Standard probability space

In probability theory, a standard probability space (called also Lebesgue-Rokhlin probability space) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940 [1] . He showed that the unit interval endowed with the Lebesgue measure has important advantages over general probability spaces, and can be used as a probability space for all practical purposes in probability theory. The dimension of the unit interval is not a concern, which 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

Vladimir Rokhlin created the theory of standard probability spaces in 1940 (see [1] , p.2); published in short in 1947, in detail in 1949 in Russian and in 1952 in English, reprinted in 1962 [1] . For modernized presentations see [2] , [3] , and Sect. 2.4 of [4] .

Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example Sect. 17 of [5] . This approach, natural for experts in descriptive set theory, is based on the isomorphism theorem for standard Borel spaces (see [5] , Theorem (15.6)) whose proof is very difficult for non-experts in descriptive set theory. The original approach of Rokhlin, based on measure theory, leads to much simpler proofs (since measure theory may neglect null sets, in contrast to descriptive set theory).

Standard probability spaces are used routinely in ergodic theory, which cannot be said on probability theory. Some probabilists hold the following opinion: only standard probability spaces are pertinent to probability theory, thus, it is a pity that the standardness is not included into the definition of probability space. Others disagree, however.

Arguments against standardness:
* the definition of standardness is technically demanding;
* the same about the theorems based on that definition;
* it is possible (and natural) to build all the probability theory without the standardness;
* events and random variables are essential, while probability spaces are auxiliary and should not be taken too seriously.

Arguments in favour of standardness:
* conditioning is easy and natural on standard probability spaces, otherwise it becomes obscure;
* the same for measure-preserving transformations between probability spaces, group actions on a probability space, etc.;
* ergodic theory uses standard probability spaces routinely and successfully;
* being unable to eliminate these (auxiliary) probability spaces, we should make them as useful as possible.

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 $extstyle\; (Omega\_1,mathcal\{F\}\_1,P\_1)$, $extstyle\; (Omega\_2,mathcal\{F\}\_2,P\_2)$ is an invertible map $extstyle\; f\; :\; Omega\_1\; o\; Omega\_2$ such that $extstyle\; f$ and $extstyle\; f^\{-1\}$ 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 $extstyle\; (Omega\_1,mathcal\{F\}\_1,P\_1)$, $extstyle\; (Omega\_2,mathcal\{F\}\_2,P\_2)$ are isomorphic $extstyle\; operatorname\{mod\}\; ,\; 0$, if there exist null sets $extstyle\; A\_1\; subset\; Omega\_1$, $extstyle\; A\_2\; subset\; Omega\_2$ such that the probability spaces $extstyle\; Omega\_1\; setminus\; A\_1$, $extstyle\; Omega\_2\; setminus\; A\_2$ are isomorphic (being endowed naturally with sigma-fields and probability measures).

Standard probability space

A probability space is standard, if it is isomorphic $extstyle\; operatorname\{mod\}\; ,\; 0$ to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination (disjoint union) of both.

See Sect. 2.4 (p. 20) of [1] ; Proposition 6 (p. 249) and Remark 2 (p. 250) in [2] ; and Theorem 4-3 in [3] . See also Sect. 17.F of [5] , and [4] (especially Sect. 2.4 and Exercise 3.1(v)).

Examples of non-standard probability spaces

A naive white noise

The space of all functions $extstyle\; f\; :\; mathbb\{R\}\; o\; mathbb\{R\}$ may be thought of as the product $extstyle\; mathbb\{R\}^mathbb\{R\}$ of a continuum of copies of the real line $extstyle\; mathbb\{R\}$. One may endow $extstyle\; mathbb\{R\}$ with a probability measure, say, the standard normal distribution $extstyle\; gamma\; =\; N(0,1)$, and treat the space of functions as the product $extstyle\; (mathbb\{R\},gamma)^mathbb\{R\}$ of a continuum of identical probability spaces $extstyle\; (mathbb\{R\},gamma)$. The product measure $extstyle\; gamma^mathbb\{R\}$ is a probability measure on $extstyle\; mathbb\{R\}^mathbb\{R\}$. Many non-experts are inclined to believe that $extstyle\; gamma^mathbb\{R\}$ describes the so-called white noise.

However, it does not. For the white noise, its integral from $0$ to $1$ should be a random variable distributed $extstyle\; N(0,1)$. In contrast, the integral (from $0$ to $1$) of $extstyle\; f\; in\; extstyle\; (mathbb\{R\},gamma)^mathbb\{R\}$ is undefined. Even worse, $extstyle\; f$ fails to be almost surely measurable. Still worse, the probability of $extstyle\; f$ being measurable is undefined. And the worst thing: if $extstyle\; X$ is a random variable distributed (say) uniformly on $extstyle\; (0,1)$ and independent of $extstyle\; f$, then $extstyle\; f(X)$ is not a random variable at all! (It lacks measurability.)

