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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. 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 are cited in and .〕 and shaped by Vladimir Rokhlin in 1940.〔Published in short in 1947, in detail in 1949 in Russian and in 1952 in English. An unpublished text of 1940 is mentioned in . "The theory of Lebesgue spaces in its present form was constructed by V. A. Rokhlin" .〕 For modernized presentations see , , and . Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example . This approach is based on the isomorphism theorem for standard Borel spaces . 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,〔"In this book we will deal exclusively with Lebesgue spaces" .〕〔"Ergodic theory on Lebesgue spaces" is the subtitle of the book .〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Standard probability space」の詳細全文を読む スポンサード リンク
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