Let $X$ be a locally compact Hausdorff space with the Borel $\sigma$-algebra $\mathscr B_X$. Suppose that $\mu$ is a positive measure, $\nu$ is a finite positive measure, and $\nu\ll\mu$.
It is known that the Radon–Nikodym theorem may well fail if $\mu$ is not $\sigma$-finite. However, assume that $\mu$ and $\nu$ are both Radon measures, though $\mu$ is not necessarily $\sigma$-finite (but $\nu$ is finite). I am trying to prove a version of the Radon–Nikodym theorem, namely that there exists some $f:X\to[0,\infty)$ such that
- $f\in L^1(\mu)$ and
- $\nu(E)=\int_Ef\,\mathrm d\mu\quad\forall E\in\mathscr B_X$.
I have already shown the existence of such an $L^1(\mu)$ function that $\leq$ holds in the second line (using the proof of the standard Radon–Nikodym theorem), but the other direction seems to be elusive.
Any hint would be greatly appreciated.