In the context of proving that the data processing inequality for $f$-divergences hold for any Markov kernel I am interested in the following statement
If $\mu$ and $\nu$ are two probability measures on a probability space $(X,\Sigma,\mathbb P)$ such that $\nu\ll\mu$ and $f$ is any function from $X$ to $[0,1]$, then for $A\in \Sigma$ \begin{align*} \int_A f~ d\mu = 0 \Rightarrow \int_A f ~d\nu = 0 \end{align*}
Does anyone knows about this ? Is it even true ? I tried to prove it using the Radon-Nikodym derivative without success.