Standard probability space explained

In probability theory, a standard probability space, also called Lebesgue - Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.

The theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940. Rokhlin showed that the unit interval endowed with the Lebesgue measure has important advantages over general probability spaces, yet can be effectively substituted for many of these in probability theory. The dimension of the unit interval is not an obstacle, as was clear already to Norbert Wiener. He constructed the Wiener process (also called Brownian motion) in the form of a measurable map from the unit interval to the space of continuous functions.

Short history

The theory of standard probability spaces was started by von Neumann in 1932^[1] and shaped by Vladimir Rokhlin in 1940.^[2] For modernized presentations see,, and .

Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example . This approach is based on the isomorphism theorem for standard Borel spaces . An alternate approach of Rokhlin, based on measure theory, neglects null sets, in contrast to descriptive set theory.Standard probability spaces are used routinely in ergodic theory.^{[3] [4]}

Definition

One of several well-known equivalent definitions of the standardness is given below, after some preparations. All probability spaces are assumed to be complete.

Isomorphism

An isomorphism between two probability spaces

style (\Omega _{1,l{F}} _{1,P} _{1)} , style (\Omega _{2,l{F}} _{2,P} _{2)} is an invertible map style f : \Omega _{1} \to \Omega _{2} such that style f and style f ^{-1} both are (measurable and) measure preserving maps.

Two probability spaces are isomorphic if there exists an isomorphism between them.

Isomorphism modulo zero

Two probability spaces

style (\Omega _{1,l{F}} _{1,P} _{1)} , style (\Omega _{2,l{F}} _{2,P} _{2)} are isomorphic style \operatorname{mod} 0 if there exist null sets style A _{1} \subset \Omega _{1} , style A _{2} \subset \Omega _{2} such that the probability spaces style \Omega _{1} \setminus A _{1} , style \Omega _{2} \setminus A _{2} are isomorphic (being endowed naturally with sigma-fields and probability measures).

Standard probability space

A probability space is standard, if it is isomorphic

style \operatorname{mod} 0 to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination (disjoint union) of both.

See,, and . See also, and . In the measure is assumed finite, not necessarily probabilistic. In atoms are not allowed.

Examples of non-standard probability spaces

A naive white noise

The space of all functions

style f : R \to R may be thought of as the product style R ^{R} of a continuum of copies of the real line style R . One may endow style R with a probability measure, say, the standard normal distribution style \gamma = N( 0 , 1 ) , and treat the space of functions as the product style ( R ,\gamma) ^{R} of a continuum of identical probability spaces style ( R ,\gamma) . The product measure style \gamma ^{R} is a probability measure on style R ^{R} . Naively it might seem that style \gamma ^{R} describes white noise.

However, the integral of a white noise function from 0 to 1 should be a random variable distributed N(0, 1). In contrast, the integral (from 0 to 1) of

style f \in style ( R ,\gamma) ^{R} is undefined. ƒ also fails to be almost surely measurable, and the probability of ƒ being measurable is undefined. Indeed, if X is a random variable distributed (say) uniformly on (0, 1) and independent of ƒ, then ƒ(X) is not a random variable at all (it lacks measurability).

A perforated interval

Let

style Z \subset ( 0 , 1 ) be a set whose inner Lebesgue measure is equal to 0, but outer Lebesgue measure is equal to 1 (thus, style Z is nonmeasurable to extreme). There exists a probability measure style m on style Z such that style m(Z \cap A) = \operatorname{mes} (A) for every Lebesgue measurable style A \subset ( 0 , 1 ) . (Here style \operatorname{mes} is the Lebesgue measure.) Events and random variables on the probability space style (Z,m) (treated style \operatorname{mod} 0 ) are in a natural one-to-one correspondence with events and random variables on the probability space style (( 0 , 1 ),\operatorname{mes}) . It might seem that the probability space style (Z,m) is as good as style (( 0 , 1 ),\operatorname{mes}) .

However, it is not. A random variable

style X defined by style X(\omega) = \omega is distributed uniformly on style ( 0 , 1 ) . The conditional measure, given style X = x , is just a single atom (at style x ), provided that style (( 0 , 1 ),\operatorname{mes}) is the underlying probability space. However, if style (Z,m) is used instead, then the conditional measure does not exist when style x \notin Z .

