We want to prove the strong law of large numbers with Birkhoff's ergodic theorem.
Let $X_k$ be an i.i.d. sequence of $\mathcal{L}^1$ random variables. This is a stochastic process with measure-preserving operation $\theta$ (the shift operator). From Birkhoff's ergodic theorem, we obtain $\frac{X_0 + \dotsb + X_{n-1}}{n} \to Y$ a.s., with $Y=\mathbb{E}[X_1 \mid \mathcal{J}_{\theta}]$ a.s.
Now, if $Y$ constant a.s., $Y= \mathbb{E}[X_1]$ a.s., and we would have the desired result. But why is $Y$ constant a.s.?