linear subspace of optional FV processes. As explained in the previous post, by comparison with the discrete-time setting, the dual optional projection can be expressed, in a non-rigorous sense, as taking the optional projection of the infinitesimal increments,

\displaystyle  dA^{\rm o}={}^{\rm o}dA. (2)

As {dA} is interpreted via the Lebesgue-Stieltjes integral {\int\cdot\,dA}, it is a random measure rather than a real-valued process. So, the optional projection of {dA} appearing in (2) does not really make sense. However, Theorem 1 does allow us to make sense of (2) in certain restricted cases. For example, if A is differentiable so that {dA=\xi\,dt} for a process {\xi}, then (9) below gives {dA={}^{\rm o}\xi\,dt}. This agrees with (2) so long as {{}^{\rm o}(\xi\,dt)} is interpreted to mean {{}^{\rm o}\xi\,dt}. Also, restricting to the jump component of the increments, {\Delta A=A-A_-}, (2) reduces to (11) below.

We defined the dual projection via expectations of integrals {\xi\cdot A} with the restriction that this is IV. An alternative approach is to first define the dual projections for IV processes, as was done in theorems 1 and 2 of the previous post, and then extend to (pre)locally IV processes by localisation of the projection. That this is consistent with our definitions follows from the fact that (pre)localisation commutes with the dual projection, as stated in (10) below.

Theorem 1

  1. A raw FV process A is optional if and only if {A^{\rm o}} exists and is equal to A.
  2. If the dual optional projection of A exists then,
    \displaystyle  (A^{\rm o})^{\rm o}=A^{\rm o}. (3)
  3. If the dual optional projections of A and B exist, and {\lambda}, {\mu} are {\mathcal F_0}-measurable random variables then,
    \displaystyle  (\lambda A+\mu B)^{\rm o}=\lambda A^{\rm o}+\mu B^{\rm o}. (4)
  4. If the dual optional projection {A^{\rm o}} exists then {{\mathbb E}[\lvert A_0\rvert\,\vert\mathcal F_0]} is almost-surely finite and
    \displaystyle  A^{\rm o}_0={\mathbb E}[A_0\,\vert\mathcal F_0]. (5)
  5. If U is a random variable and {\tau} is a stopping time, then {U1_{[\tau,\infty)}} is prelocally IV if and only if {{\mathbb E}[1_{\{\tau < \infty\}}\lvert U\rvert\,\vert\mathcal F_\tau]} is almost surely finite, in which case
    \displaystyle  \left(U1_{[\tau,\infty)}\right)^{\rm o}={\mathbb E}[1_{\{\tau < \infty\}}U\,\vert\mathcal F_\tau]1_{[\tau,\infty)}. (6)
  6. If the prelocally IV process A is nonnegative and increasing then so is {A^{\rm o}} and,
    \displaystyle  {\mathbb E}[\xi\cdot A^{\rm o}_\infty]={\mathbb E}[{}^{\rm o}\xi\cdot A_\infty] (7)

    for all nonnegative measurable {\xi} with optional projection {{}^{\rm o}\xi}. If A is merely increasing then so is {A^{\rm o}} and (7) holds for nonnegative measurable {\xi} with {\xi_0=0}.

