Talk:Helly's selection theorem
|WikiProject Mathematics||(Rated Start-class, Low-importance)|
Why tempered distribution?
In the first paragraph of "Statement of the Theorem", the article says: "where the derivative is taken in the sense of tempered distributions" Shouldn't this be replaced by "(...) sense of distributions."? In which way is it meaningful to speak about temperedness on proper subsets of the real line? — Preceding unsigned comment added by 184.108.40.206 (talk) 08:22, 26 June 2016 (UTC)
Helly certainly did not know what a tempered distribution is, but neither seems the author of the article. It is a fact that the distributional derivative of a function of one variable and locally bounded variation is a one-dimensional Radon measure, and conversely, something most graduate students will not know. But the derivative is in general not locally integrable, because most measures are not absolutely continuous with respect to Lebesgue measure. Thus the statement needs to be modified not just because it introduces unnecessary complications; it is not Helly's theorem.