If \(f_{n}\to f_{\infty}\) a.e. and if \(\mu_{n}\) are, corresponding to \(f_{n}\), the measures \[ \mu_{n}(B):= \int_{B}f_{n}(x)dx \] for all Borel sets \(B \in \mathcal{B}(\mathbb{R})\), then \[ \lVert \mu_{n}-\mu_{\infty} \rVert := \sup_{B\in \mathcal{B}}\lvert \mu_{n}(B)-\mu_{\infty}(B) \rvert \stackrel{n \to \infty}{\to} 0 . \]
\begin{proof}
\end{proof}