In probability theory, a standard probability space, also called LebesgueRokhlin 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 atomss.

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 and shaped by Vladimir Rokhlin in 1940. 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.

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 is an invertible both are (measurable and) measure preserving maps.

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

Isomorphism modulo zero

if there exist null sets are isomorphic (being endowed naturally with sigma-fields and probability measures).

Standard probability space

A probability space is standard 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

with a probability measure, say, the standard normal distribution. The product measure describes white noise.

However, the integral of a white noise function from 0 to 1 should be a random variable distributed N(0, is undefined. also fails to be almost surely measurable, and the probability of being measurable is undefined. Indeed, if X is a random variable distributed (say) uniformly on (0,1) and independent of , then (X) is not a random variable at all (it lacks measurability).

A perforated interval

be a set whose inner Lebesgue measure is equal to 0, but outer is nonmeasurable.

.

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

The formula

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

and the perforated interval corresponding to the set

another set of inner Lebesgue measure 0 but outer Lebesgue measure 1.

See also .

A criterion of standardness

(to be injective is given. The question of its existence will be addressed afterwards.

is assumed to be complete (otherwise it cannot be standard).

A single random variable

induces a pushforward measure defined by

i.e. the distribution is always a set of full outer measure,

but its inner measure can differ (see a perforated interval need not be a set of full measure

is called generating is the completion runs over all Borel sets.

Caution. means symmetric difference).

Theorem. be injective and generating, then the following two conditions are equivalent:

• is measureable with respect to the completion);

• is a standard probability space.

See also .

A random vector

A random sequence

A sequence of events

In the pioneering work are called bases (see 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

are mutually isomorphic; they all are standard measurable spaces (in other words, standard Borel spaces).

we get the property well known as being countably separated (but called separable in ).

we get the property well known as being countably generated (mod 0), see .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. is countably separated if and only if the cardinality 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

does not exceed continuum (the general case is reduced to this special case, see the caution above).

Via absolute measurability

Definition. 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. is standard if it is countably separated and perfect.

See the image measure is regular).

Via topology

Definition. is standard if there exists a topology such that

• is metrizable;

• (that is, by all open sets);

See .

Verifying the standardness

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 .

endowed with the topology of local uniform convergence) into a standard probability space.

(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.

Applications

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.

, that is, the conditional distribution. In general this is impossible (see ). However, for a standard 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

are standard , see 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.

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. See , , .

See also

Notes

References

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