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Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For ...

Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...