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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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