Invariant Measure, the Recurrence Theorem, and the Ergodic Theorem
نویسندگان
چکیده
converge a.e. to a finite limit, where fs is the characteristic function of the set E. G. D. Birkhoff's Ergodic Theorem asserts this conclusion if T is measure-preserving, in the sense that m(T~1E)=m(E) for each measurable set E. The same conclusion can be asserted under somewhat more general circumstances. We shall say that T admits a finite, equivalent, invariant measure if there is a finite measure p on S such that p = m and such that p(T~xE) =p(E) for each measurable set E. If a measurable transformation T admits such a measure, then the Individual Ergodic Theorem holds for T. For any measurable transformation T, we say that the Recurrence Theorem holds for T if, for any measurable set E and any positive integer w, it is.true that m(E — U/!,n T~'E)=0. It is known [4] that a necessary and sufficient condition that the Recurrence Theorem hold is that J" be incompressible, in the sense that E C T~XE implies m(T~lE — E)=0. A measure-preserving transformation on a finite measure space is incompressible, and consequently any transformation on a finite measure space which admits an equivalent, invariant, finite measure is also an incompressible transformation. These may be stated together.
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