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 $f_1,f_2,f_3:X\to \mathbb{C}$, $\frac{1}{N}\sum _{n=1}^N{f}_1(T^{n}x)f_2(T^{2n}x)f_3(T^{3n}x)⟶\prod _{j=1}^3{\int }_X{f}_{j}d\mu$ for μ-almost every x as $N\to \infty$ through all positive integers.