= 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.
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