On absolutely continuous measures..

Question

Is it true that if $\mu$ is a finite measure and $\nu$ is absolutely continuous w.r.t. $\mu$, then $\nu$ is a finite measure.

Answer 1

No. For example, the Lebesgue measure is absolutely continuous with respect to the gaussian probability measure $$\frac1{\sqrt{2\pi}}e^{-x^2/2}dx.$$