Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 5 / 3 / ii / a / Solution

The domination hypothesis includes $r=0$. For any approximation error $h$, linearity and the triangle inequality therefore give the uniform estimateFor a general input $g$ and an approximating $f$, this impliesThe first term converges on the dense class, while the weak type (1,1) estimate controls the exceptional set where the second is large.