= Scheffé lemma
{c}
{title2=$f_n\to f\text{ a.e.},\ \int f_n\to\int f\ \Longrightarrow\ \|f_n-f\|_1\to0\quad(f_n,f\ge0)$}
= Scheffé's lemma
{c}
{synonym}
For nonnegative <integrable> <functions> converging <almost everywhere> to an <integrable> $f$, convergence of their <integrals> implies <L1 convergence>. Indeed $\min(f_n,f)\to f$ <almost everywhere> and is dominated by $f$, so the <dominated convergence theorem> gives convergence of its <integral>. Use $|f_n-f|=f_n+f-2\min(f_n,f)$. This applies to <probability densities>, giving convergence in <total variation distance>.
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