A perforated circle is constructed similarly. Its events and random variables are the same as on the usual circle. The group of rotations acts on them naturally. However, it fails to act on the perforated circle.

See also .

A superfluous measurable set

Let

style Z \subset ( 0 , 1 ) be as in the previous example. Sets of the form style ( A \cap Z ) \cup ( B \setminus Z ), where style A and style B are arbitrary Lebesgue measurable sets, are a σ-algebra style l{F}; it contains the Lebesgue σ-algebra and style Z. The formula\displaystyle m ( ( A \cap Z ) \cup ( B \setminus Z ) ) = p \operatorname{mes} (A) + ( 1- p) \operatorname{mes} (B) gives the general form of a probability measure style m on style ( ( 0 , 1 ), l{F} ) that extends the Lebesgue measure; here style p \in [ 0 , 1 ] is a parameter. To be specific, we choose style p = 0 . 5 . It might seem that such an extension of the Lebesgue measure is at least harmless.

However, it is the perforated interval in disguise. The map

f(x) = \begin{cases} 0 . 5 x & for x \in Z, \\ 0 . 5 + 0 . 5 x & for x \in ( 0 , 1 ) \setminus Z \end{cases}

is an isomorphism between

style ( ( 0 , 1 ), l{F}, m ) and the perforated interval corresponding to the set\displaystyle Z _{1} = \{ 0 . 5 x : x \in Z \} \cup \{ 0 . 5 + 0 . 5 x : x \in ( 0 , 1 ) \setminus Z \} , another set of inner Lebesgue measure 0 but outer Lebesgue measure 1.

See also .

A criterion of standardness

Standardness of a given probability space

style (\Omega, l{F},P) is equivalent to a certain property of a measurable map style f from style (\Omega, l{F},P) to a measurable space style (X,\Sigma). The answer (standard, or not) does not depend on the choice of style (X,\Sigma) and style f . This fact is quite useful; one may adapt the choice of style (X,\Sigma) and style f to the given style (\Omega, l{F},P). No need to examine all cases. It may be convenient to examine a random variable style f : \Omega \to R , a random vector style f : \Omega \to R ^{n,} a random sequence style f : \Omega \to R ^{infty,} or a sequence of events style (A _{1,A} _{2,...)} treated as a sequence of two-valued random variables, style f : \Omega \to \{ 0 , 1 \} ^{infty.}

Two conditions will be imposed on

style f (to be injective, and generating). Below it is assumed that such style f is given. The question of its existence will be addressed afterwards.

The probability space

style (\Omega, l{F},P) is assumed to be complete (otherwise it cannot be standard).

A single random variable

A measurable function

style f : \Omega \to R induces a pushforward measure f _{*P} , – the probability measure style \mu on style R , defined by\displaystyle \mu(B) = (f _{*P)(B)} = P ( f ^{-1} (B) )    for Borel sets style B \subset R . i.e. the distribution of the random variable f . The image style f (\Omega) is always a set of full outer measure,\displaystyle \mu ^{*} ( f(\Omega) ) = inf _{B} \mu(B) = inf _{B} P(f ^{-1} (B)) = P(\Omega) = 1 , but its inner measure can differ (see a perforated interval). In other words, style f (\Omega) need not be a set of full measure style \mu.

A measurable function

style f : \Omega \to R is called generating if style l{F} is the completion with respect to P of the σ-algebra of inverse images style f ^{-1} (B), where style B \subset R runs over all Borel sets.

Caution.   The following condition is not sufficient for

style f to be generating: for every style A \in l{F} there exists a Borel set style B \subset R such that style P ( A n{\Delta} f ^{-1} (B) ) = 0 . (style \Delta means symmetric difference).

Theorem. Let a measurable function

style f : \Omega \to R be injective and generating, then the following two conditions are equivalent:\mu( style f (\Omega)) = 1 (i.e. the inner measure has also full measure, and the image style f (\Omega) is measureable with respect to the completion);(\Omega, l{F},P) is a standard probability space.

