Scheffé lemma ($f_n\to f\text{ a.e.},\int f_n\to \int f⟹\Vert f_n-f{\Vert }_1\to 0(f_n,f\ge 0)$)

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.