Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 22H (Analysis of Functions)

A point $x$ is a Lebesgue point of $f$ when $|B_r{|}^{-1}{\int }_{B_r(x)}|f(y)-f(x)|dy\to 0$ as $r\downarrow 0$.

Approximate $f$ in $L^1$ by continuous $g$. The limsup local oscillation of $f$ is bounded by the maximal function of $f-g$ plus the vanishing oscillation of $g$. The weak $(1,1)$ Hardy-Littlewood maximal inequality makes the exceptional set have measure at most $C\Vert f-g{\Vert }_1/ϵ$; let the approximation error tend to zero.

At a Lebesgue point, $f\ast g_n(x)-f(x)=\int g(u)[f(x-u/n)-f(x)]du$. Compact support and boundedness of $g$ reduce this to a constant times the mean oscillation over a ball of radius $O(1/n)$, which tends to zero.

Not for an arbitrary signed $L^1$ kernel. One can place very tall, very narrow spikes of $g$ at separated locations, with summable masses and total integral one, and matching still narrower spikes of an $L^1$ function accumulating at a Lebesgue point. Along a selected subsequence of dilations, a kernel spike lands on the matching function spike and contributes a fixed amount, while the total widths make the point a Lebesgue point. Thus compact support (or a suitable integrable radial majorant) cannot simply be dropped.