Let $\nu_n$ be a sequence of finite signed radon measure such that $\nu_n\to \nu$ strongly for a finite signed radon measure $\nu$. Let $|\nu_n|$ denote the total variation measure of $\nu_n$. We know that $|\nu_n|$ is a positive Radon measure.
My question: do I have $|\nu_n|\to |\nu|$ strongly? and do I have $||\nu|-|\mu||\leq |\nu-\mu|$ for two arbitrary finite signed measure $\nu$ and $\mu$?
I know the above statement is absolutely false if $\nu_n\to \nu$ only in weak star sense. But I somehow remembered for strong convergence it is true but I can not find the source. So, if it is true, please confirm it for me and directly me to a reference, if not... maybe a counter example?