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Suppose that $\mu$ is a Radon measure on $X$ and $\phi \in C(X,(0,\infty))$. Let $\nu(E)=\int_E \phi\,d\mu$ and let $\nu'$ be the Radon measure associated to the functional $I(f)=\int f\phi \,d \mu$ on $C_c(X)$. Then show that $\nu=\nu'$ on open sets; $\nu$ is outer regular and $\nu=\nu'$ and hence $\nu$ is a Radon measure.

I was able to show that $\nu$ and $\nu'$ agree on open sets.

I am unable to show that $\nu$ is outer regular. But assuming outer-regularity, $\nu=\nu'$ right away and hence $\nu$ is Radon.

But how do I show outer-regularity? Hint suggests the open sets $V_k=\{x: 2^k<\phi(x)<2^{k+2}\}$ with $k \in \mathbb{Z}$ cover $X$ but I don't see how that's helpful.

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  • $\begingroup$ What are your hypotheses on $X$? $\endgroup$
    – coudy
    Mar 11, 2023 at 10:04

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