 Solution

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.