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Let $\mu$ denote a finite Radon measure on $\Omega\subset \mathbb R^N$. Let $\omega(x)\geq 1$ be given where $\omega\in L^1_{\operatorname{loc}}(\Omega)$ with respect to Lebesgue measure, i.e., I know $\omega$ is a lebesgue measurable.

Next, I want to define a new Radon measure $\nu$ based on $\mu$ by $$ \nu:=\frac{1}{\omega}\mu $$ Then, what do I need on $\omega$ so that $\nu$ is a well defined finite radon measure?


I know this problem is a bit trivial but I really wish to have a clear answer. For example, do I need $\omega$ is $\mu$ measurable or sth?

Thank you!

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  • $\begingroup$ @user251257 sorry it is not. I just made an update to make it more clear. Thank you for pointing out! $\endgroup$ – spatially Sep 15 '15 at 18:53
  • $\begingroup$ @user251257 well I don't think so. I think at least $\omega$ is being $\mu$ measurable is necessary... $\endgroup$ – spatially Sep 15 '15 at 19:18
  • $\begingroup$ oh sorry. I missed that $\omega$ is just Lebesgue measurable. Yes you need either Borel or $\mu$ measurable to make the integral well defined $\endgroup$ – user251257 Sep 15 '15 at 19:22

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