= Solution
<Kolmogorov zero-one law> says that the tail sigma-algebra of <independent random variables> is trivial: every tail event has probability zero or one. The <Birkhoff ergodic theorem> says that for a probability-preserving transformation $T$ and integrable $f$, the averages $A_nf=n^{-1}\sum_{j=0}^{n-1}f\circ T^j$ converge almost surely to $\mathbb E(f\mid\mathcal I)$, where $\mathcal I$ is the invariant sigma-algebra. The <mean ergodic theorem> in $L^p$, $1\leq p<\infty$, gives the same convergence in $L^p$ for $f\in L^p$; $p=2$ is the classical von Neumann theorem, and on a probability space the $L^1$ version applies to the integrable case needed here. There is no general supremum-norm convergence assertion for $p=\infty$.
The <strong law of large numbers> for iid integrable variables is $n^{-1}\sum_{j=1}^nX_j\to\mathbb EX_1$ almost surely. To prove it, realize the sequence on its product probability space and take the left shift $T$. The shift preserves the iid product law. Birkhoff applied to $f=X_1$ gives an almost-sure invariant limit $Z$. An invariant event is, modulo null sets, measurable with respect to coordinates after every finite index, hence is a tail event. Kolmogorov's law makes the invariant sigma-algebra trivial, so $Z$ is constant almost surely. Finally the $L^1$ mean ergodic theorem permits passing <expectations> through the limit; all averages have <expectation> $\mathbb EX_1$. This identifies that constant and proves the strong law.
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