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We first present a modern simple proof of the classical ergodic Birkhoff'stheorem and Bourgain's homogeneous bilinear ergodic theorem. This proof usedthe simple fact that the shift map on integers has a simple Lebesgue spectrum.As a consequence, we establish that the homogeneous bilinear ergodic averagesalong polynomials and polynomials in primes converge almost everywhere, thatis, for any invertible measure preserving transformation $T$, acting on aprobability space $(X, \mathcal{B}, \mu)$, for any $f \in L^r(X,\mu)$ , $g \inL^{r'}(X,\mu)$ such that $\frac{1}{r}+\frac{1}{r'}= 1$, for any non-constantpolynomials $P(n),Q(n), n \in \mathbb{Z}$, taking integer values, and foralmost all $x \in X$, we have,  $$\frac{1}{N}\sum_{n=1}^{N}f(T^{P(n)}x) g(T^{Q(n)}x),$$ and$$\frac{1}{\pi_N}\sum_{\overset{p \leq N}{p\textrm{~~prime}}}f(T^{P(p)}x)g(T^{Q(p)}x),$$ converge. Here $\pi_N$ is the number of prime in $[1,N]$.

Simple proof of Bourgain bilinear ergodic theorem and its extension to polynomials and polynomials in primes