A perforated interval

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

However, it is not. A random variable $extstyle\; X$ defined by $extstyle\; X(omega)=omega$ is distributed uniformly on $extstyle\; (0,1)$. The conditional measure, given $extstyle\; X=x$, is just a single atom (at $extstyle\; x$), provided that $extstyle\; ((0,1),\; ext\{mes\})$ is the underlying probability space. However, if $extstyle\; (Z,m)$ is used instead, then the conditional measure does not exist when $extstyle\; x\; otin\; Z$.

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.

A superfluous measurable set

Let $extstyle\; Z\; subset\; (0,1)$ be as in the previous example. Sets of the form $extstyle\; (\; A\; cap\; Z\; )\; cup\; (\; B\; setminus\; Z\; ),$ where $extstyle\; A$ and $extstyle\; B$ are arbitrary Lebesgue measurable sets, are a σ-algebra $extstyle\; mathcal\{F\};$ it contains the Lebesgue σ-algebra and $extstyle\; Z.$ The formula: gives the general form of a probability measure $extstyle\; m$ on that extends the Lebesgue measure; here $extstyle\; p\; in\; [0,1]$ is a parameter. To be specific, we choose $extstyle\; p\; =\; 0.5.$ Many non-experts are inclined to believe that such an extension of the Lebesgue measure is at least harmless.

However, it is the perforated interval in disguise. The
is an isomorphism between 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.

A criterion of standardness

Standardness of a given probability space $extstyle\; (Omega,mathcal\{F\},P)$ is equivalent to a certain property of a measurable map $extstyle\; f$ from $extstyle\; (Omega,mathcal\{F\},P)$ to a measurable space $extstyle\; (X,Sigma).$ Interestingly, the answer (standard, or not) does not depend on the choice of $extstyle\; (X,Sigma)$ and $extstyle\; f$. This fact is quite useful; one may adapt the choice of $extstyle\; (X,Sigma)$ and $extstyle\; f$ to the given $extstyle\; (Omega,mathcal\{F\},P).$ No need to examine all cases. It may be convenient to examine a random variable $extstyle\; f\; :\; Omega\; o\; mathbb\{R\},$ a random vector $extstyle\; f\; :\; Omega\; o\; mathbb\{R\}^n,$ a random sequence $extstyle\; f\; :\; Omega\; o\; mathbb\{R\}^infty,$ or a sequence of events $extstyle\; (A\_1,A\_2,dots)$ treates as a sequence of two-valued random variables, $extstyle\; f\; :\; Omega\; o\; \{0,1\}^infty.$

Two conditions will be imposed on $extstyle\; f$ (to be injective, and generating). Below it is assumed that such $extstyle\; f$ is given. The question of its existence will be addressed afterwards.

The probability space $extstyle\; (Omega,mathcal\{F\},P)$ is assumed to be complete (otherwise it cannot be standard).

A single random variable

A measurable function $extstyle\; f\; :\; Omega\; o\; mathbb\{R\}$ induces a pushforward measure, --- the probability measure $extstyle\; mu$ on $extstyle\; mathbb\{R\},$ defined by: for Borel sets $extstyle\; B\; subset\; mathbb\{R\}.$(It is nothing but the distribution of the random variable.) The image $extstyle\; f\; (Omega)$ is always a set of full outer measure,: but its inner measure can differ (see "a perforated interval"). In other words, $extstyle\; f\; (Omega)$ need not be a set of full measure $extstyle\; mu.$

A measurable function $extstyle\; f\; :\; Omega\; o\; mathbb\{R\}$ is called "generating" if $extstyle\; mathcal\{F\}$ is the completion of the σ-algebra of inverse images $extstyle\; f^\{-1\}(B),$ where $extstyle\; B\; subset\; mathbb\{R\}$ runs over all Borel sets.

"Caution." The following condition is not sufficient for $extstyle\; f$ to be generating: for every $extstyle\; A\; in\; mathcal\{F\}$ there exists a Borel set $extstyle\; B\; subset\; mathbb\{R\}$ such that $extstyle\; P\; (\; A\; Delta\; f^\{-1\}(B)\; )\; =\; 0.$ ($extstyle\; Delta$ means symmetric difference).

Theorem. Let a measurable function $extstyle\; f\; :\; Omega\; o\; mathbb\{R\}$ be injective and generating, then the following two conditions are equivalent:
* $extstyle\; f\; (Omega)$ is of full measure $extstyle\; mu;$
* $(Omega,mathcal\{F\},P)\; ,$ is a standard probability space.

See also Sect. 3.1 of [4] .

