Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 25K / iii

Now $f\ge 0$ almost everywhere. The functions $\min (f_n,f)$ converge almost everywhere to $f$ and are dominated by the integrable $f$. The dominated convergence theorem gives $\int \min (f_n,f)\to \int f$. Using $|f_n-f|=f_n+f-2\min (f_n,f)$ and the assumed convergence of the nonnegative integrals yieldsThis is the nonnegative form of Scheffé's lemma: almost-everywhere convergence together with convergence of total mass implies convergence in the L1 norm.