I'm curious as to whether the converse of the Radon-Nikodym theorem holds:
Converse Theorem: let $\mu.\nu$ be measures on $(\Omega,\Sigma)$ such that
$$\nu : A\mapsto \int_A f \ d\mu$$
for some measurable function $f$. Then
- $\nu\ll\mu$, and
- $\mu$ and $\nu$ are $\sigma$-finite.
Proof: the first part is easy, as
$$\mu(A) = 0\implies \int_A f \ d\mu = 0.$$
Does the second part, regarding $\sigma$-finiteness, hold?
Best Answer
It does not. Take any measure $\mu$. Then
$$\mu(A)=\int_A~\mathrm{d}\mu.$$