Talk:Convergence of measures

Texxed. Introduced the notion of total variation measure to simplify notation and definitions. Corrected a mistake in the definition of total variation (the original one worked only for real-valued measures). Added the Riesz representation theorem and connection to Banach space theory. Precised some terminology. I have a real analysis background in measure theory: I kept the original terminology, but also added the one which is more common in real analysis. Some real analysts love to quarrel with probabilists about terminology and how measure theory should be taught: I am not one of them and open to a friendly debate with fellows probabilists :-). Camillo 13:49, 21 July 2012 (CEST)

I do not take the hint... maybe because I am both a probabilist and a (real?) analyst... do I quarrel with myself? I only know that indeed probabilists say "weak" when functional analysts say "weak$^\star$". Yes, there is a clash here. But let me ask, is the weak (no star) topology on measures really useful?

Another question. On the set of probability measures some (or all?) of these topologies (except for the strong one, of course) should coincide. (For now I am afraid to say which exactly.)

Hi Camillo -- a couple of suggestions:

• Should $\mu: \mathcal{B}\mapsto \mathbb R$ actually be $\mu: \mathcal{B}\rightarrow \mathbb R$?
• Perhaps replace $\mathcal{B}$ by $\mathscr{B}$?

--Jjg 14:35, 21 July 2012 (CEST)

Hi Jjg. I agree with the first one (just done). The second is a matter of typographical taste: I have nothing against it, but I am not keen in chasing all the $\mathcal{B}$'s to replace them... in other words, you would be welcome to do it :-)) Camillo 14:40, 21 July 2012 (CEST)