|WikiProject Mathematics||(Rated Stub-class, Low-priority)|
vague versus weak
The article needs to make clear the distinction between vague and weak convergence in the sense of probability theory. In the case of finite positive measures, (vague convergence and convergence applied to constants) <=> weak convergence, see e.g. Bauer. For signed finite or positive non-sigma-finite measures, things are much more complicated, see e.g. Mörters/Preiss. --TjrCasual (talk) 05:42, 31 August 2011 (UTC)
Amazing, I've worked with Radon measures for 30 years and this is the first time I've heard the weak-* topology on spaces of Borel regular measures called the vague topology -- or at least the first time I really noticed it. I wonder in what specific area within hard analysis/geometric measure theory/functional analysis/probability theory the term is common? 18.104.22.168 (talk) 13:50, 8 February 2017 (UTC)
Its common in functional analysis theory. The term is common for the works of Choque, Diedeunne and evtl. Bourbaki. Already is in Bauer (Integration and measure theory).
From the article:
By the Riesz representation theorem M(X) is isometric to C^0(X)*.
Isn't this the definition of M(X)? My impression is that only positive measures can be defined directly. This appears to be confirmed at Radon measures. There is a Riesz rep theorem for the signed case, of course, but its conclusion (how you can write a suitably bounded linear functional concretely in terms of positive measures and Radon-Nykodym derivatives) is not generally taken as a definition. 22.214.171.124 (talk) 13:50, 8 February 2017 (UTC)
Would be nice to bring an example of a not metrizable topology on C_0(X), since for every polish space X, the vague topology on C_0(X) is metrizable (see Bauer). — Preceding unsigned comment added by 126.96.36.199 (talk) 13:53, 18 April 2017 (UTC)