Let $\mu$ be some measure on $\mathbb{R}$ that has density $f:\mathbb{R}\to [0, \infty)$ wrt Lebesgue measure. Does this imply that $\mu$ is $\sigma$-finite?
I suppose not, because having a density wrt a $\sigma$-finite measure need not imply $\sigma$-finiteness, however, I struggle to find a counterexample. Intuitively, I suppose we'd have to somehow construct a density that has infinite integral on any set with positive Lebesgue measure, but I do not see how.