For a measure-preserving transformation, the time average of an integrable function along almost every orbit converges, and for an ergodic system equals the function's space average. Proved by George David Birkhoff, it is the founding pointwise convergence theorem of ergodic theory.
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StatementThe theorem states that for a measure preserving transformation on a probability space, the time average of an integrable function along almost every orbit converges, and when the system is ergodic that limit equals the function's space average almost everywhere. 1 Proof YearGeorge David Birkhoff published the proof in 1931 in the Proceedings of the National Academy of Sciences; John von Neumann proved a related mean ergodic theorem the same period. Classification
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Source MacTutor History of Mathematics, University of St Andrews: George David Birkhoff - MacTutor History of Mathematics
Sources
1. Birkhoff Ergodic Theorem (Wikipedia)
Wikimedia Foundationpointwise ergodic theorem section, statement sentence
More precisely, the pointwise or strong ergodic theorem states that the limit in the definition of the time average of f exists for almost every x
history section, attribution sentence
Two of the most important theorems are those of Birkhoff (1931) and von Neumann
View the Source MacTutor History of Mathematics, University of St Andrews: George David Birkhoff - MacTutor History of Mathematics
MacTutor History of Mathematics, University of St AndrewsProved By: George David Birkhoff, BiographyQuote, Proved By: George David Birkhoff, Biography
His most important work was the Ergodic Theorem he discovered in 1931
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