  7. If A has dual optional projection {A^{\rm o}} and {\xi} is an optional process such that {\xi\cdot A} is prelocally IV then, {\xi} is {A^{\rm o}}-integrable and,
    \displaystyle  (\xi\cdot A)^{\rm o}=\xi\cdot A^{\rm o}. (8)
  8. If A is an optional FV process and {\xi} is a measurable process with optional projection {{}^{\rm o}\xi} such that {\xi\cdot A} is prelocally IV then, {{}^{\rm o}\xi} is A-integrable and,
    \displaystyle  (\xi\cdot A)^{\rm o}={}^{\rm o}\xi\cdot A. (9)
  9. If A has dual optional projection {A^{\rm o}} and {\tau} is a stopping time then,
    \displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle(A^{\tau})^{\rm o}=(A^{\rm o})^{\tau},\smallskip\\ &\displaystyle(A^{\tau-})^{\rm o}=(A^{\rm o})^{\tau-}. \end{array} (10)
  10. If the dual optional projection {A^{\rm o}} exists, then its jump process is the optional projection of the jump process of A,
    \displaystyle  \Delta A^{\rm o}={}^{\rm o}\!\Delta A. (11)
  11. If A has dual optional projection {A^{\rm o}} then
    \displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle{\mathbb E}\left[\xi_0\lvert A^{\rm o}_0\rvert + \int_0^\infty\xi\,\lvert dA^{\rm o}\rvert\right]\le{\mathbb E}\left[{}^{\rm o}\xi_0\lvert A_0\rvert + \int_0^\infty{}^{\rm o}\xi\,\lvert dA\rvert\right],\smallskip\\ &\displaystyle{\mathbb E}\left[\xi_0(A^{\rm o}_0)_+ + \int_0^\infty\xi\,(dA^{\rm o})_+\right]\le{\mathbb E}\left[{}^{\rm o}\xi_0(A_0)_+ + \int_0^\infty{}^{\rm o}\xi\,(dA)_+\right],\smallskip\\ &\displaystyle{\mathbb E}\left[\xi_0(A^{\rm o}_0)_- + \int_0^\infty\xi\,(dA^{\rm o})_-\right]\le{\mathbb E}\left[{}^{\rm o}\xi_0(A_0)_- + \int_0^\infty{}^{\rm o}\xi\,(dA)_-\right], \end{array} (12)

    for all nonnegative measurable {\xi} with optional projection {{}^{\rm o}\xi}.

  12. Let {\{A^n\}_{n=1,2,\ldots}} be a sequence of right-continuous processes with variation

    \displaystyle  V^n_t=\lvert A^n_0\rvert + \int_0^t\lvert dA^n\rvert.

    If {\sum_n V^n} is prelocally IV then,

    \displaystyle  \left(\sum\nolimits_n A^n\right)^{\rm o}=\sum\nolimits_n\left(A^n\right)^{\rm o}. (13)

(more…)

8 February 19

Dual Projections

The optional and predictable projections of stochastic processes have corresponding dual projections, which are the subject of this post. I will be concerned with their initial construction here, and show that they are well-defined. The study of their properties will be left until later. In the discrete time setting, the dual projections are relatively straightforward, and can be constructed by applying the optional and predictable projection to the increments of the process. In continuous time, we no longer have discrete time increments along which we can define the dual projections. In some sense, they can still be thought of as projections of the infinitesimal increments so that, for a process A, the increments of the dual projections {A^{\rm o}} and {A^{\rm p}} are determined from the increments {dA} of A as

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dA^{\rm o}={}^{\rm o}(dA),\smallskip\\ &\displaystyle dA^{\rm p}={}^{\rm p}(dA). \end{array} (1)

Unfortunately, these expressions are difficult to make sense of in general. In specific cases, (1) can be interpreted in a simple way. For example, when A is differentiable with derivative {\xi}, so that {dA=\xi dt}, then the dual projections are given by {dA^{\rm o}={}^{\rm o}\xi dt} and {dA^{\rm p}={}^{\rm p}\xi dt}. More generally, if A is right-continuous with finite variation, then the infinitesimal increments {dA} can be interpreted in terms of Lebesgue-Stieltjes integrals. However, as the optional and predictable projections are defined for real valued processes, and {dA} is viewed as a stochastic measure, the right-hand-side of (1) is still problematic. This can be rectified by multiplying by an arbitrary process {\xi}, and making use of the transitivity property {{\mathbb E}[\xi\,{}^{\rm o}(dA)]={\mathbb E}[({}^{\rm o}\xi)dA]}. Integrating over time gives the more meaningful expressions

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle {\mathbb E}\left[\int_0^\infty \xi\,dA^{\rm o}\right]={\mathbb E}\left[\int_0^\infty{}^{\rm o}\xi\,dA\right],\smallskip\\ &\displaystyle{\mathbb E}\left[\int_0^\infty \xi\,dA^{\rm p}\right]={\mathbb E}\left[\int_0^\infty{}^{\rm p}\xi\,dA\right]. \end{array}

