= Solution
{c}
Let $C$ be the event that the sample averages have a limit in the <extended real number line>. It is measurable: the bounded sequence $\arctan S_n$ converges exactly when it is Cauchy, and the Cauchy condition is a countable combination of measurable inequalities. Limits at $\pm\pi/2$ correspond to extended limits $\pm\infty$.
For each fixed $m$, remove the first $m-1$ terms and set
$$
S_n^{(m)}=\frac1n\sum_{k=m}^nX_k\quad(n\geq m).
$$
The difference $S_n-S_n^{(m)}=n^{-1}\sum_{k<m}X_k$ tends to zero for every outcome, since the finitely many variables are real-valued. Thus these two sequences have an extended limit simultaneously. Equivalently their arctangents differ by a quantity tending to zero, because arctangent is Lipschitz. The event $C$ is therefore expressible using $X_m,X_{m+1},\ldots$ alone. Since this holds for every $m$, $C\in\mathcal T$.
Kolmogorov's law gives $\mathbb P(C)=0$ or one. In the latter case the measurable extended variable $\limsup S_n$, redefined on a <null set> if needed, is the <almost sure convergence>. In the former there is almost surely no extended-real limit. Hence \b[extended convergence occurs almost surely, or divergence occurs almost surely]. The law need not make the value of a possible limit finite.
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