“Birkhoff's Ergodic Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Birkhoff's Ergodic Theorem

释义

Birkhoff's Ergodic Theorem

Let be an ergodic Endomorphism of the Probability Space and let be a real-valued Measurable Function. Then for Almost Every , we have

(1)

. To illustrate this, take

to be the characteristic function of some Subset

so that

(2)

The left-hand side of (1) just says how often the orbit of

(that is, the points

, ...)lies in

, and the right-hand side is just the Measure of

. Thus, for an ergodic Endomorphism,``space-averages = time-averages almost everywhere.'' Moreover, if

is continuous and uniquely ergodic withBorel Probability Measure

and

is continuous, then we can replace the Almost Everywhere convergencein (1) to everywhere.

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