equivalent measures, can be one finite and one not? Let $\mu$ be a non-negative and Borel-finite measure on $\mathbb{R}$ and $\nu$ a non-negative measure on $\mathbb{R}$. If $\mu$ and $\nu$ are equivalent (one absolutely continuous with respect to the other) is it true that even $\nu$ is Borel-finite.

The other often-seen example.  Lebesgue measure on the real line, and the normalized Gaussian measure
$$
\gamma(E) = \frac{1}{\sqrt{2\pi}}\int_E \exp(-x^2/2)\;dx
$$
These two measures are mutually absolutely continuous.