A random vector

The same theorem holds for any $mathbb\{R\}^n\; ,$ (in place of $mathbb\{R\}\; ,$). A measurable function $f\; :\; Omega\; o\; mathbb\{R\}^n\; ,$ may be thought of as a finite sequence of random variables $X\_1,dots,X\_n\; :\; Omega\; o\; mathbb\{R\},\; ,$ and $f\; ,$ is generating if and only if $mathcal\{F\}\; ,$ is the completion of the σ-algebra generated by $X\_1,dots,X\_n.\; ,$

A random sequence

The theorem still holds for the space $mathbb\{R\}^infty\; ,$ of infinite sequences. A measurable function $f\; :\; Omega\; o\; mathbb\{R\}^infty\; ,$ may be thought of as an infinite sequence of random variables $X\_1,X\_2,dots\; :\; Omega\; o\; mathbb\{R\},\; ,$ and $f\; ,$ is generating if and only if $mathcal\{F\}\; ,$ is the completion of the σ-algebra generated by $X\_1,X\_2,dots.\; ,$

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 $f\; :\; Omega\; o\; \{0,1\}^infty\; ,$ and a sequence of sets $A\_1,A\_2,dots\; in\; mathcal\{F\}.\; ,$ The function $f\; ,$ is generating if and only if $mathcal\{F\}\; ,$ is the completion of the σ-algebra generated by $A\_1,A\_2,dots.\; ,$

In the pioneering work [1] sequences $A\_1,A\_2,dots\; ,$ that correspond to injective, generating $f\; ,$ are called "bases" of the probability space $(Omega,mathcal\{F\},P)\; ,$ (see Sect. 2.1 of [1] ). A basis is called complete mod 0, if $f(Omega)\; ,$ is of full measure $mu,\; ,$ see Sect. 2.2 of [1] . 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 Prop. 4 and Def. 7 in [2] .

Additional remarks

The four cases treated above are mutually equivalent, and can be united, since the measurable spaces $mathbb\{R\},\; ,$ $mathbb\{R\}^n,\; ,$ $mathbb\{R\}^infty\; ,$ 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 $extstyle\; (Omega,mathcal\{F\},P)$ to a standard measurable space $extstyle\; (X,Sigma)$ does not depend on the choice of $extstyle\; (X,Sigma).$ Taking $extstyle\; (X,Sigma)\; =\; \{0,1\}^infty$ we get the property well-known as being "countably separated" (but called "separable" in [4] ).

Existence of a generating measurable function from $extstyle\; (Omega,mathcal\{F\},P)$ to a standard measurable space $extstyle\; (X,Sigma)$ also does not depend on the choice of $extstyle\; (X,Sigma).$ Taking $extstyle\; (X,Sigma)\; =\; \{0,1\}^infty$ we get the property well-known as being "countably generated" (mod 0), see Exer. I.5 in [6] .

Every injective measurable function from a "standard" probability space to a standard measurable space is generating. See Sect. 2.5 of [1] , Corollary 2 on page 253 in [2] , Theorems 3-4, 3-5 in [3] . 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 $extstyle\; (Omega,mathcal\{F\},P)$ is countably separated if and only if the cardinality of $extstyle\; Omega$ does not exceed continuum (see Exer. 3.1(v) in [4] ). 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 $extstyle\; (Omega,mathcal\{F\},P)$ be a complete probability space such that the cardinality of $extstyle\; Omega$ does not exceed continuum (the general case is reduced to this special case, see the caution above).

Via absolute measurability

Definition. $extstyle\; (Omega,mathcal\{F\},P)$ is standard if it is countably separated, countably generated, and absolutely measurable.

See the end of Sect. 2.3 of [1] and Remark 2 on page 248 in [2] . "Absolutely measurable" means: measurable in every countably separated, countably generated probability space containing it.

Via perfectness

Definition. $extstyle\; (Omega,mathcal\{F\},P)$ is standard if it is countably separated and perfect.

See Sect. 3.1 of [4] . "Perfect" means that for every measurable function from $extstyle\; (Omega,mathcal\{F\},P)$ to $mathbb\{R\}\; ,$ the image measure is regular. (Here the image measure is defined on all sets whose inverse images belong to $extstyle\; mathcal\{F\}$, irrespective of the Borel structure of $mathbb\{R\}\; ,$).

Via topology

Definition. $extstyle\; (Omega,mathcal\{F\},P)$ is standard if there exists a topology $extstyle\; au$ on $extstyle\; Omega$ such that
* the topological space $extstyle\; (Omega,\; au)$ is metrizable;
* $extstyle\; mathcal\{F\}$ is the completion of the σ-algebra generated by $extstyle\; au$ (that is, by all open sets);
* for every $extstyle\; varepsilon\; >\; 0$ there exists a compact set $extstyle\; K$ in $extstyle\; (Omega,\; au)$ such that $extstyle\; P(K)\; ge\; 1-varepsilon.$

See Sect. 1 of [3] .