In contrast to (1), these equalities can be used to give mathematically rigorous definitions of the dual projections. As usual, we work with respect to a complete filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}, and processes are identified whenever they are equal up to evanescence. The terminology `raw IV process‘ will be used to refer to any right-continuous integrable process whose variation on the whole of {{\mathbb R}^+} has finite expectation. The use of the word `raw’ here is just to signify that we are not requiring the process to be adapted. Next, to simplify the expressions, I will use the notation {\xi\cdot A} for the integral of a process {\xi} with respect to another process A,

\displaystyle  \xi\cdot A_t\equiv\xi_0A_0+\int_0^t\xi\,dA.

Note that, whereas the integral {\int_0^t\xi\,dA} is implicitly taken over the range {(0,t]} and does not involve the time-zero value of {\xi}, I have included the time-zero values of the processes in the definition of {\xi\cdot A}. This is not essential, and could be excluded, so long as we were to restrict to processes starting from zero. The existence and uniqueness (up to evanescence) of the dual projections is given by the following result.

Theorem 1 (Dual Projections) Let A be a raw IV process. Then,

  • There exists a unique raw IV process {A^{\rm o}} satisfying
    \displaystyle  {\mathbb E}\left[\xi\cdot A^{\rm o}_\infty\right]={\mathbb E}\left[{}^{\rm o}\xi\cdot A_\infty\right] (2)

    for all bounded measurable processes {\xi}. We refer to {A^{\rm o}} as the dual optional projection of A.

  • There exists a unique raw IV process {A^{\rm p}} satisfying
    \displaystyle  {\mathbb E}\left[\xi\cdot A^{\rm p}_\infty\right]={\mathbb E}\left[{}^{\rm p}\xi\cdot A_\infty\right] (3)

    for all bounded measurable processes {\xi}. We refer to {A^{\rm p}} as the dual predictable projection of A.

Furthermore, if A is nonnegative and increasing then so are {A^{\rm o}} and {A^{\rm p}}.

(more…)

21 January 19

Pathwise Properties of Optional and Predictable Projections

Recall that the the optional and predictable projections of a process are defined, firstly, by a measurability property and, secondly, by their values at stopping times. Namely, the optional projection is measurable with respect to the optional sigma-algebra, and its value is defined at each stopping time by a conditional expectation of the original process. Similarly, the predictable projection is measurable with respect to the predictable sigma-algebra and its value at each predictable stopping time is given by a conditional expectation. While these definitions can be powerful, and many properties of the projections follow immediately, they say very little about the sample paths. Given a stochastic process X defined on a filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})} with optional projection {{}^{\rm o}\!X} then, for each {\omega\in\Omega}, we may be interested in the sample path {t\mapsto{}^{\rm o}\!X_t(\omega)}. For example, is it continuous, right-continuous, cadlag, etc? Answering these questions requires looking at {{}^{\rm o}\!X_t(\omega)} simultaneously at the uncountable set of times {t\in{\mathbb R}^+}, so the definition of the projection given by specifying its values at each individual stopping time, up to almost-sure equivalence, is not easy to work with. I did establish some of the basic properties of the projections in the previous post, but these do not say much regarding sample paths.

I will now establish the basic properties of the sample paths of the projections. Although these results are quite advanced, most of the work has already been done in these notes when we established some pathwise properties of optional and predictable processes in terms of their behaviour along sequences of stopping times, and of predictable stopping times. So, the proofs in this post are relatively simple and will consist of applications of these earlier results.

