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Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 2, Pages 380–383 (Mi tvp2605)  

Short Communications

A generalization of an ergodic theorem of Hopf

A. A. Tempel'man

Institute of Physics and Mathematics, Academy of Sciences, Lithuanian SSR

Abstract: Let $X$ be a separable locally compact semigroup; let($\Omega$, $\mathfrak G$, $m$) be a space with a $\sigma$-finite measure $m$ and let $T_x$, $x\in X$, be a dynamic system in $\Omega$ with “time” from $X$. Let, further, $p$ and $q$ be probability Borel measures on $X$ and $\lambda_n=\sum_{k=0}^np*q^{*k}$. If $f$, $g\in L_1(m)$ and $g>0$ then the limit
$$ \lim_{n\to\infty}\int_Xf(T_x\omega)\lambda_n(dx)/\int_Xg(T_x\omega)\lambda_n(dx)=h_{f,g}(\omega) $$
is shown to exist almost everywhere on $\Omega$.
$(p,q)$-conservative dynamic systems are defined as systems inducing recurrent random walks in $\Omega$ in correspondence with the measures $p$ and $q$. For such dynamic systems the equality $h_{f,g}=\mathbf E(f\mid\mathfrak F)$ is proved where $\mathbf E(f\mid\mathfrak F)$ is the conditional expectation of the function $f(\omega)$ given the $\sigma$-algebra $\mathfrak F$ of measurable invariant sets.

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English version:
Theory of Probability and its Applications, 1973, 17:2, 363–365

Bibliographic databases:

Received: 21.07.1970

Citation: A. A. Tempel'man, “A generalization of an ergodic theorem of Hopf”, Teor. Veroyatnost. i Primenen., 17:2 (1972), 380–383; Theory Probab. Appl., 17:2 (1973), 363–365

Citation in format AMSBIB
\Bibitem{Tem72}
\by A.~A.~Tempel'man
\paper A~generalization of an ergodic theorem of Hopf
\jour Teor. Veroyatnost. i Primenen.
\yr 1972
\vol 17
\issue 2
\pages 380--383
\mathnet{http://mi.mathnet.ru/tvp2605}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=301169}
\zmath{https://zbmath.org/?q=an:0265.28008}
\transl
\jour Theory Probab. Appl.
\yr 1973
\vol 17
\issue 2
\pages 363--365
\crossref{https://doi.org/10.1137/1117043}


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