See also .

A random vector

The same theorem holds for any

R ^{n} (in place of R ). A measurable function f : \Omega \to R ^{n} may be thought of as a finite sequence of random variables X _{1,...,X} _{n} : \Omega \to R , and f is generating if and only if l{F} is the completion of the σ-algebra generated by X _{1,...,X} _{n.}

A random sequence

The theorem still holds for the space

R ^{infty} of infinite sequences. A measurable function f : \Omega \to R ^{infty} may be thought of as an infinite sequence of random variables X _{1,X} _{2,...} : \Omega \to R , and f is generating if and only if l{F} is the completion of the σ-algebra generated by X _{1,X} _{2,....}

A sequence of events

In particular, if the random variables

X _{n} take on only two values 0 and 1, we deal with a measurable function f : \Omega \to \{ 0 , 1 \} ^{infty} and a sequence of sets A _{1,A} _{2,\ldots} \in l{F}. The function f is generating if and only if l{F} is the completion of the σ-algebra generated by A _{1,A} _{2,....}

In the pioneering work sequences

A _{1,A} _{2,\ldots} that correspond to injective, generating f are called bases of the probability space (\Omega, l{F},P) (see). A basis is called complete mod 0, if f(\Omega) is of full measure \mu, see . In the same section Rokhlin proved that if a probability space is complete mod 0 with respect to some basis, then it is complete mod 0 with respect to every other basis, and defines Lebesgue spaces by this completeness property. See also and .

Additional remarks

The four cases treated above are mutually equivalent, and can be united, since the measurable spaces

R , R ^{n,} R ^{infty} and \{ 0 , 1 \} ^{infty} are mutually isomorphic; they all are standard measurable spaces (in other words, standard Borel spaces).

Existence of an injective measurable function from

style (\Omega, l{F},P) to a standard measurable space style (X,\Sigma) does not depend on the choice of style (X,\Sigma). Taking style (X,\Sigma) = \{ 0 , 1 \} ^{infty} we get the property well known as being countably separated (but called separable in).

Existence of a generating measurable function from

style (\Omega, l{F},P) to a standard measurable space style (X,\Sigma) also does not depend on the choice of style (X,\Sigma). Taking style (X,\Sigma) = \{ 0 , 1 \} ^{infty} we get the property well known as being countably generated (mod 0), see .

Probability space	Countably separated	Countably generated	Standard
	Interval with Lebesgue measure
	Naive white noise
	Perforated interval

Every injective measurable function from a standard probability space to a standard measurable space is generating. See,, . This property does not hold for the non-standard probability space dealt with in the subsection "A superfluous measurable set" above.

Caution.   The property of being countably generated is invariant under mod 0 isomorphisms, but the property of being countably separated is not. In fact, a standard probability space

style (\Omega, l{F},P) is countably separated if and only if the cardinality of style \Omega does not exceed continuum (see). A standard probability space may contain a null set of any cardinality, thus, it need not be countably separated. However, it always contains a countably separated subset of full measure.

Equivalent definitions

Let

style (\Omega, l{F},P) be a complete probability space such that the cardinality of style \Omega does not exceed continuum (the general case is reduced to this special case, see the caution above).

Via absolute measurability

Definition.  

style (\Omega, l{F},P) is standard if it is countably separated, countably generated, and absolutely measurable.

See and . "Absolutely measurable" means: measurable in every countably separated, countably generated probability space containing it.

Via perfectness

Definition.  

style (\Omega, l{F},P) is standard if it is countably separated and perfect.

See . "Perfect" means that for every measurable function from

style (\Omega, l{F},P) to R the image measure is regular. (Here the image measure is defined on all sets whose inverse images belong to style l{F} , irrespective of the Borel structure of R ).

Via topology

Definition.  

style (\Omega, l{F},P) is standard if there exists a topology style \tau on style \Omega such that

• the topological space

style (\Omega,\tau) is metrizable;style l{F} is the completion of the σ-algebra generated by style \tau (that is, by all open sets);

• for every

style \varepsilon > 0 there exists a compact set style K in style (\Omega,\tau) such that style P(K) \ge 1- \varepsilon.

