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- W898739468 abstract "Let ($X, Beta, mu$) be a measure space with the $sigma$-algebra $Beta$ and the probability measure $mu$. Throughouth this article set equalities and inclusions are understood as being so modulo measure zero sets. A transformation T defined on a probability space X is said to be measure preserving if $mu(T^{-1}E) = mu(E)$ for $E in B$. It is said to be ergodic if $mu(E) = 0$ or i whenever $T^{-1}E = E$ for $E in B$. Consider the sequence ${x, Tx, T^2x,...}$ for $x in X$. One may ask the following questions: What is the relative frequency of the points $T^nx$ which visit the set Eulcorner Birkhoff Ergodic Theorem states that for an ergodic transformation T the time average $lim_{n to infty}(1/N)sum^{N-1}_{n=0}{f(T^nx)}$ equals for almost every x the space average $(1/mu(X)) int_X f(x)dmu(x)$. In the special case when f is the characteristic function $chi E$ of a set E and T is ergodic we have the following formula for the frequency of visits of T-iterates to E : $$ lim_{N to infty} frac{{n : T^n x in E, 0 leq n }{N} = mu(E) $$ for almost all $x in X$ where $cdot$ denotes cardinality of a set. For the details, see [8], [10]." @default.
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- W898739468 title "SYMBOLIC DYNAMICS AND UNIFORM DISTRIBUTION MODULO 2" @default.
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