Talk:Vague topology

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)[reply]

Wow

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? 178.39.122.125 (talk) 13:50, 8 February 2017 (UTC)[reply]

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).

Definition?

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. 178.39.122.125 (talk) 13:50, 8 February 2017 (UTC)[reply]

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 141.43.205.182 (talk) 13:53, 18 April 2017 (UTC)[reply]

What exactly are "functions vanishing at infinity" on a general topological space?

This article defines a space C_0(X) as the set of continuous functions vanishing at infinity, where X is merely a topological space. I don't think this makes sense unless X has some semi-norm defined on it. — Preceding unsigned comment added by Yossilonke (talk • contribs) 05:31, 24 August 2020 (UTC)[reply]

It makes sense for locally compact Hausdorff spaces, as the linked Wikipedia on vanishing at infinity defines it. For a general toplogical spaces, the definition still works but none of the theory holds. — Preceding unsigned comment added by Juto20 (talk • contribs) 17:06, 19 December 2023 (UTC)[reply]

The statemant is not true.

The Riesz representation theorem says, the every _continuous_ linear functional can be represented by a regular Radon measure. But there is no isometry.

Take for example an Lebesgue measure, that is regular Radon. The Lebesgue measure represents no continuous functional on $C_0(X)$. One take the function $x:t\mapsto 1/(1+|t|)$ on $\R$ as a counterexample. The integral $\int_\R x(t) dt$ diverges meanwhile $\|x\|_{C_0} = \sup_\R |x(t)| = 1$. 37.4.248.139 (talk) 23:01, 1 October 2024 (UTC)[reply]

The isometrical isomorphy is to _bounded_ regular Radon measures. See for example Roubíček

"Relaxation in Optimization Theory and Variational Calculus", Thm. 1.32 37.4.248.139 (talk) 23:06, 1 October 2024 (UTC)[reply]