Talk:Radon measure

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[edit] Equations are atrocious

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.

[edit] Radon distance - a metric?

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)

[edit] Measures

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)