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.
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.
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) harv error: no target: CITEREFRudolf1990 (help).
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,
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 (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.
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 (Rudolph 1990, page 17).
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 (Rudolph 1990, Exercise 2.11 on page 18).
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 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
i.e. the distribution of the random variable . The image is always a set of full outer measure,
A measurable function is called generating if is the completion with respect to 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:
- (i.e. the inner measure has also full measure, and the image is measureable with respect to the completion);
- is a standard probability space.
See also (Itô 1984, Sect. 3.1).
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 (Rokhlin 1952) sequences that correspond to injective, generating are called bases of the probability space (see Rokhlin 1952, Sect. 2.1). A basis is called complete mod 0, if is of full measure see (Rokhlin 1952, Sect. 2.2). 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 (Haezendonck 1973, Prop. 4 and Def. 7) and (Rudolph 1990, Sect. 2.3, especially Theorem 2.2).
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 Itô 1984).
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 (Durrett 1996, Exer. I.5).
|Probability space||Countably separated||Countably generated||Standard|
|Interval with Lebesgue measure||Yes||Yes||Yes|
|Naive white noise||No||No||No|
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 is countably separated if and only if the cardinality of does not exceed continuum (see Itô 1984, Exer. 3.1(v)). 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.
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 (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.
Definition. is standard if it is countably separated and perfect.
See (Itô 1984, Sect. 3.1). "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 ).
Definition. is standard if there exists a topology on 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 (de la Rue 1993, Sect. 1).
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.)
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.
The product of two standard probability spaces is a standard probability space.
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).
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 Durrett 1996, Sect. 4.1(c)). However, for a standard probability space this is possible, and well known as canonical system of measures (see Rokhlin 1952, Sect. 3.1), which is basically the same as conditional probability measures (see Itô 1984, Sect. 3.5), disintegration of measure (see Kechris 1995, Exercise (17.35)), and regular conditional probabilities (see Durrett 1996, Sect. 4.1(c)).
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 (de la Rue 1993, Theorem 3-2). If is also one-to-one then every satisfies , . Therefore, is measurable (and measure preserving). See (Rokhlin 1952, Sect. 2.5 (p. 20)) and (de la Rue 1993, Theorem 3-5). See also (Haezendonck 1973, Proposition 9 (and Remark after it)).
"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 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 (Rokhlin 1952, Sect. 2.6 (p. 23) and 3.2), (Kechris 1995, Sect. 17.F), (Petersen 1983, Theorem 4.7 on page 17).
- (von Neumann 1932) and (Halmos & von Neumann 1942) are cited in (Rokhlin 1952, page 2) and (Petersen 1983, page 17).
- Published in short in 1947, in detail in 1949 in Russian and in 1952 (Rokhlin 1952) in English. An unpublished text of 1940 is mentioned in (Rokhlin 1952, page 2). "The theory of Lebesgue spaces in its present form was constructed by V. A. Rokhlin" (Sinai 1994, page 16).
- "In this book we will deal exclusively with Lebesgue spaces" (Petersen 1983, page 17).
- "Ergodic theory on Lebesgue spaces" is the subtitle of the book (Rudolph 1990).
- Rokhlin, V. A. (1952), On the fundamental ideas of measure theory (PDF), Translations, 71, American Mathematical Society, pp. 1–54. Translated from Russian: Рохлин, В. А. (1949), "Об основных понятиях теории меры", Математический Сборник (Новая Серия), 25 (67): 107–150.
- von Neumann, J. (1932), "Einige Sätze über messbare Abbildungen", Annals of Mathematics, Second Series, 33: 574–586, doi:10.2307/1968536.
- Halmos, P. R.; von Neumann, J. (1942), "Operator methods in classical mechanics, II", Annals of Mathematics, Second Series, Annals of Mathematics, 43 (2): 332–350, doi:10.2307/1968872, JSTOR 1968872.
- Haezendonck, J. (1973), "Abstract Lebesgue–Rohlin spaces", Bulletin de la Société Mathématique de Belgique, 25: 243–258.
- de la Rue, T. (1993), "Espaces de Lebesgue", Séminaire de Probabilités XXVII, Lecture Notes in Mathematics, 1557, Springer, Berlin, pp. 15–21.
- Petersen, K. (1983), Ergodic theory, Cambridge Univ. Press.
- Itô, K. (1984), Introduction to probability theory, Cambridge Univ. Press.
- Rudolph, D. J. (1990), Fundamentals of measurable dynamics: Ergodic theory on Lebesgue spaces, Oxford: Clarendon Press.
- Sinai, Ya. G. (1994), Topics in ergodic theory, Princeton Univ. Press.
- Kechris, A. S. (1995), Classical descriptive set theory, Springer.
- Durrett, R. (1996), Probability: theory and examples (Second ed.).
- Wiener, N. (1958), Nonlinear problems in random theory, M.I.T. Press.