Standard probability space

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, a standard probability space (called also Lebesgue–Rokhlin probability space or just Lebesgue space; the latter term is ambiguous) is a probability space

Probability space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. He showed that the unit interval

Unit interval

In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...

endowed with the Lebesgue measure

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

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

John von Neumann

John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

in 1932 and shaped by Vladimir Rokhlin in 1940. The dimension of the unit interval is not a concern, which was clear already to Norbert Wiener

Norbert Wiener

Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...

. He constructed the Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

(also called Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

) in the form of a measurable map

Map (mathematics)

In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...

from the unit interval to the space of continuous functions

Function space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

Short history

The theory of standard probability spaces was started by von Neumann

John 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

Descriptive set theory

In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...

, via standard Borel spaces

Borel algebra

In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...

, see for example . This approach, natural for experts in descriptive set theory, is based on the isomorphism theorem for standard Borel spaces 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 set

Null set

In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...

s, in contrast to descriptive set theory).

Standard probability spaces are used routinely in ergodic theory

Ergodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

, 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
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...

and random variables are essential, while probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

s are auxiliary and should not be taken too seriously.

Arguments in favour of standardness:

conditioning
Conditioning (probability)
Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations and conditional distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory...

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
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

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 space

Probability space

s are assumed to be complete

Complete measure

In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...

Isomorphism

An isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...

between two probability spaces

is an invertible

Inverse function

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

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 set

Null set

In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...

such that the probability spaces

are isomorphic (being endowed naturally with sigma-fields and probability measures).

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

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

Product measure

In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space...

is a probability measure on

. Many non-experts are inclined to believe that

describes the so-called white noise

White noise

White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...

.

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

Almost surely

In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...

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

Inner measure

In mathematics, in particular in measure theory, an inner measure is a function on the set of all subsets of members of a given σ-algebra with values in the extended real numbers satisfying some technical conditions...

Lebesgue measure is equal to 0, but outer

Outer measure

In mathematics, in particular in measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. A general theory of outer measures was first introduced by...

Lebesgue measure – to 1 (thus,

is nonmeasurable to extreme). There exists a probability measure

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 .

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

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 .

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

treates as a sequence of two-valued random variables,

Two conditions will be imposed on

(to be injective

Injective function

In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...

, 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

Complete measure

In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...

(otherwise it cannot be standard).

A single random variable

A measurable function

induces a pushforward measure

Pushforward measure

In measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...

, --- the probability measure

defined by

for Borel sets

(It is nothing but the distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

of the random variable.) The image

is always a set of full outer measure,

but its inner measure

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

Symmetric difference

In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....

).

Theorem. Let a measurable function

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

is of full measure
is a standard probability space.

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 sequences

that correspond to injective, generating

are called bases of the probability space

(see ). A basis is called complete mod 0, if

is of full measure

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

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 ).

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 .

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

is countably separated if and only if the cardinality of

does not exceed continuum

Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by |\mathbb R| or \mathfrak c ....

(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

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 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 . "Perfect" means that for every measurable function from

the image measure is regular

Regular measure

In mathematics, a regular measure on a topological space is a measure for which every measurable set is "approximately open" and "approximately closed".-Definition:...

. (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

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

such that

the topological space is metrizable;
is the completion of the σ-algebra generated by (that is, by all open sets);
for every there exists a compact set in such that

See .

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

Polish space

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...

, see , , , and .

For example, the Wiener measure turns the Polish space

(of all continuous functions

endowed with the topology

Topological space

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

Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...

) into a standard probability space.

(Thus, the idea of dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

, very natural for topological space

Topological space

s, is utterly inappropriate for standard probability spaces.)

The product

Product measure

In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space...

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

Probability measure

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...

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

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

Conditional expectation

In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....

. In the non-discrete setup, conditioning is often treated indirectly, since the condition may have probability 0, see conditional expectation

Conditional expectation

In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....

. 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

Conditional expectation

In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....

); Hölder's inequality

Hölder's inequality

In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....

; 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

Conditional distribution

Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value...

given

. In general this is impossible (see ). However, for a standard probability space

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

Disintegration theorem

In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures...

(see ), and regular conditional probabilities

Regular conditional probability

Regular conditional probability is a concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions...

(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

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 . If

is also one-to-one then every

satisfies

. Therefore

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

Complete Boolean algebra

In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum . Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing...

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 , , .

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