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If $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, for what kind of functions $f$ does $\int_\Omega f d\mu$ make sense for all Radon measures $\mu$ with $\mu(\Omega) < \infty$?

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I mean, precisely those for which $f \in L^1(\mu)$. Since $\mu(\Omega) < \infty$, this includes for example all bounded $\mu$-measurable functions. If $\mu$ is also a Borel measure then every continuous function is $\mu$-measurable, although this last requirement almost forces $\mu$ to be equal to Lebesgue measure. – user12014 Jun 11 '12 at 1:06
@PZZ Thank you for your prompt reply. I didn't phrase my question very well. Please read the corrected question. – Stefan Smith Jun 11 '12 at 1:43
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Perhaps you want the so-called "universally measurable" functions. A set is universally measurable iff it is measurable for every Radon measure. – GEdgar Jun 11 '12 at 1:45
@PZZ I Googled "universally measurable" and found plenty of information on it. In more generality then I wanted. Can you recommend a reference that refers to Radon measures? – Stefan Smith Jun 11 '12 at 2:00
@user20520 Oh, I see. I've never encountered universally measurable sets of functions, so unfortunately I cannot recommend a text. – user12014 Jun 11 '12 at 3:44

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