Consider a measure space $(E,\mathcal{B}(E),\mu)$ and let $f:E\to E$ be a measurable mapping. The push forward of the measure $\mu$ by the mapping $f$ is defined to be $$ f_\#\mu(A) = \mu(f^{-1}(A)) \quad \text{for all } A\in\mathcal{B}(E). $$
My question is the following: what regularity do we need to assume on $f$ to make sure that $f_\#\mu\ll\mu$?
I was writing this question when I found this other post that helped a lot. But now I am wondering, for the Lebesgue measure on $\mathbb{R}^n$, instead of locally Lipschitz mappings, could we relax this condition to locally Hölder with some $\alpha \in (0,1)$?
Thanks in advance!
