= Solution
Let $T(x)=x+\alpha\pmod1$ with irrational $\alpha$. This <irrational rotation of the circle> preserves <Lebesgue measure> and is ergodic. Apply the <Birkhoff ergodic theorem> to the <indicator function> $\mathbf1_A$. Its limit
$$
\mathbb E[\mathbf1_A\mid\mathcal I]
$$
is invariant, hence almost everywhere constant by <ergodic measure-preserving transformation>[ergodicity]. Its integral must equal that of $\mathbf1_A$, so the constant is $\mu(A)$. Therefore
$$
\boxed{
\frac{S_n(\mathbf1_A)(x)}n
=\frac1n\sum_{j=0}^{n-1}\mathbf1_A(T^jx)
\longrightarrow\mu(A)
}
$$
for <Lebesgue measure>-almost every $x\in(0,1]$.
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