Verifying the standardness

Every probability distribution on the space $extstyle\; mathbb\{R\}^n$ 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 Sect. 2.7 (p. 24) of [1] ; Example 1 (p. 248) in [2] ; Theorem 2-3 in [3] ; and Theorem 2.4.1 in [4] .

For example, the Wiener measure turns the Polish space $extstyle\; C\; [0,infty)$ (of all continuous functions $extstyle\; [0,infty)\; o\; mathbb\{R\},$ 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 $extstyle\; mathbb\{R\}^infty$ (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 Sect. 3.4 of [1] , Proposition 12 in [2] , and Theorem 2.4.3 in [4] .

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 Sect. 2.3 (p. 14) of [1] and Proposition 5 in [2] .

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 $extstyle\; Y$ on a probability space $extstyle\; (Omega,mathcal\{F\},P)$, it is natural to try constructing a conditional measure $extstyle\; P\_y$, that is, the conditional distribution of $extstyle\; omega\; in\; Omega$ given $extstyle\; Y(omega)=y$. In general this is impossible (see Sect. 4.1(c) in [6] ). However, for a "standard" probability space $extstyle\; (Omega,mathcal\{F\},P)$ this is possible, and well-known as "canonical system of measures" (see Sect. 3.1 of [1] ), which is basically the same as "conditional probability measures" (see Sect. 3.5 in [4] ), "disintegration of measure" (see Exercise (17.35) in [5] ), and "regular conditional probabilities" (see Sect. 4.1(c) in [6] ).

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 $extstyle\; (Omega\_1,mathcal\{F\}\_1,P\_1)$, $extstyle\; (Omega\_2,mathcal\{F\}\_2,P\_2)$ and a measure preserving map $extstyle\; f\; :\; Omega\_1\; o\; Omega\_2$, the image $extstyle\; f(Omega\_1)$ need not cover the whole $extstyle\; Omega\_2$, it may miss a null set. It may seem that $extstyle\; P\_2(f(Omega\_1))$ has to be equal to 1, but it is not so. The outer measure of $extstyle\; f(Omega\_1)$ is equal to 1, but the inner measure may differ. However, if the probability spaces $extstyle\; (Omega\_1,mathcal\{F\}\_1,P\_1)$, $extstyle\; (Omega\_2,mathcal\{F\}\_2,P\_2)$ are "standard " then $extstyle\; P\_2(f(Omega\_1))=1$, see Theorem 3-2 in [3] . If $extstyle\; f$ is also one-to-one then every $extstyle\; A\; in\; mathcal\{F\}\_1$ satisfies $extstyle\; f(A)\; in\; mathcal\{F\}\_2$, $extstyle\; P\_2(f(A))=P\_1(A)$. Therefore $extstyle\; f^\{-1\}$ is measurable (and measure preserving). See Sect. 2.5 (p. 20) of [1] and Theorem 3-5 in [3] . See also Proposition 9 in [2] (and Remark after it).

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 sstructure). Every measure preserving map $extstyle\; f\; :\; Omega\_1\; o\; Omega\_2$ leads to a homomorphism $extstyle\; F$ of measure algebras; basically, $extstyle\; F(B)\; =\; f^\{-1\}(B)$ for $extstyle\; Binmathcal\{F\}\_2$.

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 $extstyle\; F$ corresponds to some $extstyle\; f$. See Sect. 2.6 (p. 23) and 3.2 of [1] and Sect. 17.F of [5] .

Further reading

* [1] V.A. Rohlin, "On the fundamental ideas of measure theory", Translations (American Mathematical Society) Series 1, Vol. 10, 1-54 (1962). Translated from Russian: В.А. Рохлин, "Об основных понятиях теории меры", Математический Сборник (новая серия) 25(67), 107-150 (1949).
* [2] J. Haezendonck, "Abstract Lebesgue-Rohlin spaces", Bulletin de la Societe Mathematique de Belgique 25, 243-258 (1973).
* [3] T. de la Rue, "Espaces de Lebesgue", Lecture Notes in Mathematics (Seminaire de Probabilites XXVII), Springer, Berlin, 1557, 15-21 (1993).
* [4] K. Itô, "Introduction to probability theory", Cambridge Univ. Press 1984.
* [5] A.S. Kechris, "Classical descriptive set theory", Springer 1995.
* [6] R. Durrett, "Probability: theory and examples" (second edition), 1996.
* [7] N. Wiener, "Nonlinear problems in random theory", M.I.T. Press 1958.

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