$(\Omega, \mathcal{F},\mu)$ 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!