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.
Contents
- Short history
- Definition
- Isomorphism
- Isomorphism modulo zero
- Standard probability space
- A naive white noise
- A perforated interval
- A superfluous measurable set
- A criterion of standardness
- A single random variable
- A random vector
- A random sequence
- A sequence of events
- Additional remarks
- Equivalent definitions
- Via absolute measurability
- Via perfectness
- Via topology
- Verifying the standardness
- Regular conditional probabilities
- Measure preserving transformations
- References
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 (Haezendonck 1973), (de la Rue 1993), (Itô 1984, Sect. 2.4) and (Rudolf 1990, Chapter 2).
Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example (Kechris 1995, Sect. 17). This approach is based on the isomorphism theorem for standard Borel spaces (Kechris 1995, Theorem (15.6)). 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 between two probability spaces
Two probability spaces are isomorphic, if there exists an isomorphism between them.
Isomorphism modulo zero
Two probability spaces
Standard probability space
A probability space is standard, if it is isomorphic
See (Rokhlin 1952, Sect. 2.4 (p. 20)), (Haezendonck 1973, Proposition 6 (p. 249) and Remark 2 (p. 250)), and (de la Rue 1993, Theorem 4-3). See also (Kechris 1995, Sect. 17.F), and (Itô 1984, especially Sect. 2.4 and Exercise 3.1(v)). In (Petersen 1983, Definition 4.5 on page 16) the measure is assumed finite, not necessarily probabilistic. In (Sinai 1994, Definition 1 on page 16) atoms are not allowed.
A naive white noise
The space of all functions
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
A perforated interval
Let
However, it is not. A random variable
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 (Rudolph 1990, page 17).
A superfluous measurable set
Let
gives the general form of a probability measure
However, it is the perforated interval in disguise. The map
is an isomorphism between
another set of inner Lebesgue measure 0 but outer Lebesgue measure 1.
See also (Rudolph 1990, Exercise 2.11 on page 18).
A criterion of standardness
Standardness of a given probability space
Two conditions will be imposed on
The probability space
A single random variable
A measurable function
(It is nothing but the distribution of the random variable.) The image
but its inner measure can differ (see a perforated interval). In other words,
A measurable function
Caution. The following condition is not sufficient for
Theorem. Let a measurable function
See also (Itô 1984, Sect. 3.1).
A random vector
The same theorem holds for any
A random sequence
The theorem still holds for the space
A sequence of events
In particular, if the random variables
In the pioneering work (Rokhlin 1952) sequences
Additional remarks
The four cases treated above are mutually equivalent, and can be united, since the measurable spaces
Existence of an injective measurable function from
Existence of a generating measurable function from
Every injective measurable function from a standard probability space to a standard measurable space is generating. See (Rokhlin 1952, Sect. 2.5), (Haezendonck 1973, Corollary 2 on page 253), (de la Rue 1993, Theorems 3-4 and 3-5). 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
Equivalent definitions
Let
Via absolute measurability
Definition.
See (Rokhlin 1952, the end of Sect. 2.3) and (Haezendonck 1973, Remark 2 on page 248). "Absolutely measurable" means: measurable in every countably separated, countably generated probability space containing it.
Via perfectness
Definition.
See (Itô 1984, Sect. 3.1). "Perfect" means that for every measurable function from
Via topology
Definition.
See (de la Rue 1993, Sect. 1).
Verifying the standardness
Every probability distribution on the space
The same holds on every Polish space, see (Rokhlin 1952, Sect. 2.7 (p. 24)), (Haezendonck 1973, Example 1 (p. 248)), (de la Rue 1993, Theorem 2-3), and (Itô 1984, Theorem 2.4.1).
For example, the Wiener measure turns the Polish space
Another example: for every sequence of random variables, their joint distribution turns the Polish 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 (Rokhlin 1952, Sect. 3.4), (Haezendonck 1973, Proposition 12), and (Itô 1984, Theorem 2.4.3).
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 (Rokhlin 1952, Sect. 2.3 (p. 14)) and (Haezendonck 1973, Proposition 5).
Every probability measure 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. 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
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
"There is a coherent way to ignore the sets of measure 0 in a measure space" (Petersen 1983, page 15). 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
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