State the Dominated Convergence Theorem.

Hence or otherwise prove Kronecker's Lemma: if $\left(a_{j}\right)$ is a sequence of non-negative reals such that

$$\sum_{j=1}^{\infty} \frac{a_{j}}{j}<\infty$$

then

$$n^{-1} \sum_{j=1}^{n} a_{j} \rightarrow 0 \quad(n \rightarrow \infty)$$

Let $\xi_{1}, \xi_{2}, \ldots$ be independent $N(0,1)$ random variables and set $S_{n}=\xi_{1}+\ldots+\xi_{n}$. Let $\mathcal{F}_{0}$ be the collection of all finite unions of intervals of the form $(a, b)$, where $a$ and $b$ are rational, together with the whole line $\mathbb{R}$. Prove that with probability 1 the limit

$$m(B) \equiv \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{j=1}^{n} I_{B}\left(S_{j}\right)$$

exists for all $B \in \mathcal{F}_{0}$, and identify it. Is it possible to extend $m$ defined on $\mathcal{F}_{0}$ to a measure on the Borel $\sigma$-algebra of $\mathbb{R}$ ? Justify your answer.