See .

Verifying the standardness

Every probability distribution on the space

style R ^{n} turns it into a standard probability space. (Here, a probability distribution means a probability measure defined initially on the Borel sigma-algebra and completed.)

The same holds on every Polish space, see,,, and .

For example, the Wiener measure turns the Polish space

style C[ 0 , inf ty) (of all continuous functions style [ 0 , inf ty) \to R , endowed with the topology of local uniform convergence) into a standard probability space.

Another example: for every sequence of random variables, their joint distribution turns the Polish space

style R ^{infty} (of sequences; endowed with the product topology) into a standard probability space.

(Thus, the idea of dimension, very natural for topological spaces, is utterly inappropriate for standard probability spaces.)

The product of two standard probability spaces is a standard probability space.

The same holds for the product of countably many spaces, see,, and .

A measurable subset of a standard probability space is a standard probability space. It is assumed that the set is not a null set, and is endowed with the conditional measure. See and .

Every probability measure on a standard Borel space turns it into a standard probability space.

Using the standardness

Regular conditional probabilities

In the discrete setup, the conditional probability is another probability measure, and the conditional expectation may be treated as the (usual) expectation with respect to the conditional measure, see conditional expectation. In the non-discrete setup, conditioning is often treated indirectly, since the condition may have probability 0, see conditional expectation. As a result, a number of well-known facts have special 'conditional' counterparts. For example: linearity of the expectation; Jensen's inequality (see conditional expectation); Hölder's inequality; the monotone convergence theorem, etc.

Given a random variable

style Y on a probability space style (\Omega, l{F},P) , it is natural to try constructing a conditional measure style P _{y} , that is, the conditional distribution of style \omega \in \Omega given style Y(\omega) = y . In general this is impossible (see). However, for a standard probability space style (\Omega, l{F},P) this is possible, and well known as canonical system of measures (see), which is basically the same as conditional probability measures (see), disintegration of measure (see), and regular conditional probabilities (see).

The conditional Jensen's inequality is just the (usual) Jensen's inequality applied to the conditional measure. The same holds for many other facts.

Measure preserving transformations

Given two probability spaces

style (\Omega _{1,l{F}} _{1,P} _{1)} , style (\Omega _{2,l{F}} _{2,P} _{2)} and a measure preserving map style f : \Omega _{1} \to \Omega _{2} , the image style f(\Omega _{1)} need not cover the whole style \Omega _{2} , it may miss a null set. It may seem that style P _{2(f(\Omega} _{1))} has to be equal to 1, but it is not so. The outer measure of style f(\Omega _{1)} is equal to 1, but the inner measure may differ. However, if the probability spaces style (\Omega _{1,l{F}} _{1,P} _{1)} , style (\Omega _{2,l{F}} _{2,P} _{2)} are standard then style P _{2(f(\Omega} _{1))=1} , see . If style f is also one-to-one then every style A \in l{F} _{1} satisfies style f(A) \in l{F} _{2} , style P _{2(f(A))=P} _{1(A)} . Therefore, style f ^{-1} is measurable (and measure preserving). See and . See also .

"There is a coherent way to ignore the sets of measure 0 in a measure space" . Striving to get rid of null sets, mathematicians often use equivalence classes of measurable sets or functions. Equivalence classes of measurable subsets of a probability space form a normed complete Boolean algebra called the measure algebra (or metric structure). Every measure preserving map

style f : \Omega _{1} \to \Omega _{2} leads to a homomorphism style F of measure algebras; basically, style F(B) = f ^{-1} (B) for style B\in l{F} _{2} .

It may seem that every homomorphism of measure algebras has to correspond to some measure preserving map, but it is not so. However, for standard probability spaces each

style F corresponds to some style f . See,, .

Notes

1. and are cited in and .
2. Published in short in 1947, in detail in 1949 in Russian and in 1952 in English. An unpublished text of 1940 is mentioned in . "The theory of Lebesgue spaces in its present form was constructed by V. A. Rokhlin" .
3. "In this book we will deal exclusively with Lebesgue spaces" .
4. "Ergodic theory on Lebesgue spaces" is the subtitle of the book .

References

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