Standard probability space

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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. 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 theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940. The dimension of the unit interval is not a concern, 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 Template:Harv, Template:Harv, Template:Harv and Template:Harv.

Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example Template:Harv. This approach is based on the isomorphism theorem for standard Borel spaces Template:Harv. 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 , is an invertible map such that and 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 , are isomorphic , if there exist null sets , such that the probability spaces , are isomorphic (being endowed naturally with sigma-fields and probability measures).

Standard probability space

A probability space is standard, if it is isomorphic to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination (disjoint union) of both.

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

Examples of non-standard probability spaces

A naive white noise

The space of all functions may be thought of as the product of a continuum of copies of the real line . One may endow with a probability measure, say, the standard normal distribution , and treat the space of functions as the product of a continuum of identical probability spaces . The product measure is a probability measure on . Many non-experts are inclined to believe that 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 N(0, 1). In contrast, the integral (from 0 to 1) of is undefined. Even worse, ƒ fails to be almost surely measurable. Still worse, the probability of ƒ being measurable is undefined. And the worst thing: 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 be a set whose inner Lebesgue measure is equal to 0, but outer Lebesgue measure is equal to 1 (thus, is nonmeasurable to extreme). There exists a probability measure on such that for every Lebesgue measurable . (Here is the Lebesgue measure.) Events and random variables on the probability space (treated ) are in a natural one-to-one correspondence with events and random variables on the probability space . Many non-experts are inclined to conclude that the probability space is as good as .

However, it is not. A random variable defined by is distributed uniformly on . The conditional measure, given , is just a single atom (at ), provided that is the underlying probability space. However, if is used instead, then the conditional measure does not exist when .

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 Template:Harv.

A superfluous measurable set

Let be as in the previous example. Sets of the form where and are arbitrary Lebesgue measurable sets, are a σ-algebra it contains the Lebesgue σ-algebra and The formula

gives the general form of a probability measure on that extends the Lebesgue measure; here is a parameter. To be specific, we choose 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 map

is an isomorphism between and the perforated interval corresponding to the set

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

See also Template:Harv.

A criterion of standardness

Standardness of a given probability space is equivalent to a certain property of a measurable map from to a measurable space Interestingly, the answer (standard, or not) does not depend on the choice of and . This fact is quite useful; one may adapt the choice of and to the given No need to examine all cases. It may be convenient to examine a random variable a random vector a random sequence or a sequence of events treated as a sequence of two-valued random variables,

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

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

A single random variable

A measurable function induces a pushforward measure, – the probability measure on defined by

   for Borel sets

(It is nothing but the distribution of the random variable.) The image is always a set of full outer measure,

but its inner measure can differ (see a perforated interval). In other words, need not be a set of full measure

A measurable function is called generating if is the completion of the σ-algebra of inverse images where runs over all Borel sets.

Caution.   The following condition is not sufficient for to be generating: for every there exists a Borel set such that ( means symmetric difference).

Theorem. Let a measurable function be injective and generating, then the following two conditions are equivalent:

See also Template:Harv.

A random vector

The same theorem holds for any (in place of ). A measurable function may be thought of as a finite sequence of random variables and is generating if and only if is the completion of the σ-algebra generated by

A random sequence

The theorem still holds for the space of infinite sequences. A measurable function may be thought of as an infinite sequence of random variables and is generating if and only if is the completion of the σ-algebra generated by

A sequence of events

In particular, if the random variables take on only two values 0 and 1, we deal with a measurable function and a sequence of sets The function is generating if and only if is the completion of the σ-algebra generated by

In the pioneering work Template:Harv sequences that correspond to injective, generating are called bases of the probability space (see Template:Harvnb). A basis is called complete mod 0, if is of full measure see Template:Harv. 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 Template:Harv and Template:Harv.

Additional remarks

The four cases treated above are mutually equivalent, and can be united, since the measurable spaces and are mutually isomorphic; they all are standard measurable spaces (in other words, standard Borel spaces).

Existence of an injective measurable function from to a standard measurable space does not depend on the choice of Taking we get the property well known as being countably separated (but called separable in Template:Harvnb).

Existence of a generating measurable function from to a standard measurable space also does not depend on the choice of Taking we get the property well known as being countably generated (mod 0), see Template:Harv.

Probability space Countably separated Countably generated Standard
Template:Rh | Interval with Lebesgue measure Yes Yes Yes
Template:Rh | Naive white noise No No No
Template:Rh | Perforated interval Yes Yes No

Every injective measurable function from a standard probability space to a standard measurable space is generating. See Template:Harv, Template:Harv, Template:Harv. 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 is countably separated if and only if the cardinality of does not exceed continuum (see Template:Harvnb). 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 be a complete probability space such that the cardinality of 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 Template:Harv and Template:Harv. "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 Template:Harv. "Perfect" means that for every measurable function from to the image measure is regular. (Here the image measure is defined on all sets whose inverse images belong to , irrespective of the Borel structure of ).

Via topology

Definition.   is standard if there exists a topology on such that

See Template:Harv.

Verifying the standardness

Every probability distribution on the space 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 Template:Harv, Template:Harv, Template:Harv, and Template:Harv.

For example, the Wiener measure turns the Polish space (of all continuous functions 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 (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 Template:Harv, Template:Harv, and Template:Harv.

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 Template:Harv and Template:Harv.

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 on a probability space , it is natural to try constructing a conditional measure , that is, the conditional distribution of given . In general this is impossible (see Template:Harvnb). However, for a standard probability space this is possible, and well known as canonical system of measures (see Template:Harvnb), which is basically the same as conditional probability measures (see Template:Harvnb), disintegration of measure (see Template:Harvnb), and regular conditional probabilities (see Template:Harvnb).

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 , and a measure preserving map , the image need not cover the whole , it may miss a null set. It may seem that has to be equal to 1, but it is not so. The outer measure of is equal to 1, but the inner measure may differ. However, if the probability spaces , are standard then , see Template:Harv. If is also one-to-one then every satisfies , . Therefore is measurable (and measure preserving). See Template:Harv and Template:Harv. See also Template:Harv.

"There is a coherent way to ignore the sets of measure 0 in a measure space" Template:Harv. 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 leads to a homomorphism of measure algebras; basically, for .

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 corresponds to some . See Template:Harv, Template:Harv, Template:Harv.

Notes

  1. Template:Harv and Template:Harv are cited in Template:Harv and Template:Harv.
  2. Published in short in 1947, in detail in 1949 in Russian and in 1952 in English, reprinted in 1962 Template:Harv. An unpublished text of 1940 is mentioned in Template:Harv. "The theory of Lebesgue spaces in its present form was constructed by V. A. Rokhlin" Template:Harv.
  3. "In this book we will deal exclusively with Lebesgue spaces" Template:Harv.
  4. "Ergodic theory on Lebesgue spaces" is the subtitle of the book Template:Harv.

References

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