Lp duality on an arbitrary measure space ($(L^p{)}^{\ast }\cong L^q,1<p<\infty$)

For conjugate exponents $p,q\in (1,\infty )$, every bounded linear functional on an Lp space is uniquely $h\mapsto \int hgd\mu$ for $g\in L^q$, with functional norm $\Vert g{\Vert }_q$. This holds on arbitrary measure spaces. Apply the Radon-Nikodym theorem on finite-measure pieces, then use support localization of an Lp functional to obtain one global density without assuming sigma-finiteness of the entire measure.

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.