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$(\Omega, \mathcal{F},\mu)$ is a measure space, and $f: \Omega \to \Omega$ is a measurable mapping. Let $\nu$ be the measure on the same measurable space induced from $\mu$ by $f$ .

I wonder if there are

  • conditions/characterizations for $\mu$ to be absolutely continuous with respect to $\nu$,
  • conditions/characterizations for $\nu$ to be absolutely continuous with respect to $\mu$?

Motivations are when $(\Omega, \mathcal{F},\mu) = (\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n),m)$ the Lebesgue measure space, for a transformation on it, there are absolute value of Jacobian of the transformation

  • in change of variable formula for integral, and
  • in determining the probability density function of a continuous random variable after some transformation.

Thanks and regards!

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1 Answer

I think that in the generality of abstract measure spaces you will not find characterizations much better than "$\mu$ is absolutely continuous with respect to $\nu$ if and only if $\mu$ is absolutely continuous with respect to $\nu$".

This is related to Lusin's condition (N) which says: for any set $E$ of measure $0$ the image $f(E)$ has measure $0$. The paper I linked discusses whether it suffices to look only at the closed sets. There is a corresponding condition $(N^{-1})$: for any set $E$ of positive measure the image $f(E)$ also has positive measure.

The analysis of these conditions for maps in Euclidean spaces involves quite a bit of real analysis and some topology: a representative example is Mappings of finite distortion: condition N by Kauhanen, Koskela, and Malý.

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