Talk:Borel measure

This was quite demanding: the terminology is very different according to different authors and I tried to be as complete as possible. In fact, since I know quite well the books using terminologies (B) and (C) I could provide precise references, whereas I lack knowledge of the (A)-terminology (the one used in the original article): any suggestion for precise references?Camillo 11:06, 15 August 2012 (CEST)

Indeed, encyclopedia tends to combine different approaches that often do not meet otherwise... --Boris Tsirelson 11:16, 15 August 2012 (CEST)

Also, I decided to completely leave out the following comment at the bottom of the original article:

Often, by the Borel measure on the real line one understands the measure defined on the Borel sets such that its value on an arbitrary segment is equal to the length of that segment.

This statement looks really weird to me: I think nowadays everybody calls it Lebesgue measure, even if one restricts it to the Borel sets... or am I missing something? Camillo 11:06, 15 August 2012 (CEST)

I agree. Really, when I have to be precise, I call it "Lebesgue measure restricted to Borel sets". --Boris Tsirelson 11:16, 15 August 2012 (CEST)

Borel regular measures

"(C) Borel (outer) measures such that for any $A\subset X$ there is a Borel set $B$ with $\mu(A)=\mu(B)$ (cp. with Definition 1.5(3) of [Ma])." — ?? Hope the Def in Ma is not exactly this. "$\mu(A)=\mu(B)$" does not mean much without something like "$A\subset B$" or "$B\subset A$"... --Boris Tsirelson (talk) 14:05, 23 September 2012 (CEST)

Comments

"for any open set $U$ there is a continuous function $f$ with $f^{-1} (]0, \infty[) = U$ (cp. with Separation axiom)" — Related a bit to separation axioms, but more strongly related to "every open set is an F_σ set"; the latter holds in all metrizable spaces, but not all topological spaces. --Boris Tsirelson (talk) 14:11, 23 September 2012 (CEST)

Different authors

It would be nice to know who among the sources in "References" uses (A), and (B), and (C). --Boris Tsirelson (talk) 14:15, 23 September 2012 (CEST)