Scheffé's lemma

Remarks

Proof idea: the direct implication follows from $\abs{\int \abs{f_n} - \int \abs{f}} \le \int \abs{f_n - f}$. For the converse, the functions $g_n = \abs{f_n} + \abs{f} - \abs{f_n - f}$ are non-negative and tend to $2\abs{f}$ almost everywhere. Fatou's lemma gives $2 \int \abs{f} \le 2 \int \abs{f} - \limsup_n \int \abs{f_n - f}$, and $\int \abs{f}$ is finite.

Example: let $p_n$ and $p$ be probability densities on $\R$ with $p_n \to p$ almost everywhere. Then $\int p_n = \int p = 1$, so $\int \abs{p_n - p} \to 0$. The distributions with densities $p_n$ then converge to the one with density $p$ in total variation.

The convergence of the norms is needed. For $f_n = n \mathbf{1}_{(0, 1/n)}$ on $\R$, $f_n \to 0$ everywhere but $\int \abs{f_n - 0} = 1$.

Going deeper

For a student

Scheffé's lemma is a quick criterion for convergence in $L^1$. Its main use is in probability and statistics: pointwise convergence of densities implies convergence of the distributions in total variation, hence in distribution. It is close in spirit to the Vitali convergence theorem, but it holds on any measure space and its proof is a direct application of Fatou's lemma.