Talk:Borel measure

Completion

The article states:

The Borel measure is not complete.

The article on complete measure states that completions are unique. What is the completion of the Borel measure (is it the Lebesgue measure)? linas 00:53, 23 November 2005 (UTC)[reply]

Yes— at least in the case of the real line that is how Lebesgue measure is defined. - Gauge 22:41, 22 January 2006 (UTC)[reply]

Any measure

I have understood the term "Borel measure" to mean any measure on the sigma-algebra of Borel sets in a topological space. Does this jar with what others here have seen? Michael Hardy 21:54, 1 June 2006 (UTC)[reply]

I understand a "Borel measure" to be any measure on the Borel sigma-algebra which is finite on compact sets, so I would say that this article seems misleading and should be expanded. Madmath789 15:05, 17 June 2006 (UTC)[reply]

I agree (see e.g. http://planetmath.org/encyclopedia/BorelMeasure.html), so I have corrected the text for it. The article still need expanding though. Simon Lacoste-Julien (talk) 06:33, 17 April 2010 (UTC)[reply]

following up on comment by Michael Hardy: the requirement that compact sets have finite measure, along with the measure of a set can be approximated from above by those of open sets, are for "regular" measures, stronger than mere Borel. Mct mht 21:13, 18 June 2006 (UTC)[reply]

An unstated assumption?

I might see a problem with the text which reads
"To clarify, when one says that the Lebesgue measure $\lambda$ is an extension of the Borel measure $\mu$ , it means that every Borel-measurable set E is also a Lebesgue-measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., $\lambda (E)=\mu (E)$ for every Borel measurable set)."
Yet here (http://www.math.cornell.edu/~neldredge/7770/7770-lecture-notes.pdf) mention that only Borel measures may exist in infinite dimensional spaces. I am not sure if I understand correctly, but it appears that completion of infinite dimension sets may be infeasible. — Preceding unsigned comment added by Mouse7mouse9 (talk • contribs) 07:07, 10 March 2015 (UTC)[reply]

I found Infinite-dimensional Lebesgue measure, which shows that the lebesgue measure cannot exist in infinite dimensional spaces, therefore not all Borel measurable sets have a Lebesgue measure. The article needs to be corrected to account for finite dimensions. Mouse7mouse9 07:25, 10 March 2015 (UTC)

That statement was made in a section titled "the real number line", which is one-dimensional. Thus, infinite dimensions should not have entered the conversation. Never-the-less, I added a clarifying remark, anyway. 67.198.37.16 (talk) 20:18, 4 December 2023 (UTC)[reply]