Talk:Radon measure

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 Field: Analysis

Equations are atrocious[edit]

The author has left a lot of work for others, as the mathematics is presented in LaTeX and needs to be formatted properly in order to be readable online.

Radon distance - a metric?[edit]

In the article, the Radon distance is defined as a supremum of integrals of bounded continuous functions. However, if the measure m1 is not finite and if the measure m2 is the zero measure, we could take f to be the constant function equal to 1 and then the distance of m1 and m2 is m1(X) which is plus infinity. This is not suitable for a metric! Therefore, this part of the article needs to be corrected (e.g. by considering only finite Radon measures). ASlateff 128.131.37.74 13:18, 11 June 2007 (UTC)

Integration[edit]

The section when Integration is defined is confusing. I do not understand what the vectorspace F has to do with anything. —Preceding unsigned comment added by MathHisSci (talkcontribs) 17:02, 24 January 2010 (UTC)

Recent edits clarify things, thank you. MathHisSci (talk) 21:34, 6 February 2010 (UTC)

Measures[edit]

In the section "Measures", there is some confusion about whether \mathcal{K}(X) refers to the set of all continuous real-valued functions or only to those with compact support. The latter is probably true. In that case, it may be useful to point out that without positivity, Radon measures are not necessarily measures: I think that the mapping

f\mapsto\int_\R f(x)\,\sin(x)\,dx

is a (complex-valued) continuous linear functional on \mathcal{K}(\R), which represents a signed measure on any compact subset of \R, but not on \R itself because \mu(\R) cannot be defined.--146.107.3.4 (talk) 09:42, 10 December 2007 (UTC)

You're right, \mathcal{K}(X) refers to continuous functions with compact support on X. Thanks for attentiveness — I fixed that one now.
As for non-positive linear functionals and measures, it is a question of conflicting terminologies: Bourbaki and other proponents of the described approach to measure theory generally call measures what others call signed measures, while using the term positive measure for what others call simply a measure. I have clarified this in the introduction to the section in the article. Stca74 (talk) 12:23, 11 December 2007 (UTC)

error in Examples[edit]

Note for an expert in the field: In the Example section the second example of those which are not Radon measures is incomplete and is in error. Somebody needs to fix it. HowiAuckland (talk) 20:36, 5 August 2010 (UTC)

The article falsely claimed that the space of probability measures was always compact under the Wasserstein metric. This is true only if the underlying space is also compact (counterexamples are trivial to construct); I just fixed that. As a general remark, there are not enough references in this article. Hairer (talk) 12:41, 2 October 2011 (UTC)

Direct limit of topological spaces vs. direct limit of locally convex topological vector spaces[edit]

Within the article, \mathcal{K}(X) has been defined as a "direct limit of topological spaces ... equipped with the direct limit topology". Shouldn't \mathcal{K}(X) be defined as a direct limit of locally convex topological vector spaces, carrying the coarsest locally convex topology making the injections continuous? Bourbaki's Topological Vector Spaces discusses this topology, which is called the final locally convex topology, and it does not appear to coincide with the final topology (i.e. the finest topology making the injections continuous). Is there some reason why the two topologies would coincide in this case? 174.91.40.227 (talk) 23:10, 16 May 2011 (UTC) 174.91.40.227 (talk) 23:14, 16 May 2011 (UTC)

You're right, locally convexity needs to be imposed. This can be fixed easily. For the base of the topology, instead of taking subsets whose preimages are open, take convex balanced subsets. I assume by "coarsest", you mean finest.
Incidentally, tt is really an overkill to be talking about LCTVS here. Much simpler, and with no cost, to take the Banach completion and simply consider the Banach space of continuous functions vanishing at infinity. Mct mht (talk) 15:00, 31 March 2013 (UTC)