  Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$
$1\leq i\leq H,$ be commuting invertible measure preserving transformations on
this measure space. We prove the following pointwise results;
  The averages
  $$\frac{1}{N}\sum_{n=1}^N f_1(T_1^nx)f_2(T_2^nx)\cdots f_H(T_H^nx)$$ converge
a.e. for every function $f_i \in L^{\infty}(\mu)$ .\\ As a consequence if $T_i
= T^i$ for $1\leq i \leq H$ where $T$ is an invertible measure preserving
transformation on $(X, \mathcal{A}, \mu)$ then the averages
  $$\frac{1}{N}\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)...f_H(T^{Hn}x)$$ converge
a.e.
  This solves a long open question on the pointwise convergence of
nonconventional ergodic averages. For $H=2$ it provides another proof of J.
Bourgain's a.e. double recurrence theorem.
