Let $M$ be the space of finite signed measures on $\mathbb{R}$, equipped with the topology of weak convergence of measures. I would like to know if taking the total variation of a measure is a measurable operation with respect to this topology. That is, is $\mu \mapsto |\mu|$ a Borel measurable map from $M$ to $M$?
I have read When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide? where it is mentioned that $\mu \mapsto ||\mu||$ is Borel measurable in a similar setting, but no proof or reference is given. A reference would be greatly appreciated.
