Recurrence and Ergodic Theorems
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Theorem 1 (Bergelson (1985), Theorem 1.1) Let \((X,\SigmaAlgebra{B},\Measure)\) be a probability space and suppose that \(B_n\in\SigmaAlgebra{B}\) such that \(\Measure(B_n)=b>0\) for all \(n\in\mathbb{N}\).
Then there exists a positively dense index set \(I\subset\mathbb{N}\) such that, for any finite subset \(F\subseteq I\), we have \[\Measure\left(\bigcap_{i\in F}B_i \right)>0. \]
Theorem 2 (cf. Lindenstrauss (2001), Theorem 1.2) Let \(\AmenableGroup\) be a discrete amenable group acting on a measure space \((X,\SigmaAlgebra{B},\Measure)\) by measure preserving transformation and let \(\Folner=(\Folner[N])_{N\in\mathbb{N}}\) be a tempered Følner sequence.
Then, for any \(f\in\text{L}^1(\Measure)\), there is a \(\AmenableGroup\)-invariant \(\bar{f}\in\text{L}^1(\Measure)\) such that \[\lim_{N\rightarrow\infty}\frac{1}{\CountingMeasure{\Folner[N]}}\sum_{\AmenableGroupElement\in\Folner[N]}f(\GroupAction{\AmenableGroupElement}{x})=\bar{f}(x) \] for \(\Measure\)-almost every \(x\in X\). In particular, if the \(\AmenableGroup\) action is ergodic, then \[\lim_{N\rightarrow\infty}\frac{1}{\CountingMeasure{\Folner[N]}}\sum_{\AmenableGroupElement\in\Folner[N]}f(\GroupAction{\AmenableGroupElement}{x})=\int f(x)\ d\Measure(x) \] for \(\Measure\) almost every \(x\).
Corollary 1 (cf. Host (2019), Corollary 8) Let \((X,\AmenableGroup)\) be a topological dynamical system where \(\AmenableGroup\) is an amenable group, \(\Measure\) an ergodic measure on \(X\) and \(\Folner\) a tempered Følner sequence. Then \(\Measure\)-almost every \(x\in X\) is generic for \(\Measure\) along \(\Folner\).