A generalization of an ergodic theorem of Hopf

A. A. Tempel'man

Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 2, pp. 380-383

Résumé

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 $$ \lim_{n\to\infty}\int_Xf(T_x\omega)\lambda_n(dx)\bigg/\int_Xg(T_x\omega)\lambda_n(dx)=h_{f,g}(\omega) $$ 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.