= Support localization of an Lp functional
{title2=$g=0\text{ outside a sigma-finite }S$}
For $1<p<\infty$, local <Radon-Nikodym derivatives> $g_E$ representing an $L^p$ functional on finite-measure sets have uniformly bounded $L^q$ mass. Choose a sequence of finite-measure sets approaching the supremum of that mass, and glue their compatible densities on their union $S$. Any finite-measure set outside $S$ must have zero local density, since otherwise it would increase the supremum. Every <Lp space> function is approximable by <simple functions> of finite-measure support, so the glued density represents the functional on the whole space.
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