# 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.
Take a convergent series of positive numbers. You can turn this into a measure on the powerset of the natural numbers that is equivalent with counting measure. Both measures can easily be extended to all of $\mathbb{R}$ with the Borel $\sigma$-algebra.