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.
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 (Rudolph 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 ( Ω 1 , F 1 , P 1 ) {\displaystyle \textstyle (\Omega _{1},{\mathcal {F}}_{1},P_{1})} , ( Ω 2 , F 2 , P 2 ) {\displaystyle \textstyle (\Omega _{2},{\mathcal {F}}_{2},P_{2})} is an invertible map f : Ω 1 → Ω 2 {\displaystyle \textstyle f:\Omega _{1}\to \Omega _{2}} such that f {\displaystyle \textstyle f} and f − 1 {\displaystyle \textstyle f^{-1}} 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 ( Ω 1 , F 1 , P 1 ) {\displaystyle \textstyle (\Omega _{1},{\mathcal {F}}_{1},P_{1})} , ( Ω 2 , F 2 , P 2 ) {\displaystyle \textstyle (\Omega _{2},{\mathcal {F}}_{2},P_{2})} are isomorphic mod 0 {\displaystyle \textstyle \operatorname {mod} \,0} if there exist null sets A 1 ⊂ Ω 1 {\displaystyle \textstyle A_{1}\subset \Omega _{1}} , A 2 ⊂ Ω 2 {\displaystyle \textstyle A_{2}\subset \Omega _{2}} such that the probability spaces Ω 1 ∖ A 1 {\displaystyle \textstyle \Omega _{1}\setminus A_{1}} , Ω 2 ∖ A 2 {\displaystyle \textstyle \Omega _{2}\setminus A_{2}} are isomorphic (being endowed naturally with sigma-fields and probability measures).
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