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.

  • $\begingroup$ In the norm topology of $M$, yes. Which is enough for proving $\mu \mapsto \|\mu\|$ is Borel. In fact, that map is even continuous for the norm topology. But in the weak* topology? Doubtful... $\endgroup$ – Gerald Edgar Mar 2 '17 at 12:14

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