Before proceeding, let us consider what kind of properties it is reasonable to expect of the projections. Firstly, it does not seem reasonable to expect the optional projection {{}^{\rm o}\!X} or the predictable projection {{}^{\rm p}\!X} to satisfy properties not held by the original process X. Therefore, in this post, we will be concerned with the sample path properties which are preserved by the projections. Consider a process with constant paths. That is, {X_t=U} at all times t, for some bounded random variable U. This has about as simple sample paths as possible, so any properties preserved by the projections should hold for the optional and predictable projections of X. However, we know what the projections of this process are. Letting M be the martingale defined by {M_t={\mathbb E}[U\,\vert\mathcal F_t]} then, assuming that the underlying filtration is right-continuous, M has a cadlag modification and, furthermore, this modification is the optional projection of X. So, assuming that the filtration is right-continuous, the optional projection of X is cadlag, meaning that it is right-continuous and has left limits everywhere. So, we can hope that the optional projection preserves these properties. If the filtration is not right-continuous, then M need not have a cadlag modification, so we cannot expect optional projection to preserve right-continuity in this case. However, M does still have a version with left and right limits everywhere, which is the optional projection of X. So, without assuming right-continuity of the filtration, we may still hope that the optional projection preserves the existence of left and right limits of a process. Next, the predictable projection is equal to the left limits, {{}^{\rm p}\!X_t=M_{t-}}, which is left-continuous with left and right limits everywhere. Therefore, we can hope that predictable projections preserve left-continuity and the existence of left and right limits. The existence of cadlag martingales which are not continuous, such as the compensated Poisson process, imply that optional projections do not generally preserve left-continuity and the predictable projection does not preserve right-continuity.

Recall that I previously constructed a version of the optional projection and the predictable projection for processes which are, respectively, right-continuous and left-continuous. This was done by defining the projection at each deterministic time and, then, enforcing the respective properties of the sample paths. We can use the results in those posts to infer that the projections do indeed preserve these properties, although I will now more direct proofs in greater generality, and using the more general definition of the optional and predictable projections.

We work with respect to a complete filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}. As usual, we say that the sample paths of a process satisfy any stated property if they satisfy it up to evanescence. Since integrability conditions will be required, I mention those now. Recall that a process X is of class (D) if the set of random variables {X_\tau}, over stopping times {\tau}, is uniformly integrable. It will be said to be locally of class (D) if there is a sequence {\tau_n} of stopping times increasing to infinity and such that {1_{\{\tau_n > 0\}}1_{[0,\tau_n]}X} is of class (D) for each n. Similarly, it will be said to be prelocally of class (D) if there is a sequence {\tau_n} of stopping times increasing to infinity and such that {1_{[0,\tau_n)}X} is of class (D) for each n.

Theorem 1 Let X be pre-locally of class (D), with optional projection {{}^{\rm o}\!X}. Then,

  • if X has left limits, so does {{}^{\rm o}\!X}.
  • if X has right limits, so does {{}^{\rm o}\!X}.

Furthermore, if the underlying filtration is right-continuous then,

  • if X is right-continuous, so is {{}^{\rm o}\!X}.
  • if X is cadlag, so is {{}^{\rm o}\!X}.

(more…)

20 January 19

Properties of Optional and Predictable Projections

Having defined optional and predictable projections in an earlier post, I now look at their basic properties. The first nontrivial property is that they are well-defined in the first place. Recall that existence of the projections made use of the existence of cadlag modifications of martingales, and uniqueness relied on the section theorems. By contrast, once we accept that optional and predictable projections are well-defined, everything in this post follows easily. Nothing here requires any further advanced results of stochastic process theory.

Optional and predictable projections are similar in nature to conditional expectations. Given a probability space {(\Omega,\mathcal F,{\mathbb P})} and a sub-sigma-algebra {\mathcal G\subseteq\mathcal F}, the conditional expectation of an ({\mathcal F}-measurable) random variable X is a {\mathcal G}-measurable random variable {Y={\mathbb E}[X\,\vert\mathcal G]}. This is defined whenever the integrability condition {{\mathbb E}[\lvert X\rvert\,\vert\mathcal G] < \infty} (a.s.) is satisfied, only depends on X up to almost-sure equivalence, and Y is defined up to almost-sure equivalence. That is, a random variable {X^\prime} almost surely equal to X has the same conditional expectation as X. Similarly, a random variable {Y^\prime} almost-surely equal to Y is also a version of the conditional expectation {{\mathbb E}[X\,\vert\mathcal G]}.

