 Support localization of an Lp functional

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.