  We first present a modern simple proof of the classical ergodic Birkhoff's
theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used
the simple fact that the shift map on integers has a simple Lebesgue spectrum.
As a consequence, we establish that the homogeneous bilinear ergodic averages
along polynomials and polynomials in primes converge almost everywhere, that
is, for any invertible measure preserving transformation $T$, acting on a
probability space $(X, \mathcal{B}, \mu)$, for any $f \in L^r(X,\mu)$ , $g \in
L^{r'}(X,\mu)$ such that $\frac{1}{r}+\frac{1}{r'}= 1$, for any non-constant
polynomials $P(n),Q(n), n \in \mathbb{Z}$, taking integer values, and for
almost 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 $[1,N]$.