The setup with projections of stochastic processes is similar. We start with a filtered probability space {(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge0},{\mathbb P})}, and a (real-valued) stochastic process is a map

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

which we assume to be jointly-measurable. That is, it is measurable with respect to the Borel sigma-algebra {\mathcal B({\mathbb R})} on the image, and the product sigma-algebra {\mathcal B({\mathbb R})\otimes\mathcal F} on the domain. The optional and predictable sigma-algebras are contained in the product,

\displaystyle  \mathcal P\subseteq\mathcal O\subseteq \mathcal B({\mathbb R})\otimes\mathcal F.

We do not have a reference measure on {({\mathbb R}^+\times\Omega,\mathcal B({\mathbb R})\otimes\mathcal F)} in order to define conditional expectations with respect to {\mathcal O} and {\mathcal P}. However, the optional projection {{}^{\rm o}\!X} and predictable projection {{}^{\rm p}\!X} play similar roles. Assuming that the necessary integrability properties are satisfied, then the projections exist. Furthermore, the projection only depends on the process X up to evanescence (i.e., up to a zero probability set), and {{}^{\rm o}\!X} and {{}^{\rm p}\!X} are uniquely defined up to evanescence.

In what follows, we work with respect to a complete filtered probability space. Processes are always only considered up to evanescence, so statements involving equalities, inequalities, and limits of processes are only required to hold outside of a zero probability set. When we say that the optional projection of a process exists, we mean that the integrability condition in the definition of the projection is satisfied. Specifically, that {{\mathbb E}[1_{\{\tau < \infty\}}\lvert X_\tau\rvert\,\vert\mathcal F_\tau]} is almost surely finite. Similarly for the predictable projection.

The following lemma gives a list of initial properties of the optional projection. Other than the statement involving stopping times, they all correspond to properties of conditional expectations.

Lemma 1

  1. X is optional if and only if {{}^{\rm o}\!X} exists and is equal to X.
  2. If the optional projection of X exists then,
    \displaystyle  {}^{\rm o}({}^{\rm o}\!X)={}^{\rm o}\!X. (1)
  3. If the optional projections of X and Y exist, and {\lambda,\mu} are {\mathcal{F}_0}-measurable random variables, then,
    \displaystyle  {}^{\rm o}(\lambda X+\mu Y) = \lambda\,^{\rm o}\!X + \mu\,^{\rm o}Y. (2)
  4. If the optional projection of X exists and U is an optional process then,
    \displaystyle  {}^{\rm o}(UX) = U\,^{\rm o}\!X (3)
  5. If the optional projection of X exists and {\tau} is a stopping time then, the optional projection of the stopped process {X^\tau} exists and,
    \displaystyle  1_{[0,\tau]}{}^{\rm o}(X^\tau)=1_{[0,\tau]}{}^{\rm o}\!X. (4)
  6. If {X\le Y} and the optional projections of X and Y exist then, {{}^{\rm o}\!X\le{}^{\rm o}Y}.

(more…)

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)}.

[caption_id=”sectionpic” align=”aligncenter” width=”450″] measurable sectionFigure 1: A section of a measurable set[/caption] 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)}.

(more…)

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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6 January 19

Essential Suprema

Filed under: Probability Theory — George Lowther @ 2:00 PM
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Given a sequence {X_1,X_2,\ldots} of real-valued random variables defined on a probability space {(\Omega,\mathcal F,{\mathbb P})}, it is a standard result that the supremum

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle X\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\},\smallskip\\ &\displaystyle X(\omega)=\sup_nX_n(\omega). \end{array}

is measurable. To ensure that this is well-defined, we need to allow X to have values in {{\mathbb R}\cup\{\infty\}}, so that {X(\omega)=\infty} whenever the sequence {X_n(\omega)} is unbounded above. The proof of this fact is simple. We just need to show that {X^{-1}((-\infty,a])} is in {\mathcal F} for all {a\in{\mathbb R}}. Writing,

\displaystyle  X^{-1}((-\infty,a])=\bigcap_nX_n^{-1}((-\infty,a]),

the properties that {X_n} are measurable and that the sigma-algebra {\mathcal F} is closed under countable intersections gives the result.

