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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.

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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.

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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.

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