In how many ways Borel strong law of large number can be proved: (I was wandering to get a collection)
The statement is that "Almost all $x \in [0.1)$ are normal to $2$; i.e for almost every $x \in [0.1)$, the frequency of $1$'s in the binary expansion of $x$ is $1/2$.
Here is one proof using ergodicity; let $L$ be the full measure set of points with unique binary expansion.
Now $x \in L$
$x=.a_0a_1...=\sum_{i=0}^\infty \frac{a_i}{2^{i+1}}$ where $a_i \in \{0,1\}$ then define $Tx=\sum_{i=1}^\infty \frac{a_i}{2^{i+1}}$ i.e $T:[0,1) \to [0,1)$ s.t $x \mapsto 2x(mod 1)$
Check that $T$ is ergodic w.r.t Lebesgue measure. Then taking $f=\chi_{[\frac 12,1)}$
The frequency of $1$'s in the binary expansion of $x= lim_{n \to \infty}\frac 1n |\{0 \leq j \leq n-1| a_j=1\}|=lim_{n \to \infty}\frac 1n |\{0 \leq j \leq n-1| T^j(x) \in[\frac 12,1)\}|=lim_{n \to \infty}\frac 1n\sum_{j=0}^{n-1}f(T^jx)=lim_{n \to \infty}\frac 1n\sum_{j=0}^{n-1}\chi_{[\frac 12,1)}(T^jx) \to \int_Xf=\frac 12$