The measurability of the suprema of sequences of random variables is a vital property, used throughout probability theory. However, once we start looking at uncountable collections of random variables things get more complicated. Given a, possibly uncountable, collection of random variables {\mathcal S}, the supremum {S=\sup\mathcal S} is,

\displaystyle  S(\omega)=\sup\left\{X(\omega)\colon X\in\mathcal S\right\}. (1)

However, there are a couple of reasons why this is often not a useful construction:

  • The supremum need not be measurable. For example, consider the probability space {\Omega=[0,1]} with {\mathcal F} the collection of Borel or Lebesgue subsets of {\Omega}, and {{\mathbb P}} the standard Lebesgue measure. For any {a\in[0,1]} define the random variable {X_a(\omega)=1_{\{\omega=a\}}} and, for a subset A of {[0,1]}, consider the collection of random variables {\mathcal S=\{X_a\colon a\in A\}}. Its supremum is

    \displaystyle  S(\omega)=1_{\{\omega\in A\}}

    which is not measurable if A is a non-measurable set (e.g., a Vitali set).

  • Even if the supremum is measurable, it might not be a useful quantity. Letting {X_a} be the random variables on {(\Omega,\mathcal F,{\mathbb P})} constructed above, consider {\mathcal S=\{X_a\colon a\in[0,1]\}}. Its supremum is the constant function {S=1}. As every {X\in\mathcal S} is almost surely equal to 0, it is almost surely bounded above by the constant function {Y=0}. So, the supremum {S=1} is larger than we may expect, and is not what we want in many cases.

The essential supremum can be used to correct these deficiencies, and has been important in several places in my notes. See, for example, the proof of the debut theorem for right-continuous processes. So, I am posting this to use as a reference. Note that there is an alternative use of the term `essential supremum’ to refer to the smallest real number almost surely bounding a specified random variable, which is the one referred to by Wikipedia. This is different from the use here, where we look at a collection of random variables and the essential supremum is itself a random variable.

The essential supremum is really just the supremum taken within the equivalence classes of random variables under the almost sure ordering. Consider the equivalence relation {X\cong Y} if and only if {X=Y} almost surely. Writing {[X]} for the equivalence class of X, we can consider the ordering given by {[X]\le[Y]} if {X\le Y} almost surely. Then, the equivalence class of the essential supremum of a collection {\mathcal S} of random variables is the supremum of the equivalence classes of the elements of {\mathcal S}. In order to avoid issues with unbounded sets, we consider random variables taking values in the extended reals {\bar{\mathbb R}={\mathbb R}\cup\{\pm\infty\}}.

Definition 1 An essential supremum of a collection {\mathcal S} of {\bar{\mathbb R}}-valued random variables,

\displaystyle  S = {\rm ess\,sup\,}\mathcal{S}

is the least upper bound of {\mathcal{S}}, using the almost-sure ordering on random variables. That is, S is an {\bar{\mathbb R}}-valued random variable satisfying

  • upper bound: {S\ge X} almost surely, for all {X\in\mathcal S}.
  • minimality: for all {\bar{\mathbb R}}-valued random variables Y satisfying {Y\ge X} almost surely for all {X\in\mathcal S}, we have {Y\ge 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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1 January 19

Choquet’s Capacitability Theorem and Measurable Projection

In this post I will give a proof of the measurable projection theorem. Recall that this states that for a complete probability space {(\Omega,\mathcal F,{\mathbb P})} and a set S in the product sigma-algebra {\mathcal F\otimes\mathcal B({\mathbb R})}, the projection, {\pi_\Omega(S)}, of S onto {\Omega}, is in {\mathcal F}. The previous post on analytic sets made some progress towards this result. Indeed, using the definitions and results given there, it follows quickly that {\pi_\Omega(S)} is {\mathcal F}-analytic. To complete the proof of measurable projection, it is necessary to show that analytic sets are measurable. This is a consequence of Choquet’s capacitability theorem, which I will prove in this post. Measurable projection follows as a simple consequence.

