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- Jun 20, 2014

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Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a Radon measure on $X$. Recall that the complement of the support of $\mu$ is the union of all open subsets of $X$ of $\mu$-measure zero. Show that the support of $\mu$ is the set of all $x\in X$ such that for all compactly supported continuous functions $f : X\to [0,1]$ with $f(x) > 0$, the integral $\int_X f\, d\mu > 0$.

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