Let $f: X \to \mathbb{R}$ be a continuous function over a compact metric space $X$. Assume that $\mu$ and $\nu$ are two Borel probability measures on $X$.
Suppose that $\mu << \nu$. Is it true that $$f_{\ast}\mu << f_{\ast} \nu?$$ We recall $f_{\ast}\mu(A)=\mu(f^{-1} (A)).$
If measures are invariant, then the result is obvious.