The condition that the underlying probability space is complete is necessary and, if this condition was dropped, then the result would no longer hold. Recall that, if {(\Omega,\mathcal F,{\mathbb P})} is a probability space, then the completion, {\mathcal F_{\mathbb P}}, of {\mathcal F} with respect to {{\mathbb P}} consists of the sets {A\subseteq\Omega} such that there exists {B,C\in\mathcal F} with {B\subseteq A\subseteq C} and {{\mathbb P}(B)={\mathbb P}(C)}. The probability space is complete if {\mathcal F_{\mathbb P}=\mathcal F}. More generally, {{\mathbb P}} can be uniquely extended to a measure {\bar{\mathbb P}} on the sigma-algebra {\mathcal F_{\mathbb P}} by setting {\bar{\mathbb P}(A)={\mathbb P}(B)={\mathbb P}(C)}, where B and C are as above. Then {(\Omega,\mathcal F_{\mathbb P},\bar{\mathbb P})} is the completion of {(\Omega,\mathcal F,{\mathbb P})}.

In measurable projection, then, it needs to be shown that if {A\subseteq\Omega} is the projection of a set in {\mathcal F\otimes\mathcal B({\mathbb R})}, then A is in the completion of {\mathcal F}. That is, we need to find sets {B,C\in\mathcal F} with {B\subseteq A\subseteq C} with {{\mathbb P}(B)={\mathbb P}(C)}. In fact, it is always possible to find a {C\supseteq A} in {\mathcal F} which minimises {{\mathbb P}(C)}, and its measure is referred to as the outer measure of A. For any probability measure {{\mathbb P}}, we can define an outer measure on the subsets of {\Omega}, {{\mathbb P}^*\colon\mathcal P(\Omega)\rightarrow{\mathbb R}^+} by approximating {A\subseteq\Omega} from above,

\displaystyle  {\mathbb P}^*(A)\equiv\inf\left\{{\mathbb P}(B)\colon B\in\mathcal F, A\subseteq B\right\}. (1)

Similarly, we can define an inner measure by approximating A from below,

\displaystyle  {\mathbb P}_*(A)\equiv\sup\left\{{\mathbb P}(B)\colon B\in\mathcal F, B\subseteq A\right\}.

It can be shown that A is {\mathcal F}-measurable if and only if {{\mathbb P}_*(A)={\mathbb P}^*(A)}. We will be concerned primarily with the outer measure {{\mathbb P}^*}, and will show that that if A is the projection of some {S\in\mathcal F\otimes\mathcal B({\mathbb R})}, then A can be approximated from below in the following sense: there exists {B\subseteq A} in {\mathcal F} for which {{\mathbb P}^*(B)={\mathbb P}^*(A)}. From this, it will follow that A is in the completion of {\mathcal F}.

It is convenient to prove the capacitability theorem in slightly greater generality than just for the outer measure {{\mathbb P}^*}. The only properties of {{\mathbb P}^*} that are required is that it is a capacity, which we now define. Recall that a paving {\mathcal E} on a set X is simply any collection of subsets of X, and we refer to the pair {(X,\mathcal E)} as a paved space.

Definition 1 Let {(X,\mathcal E)} be a paved space. Then, an {\mathcal E}-capacity is a map {I\colon\mathcal P(X)\rightarrow{\mathbb R}} which is increasing, continuous along increasing sequences, and continuous along decreasing sequences in {\mathcal E}. That is,

  • if {A\subseteq B} then {I(A)\le I(B)}.
  • if {A_n\subseteq X} is increasing in n then {I(A_n)\rightarrow I(\bigcup_nA_n)} as {n\rightarrow\infty}.
  • if {A_n\in\mathcal E} is decreasing in n then {I(A_n)\rightarrow I(\bigcap_nA_n)} as {n\rightarrow\infty}.

As was claimed above, the outer measure {{\mathbb P}^*} defined by (1) is indeed a capacity.

Lemma 2 Let {(\Omega,\mathcal F,{\mathbb P})} be a probability space. Then,

  • {{\mathbb P}^*(A)={\mathbb P}(A)} for all {A\in\mathcal F}.
  • For all {A\subseteq\Omega}, there exists a {B\in\mathcal F} with {A\subseteq B} and {{\mathbb P}^*(A)={\mathbb P}(B)}.
  • {{\mathbb P}^*} is an {\mathcal F}-capacity.

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