Pointwise convergence of triple ergodic averages for mixing transformations
We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space $(X,\mathcal F,\mu)$, with measurable inverse. For every triple of bounded measurable functions $f_1,f_2,f_3:X\to\mathbb C$, $\displaystyle \frac1N\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)f_3(T^{3n}x) \longrightarrow \prod_{j=1}^3\int_X f_j\,d\mu$ for μ -almost every x as $N\to\infty$ through all positive integers.
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