Strongly Mixing Sequences of Measure Preserving Transformations

We call a sequence (Tn) of measure preserving transformations strongly mixing if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$P(T_n^{ - 1} A \cap B)$$ \end{document} tends to P(A)P(B) for arbitrary measurable A, B. We investigate whether one can pass to a suitable subsequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$T_{n_k } $$ \end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{1}{K}\sum\limits_{k = 1}^k {f(T_{n_k } )} \to \smallint fdP$$ \end{document} almost surely for all (or "many") integrable f.