Namespaces
Variants
Actions

Difference between revisions of "Talk:Borel measure"

From Encyclopedia of Mathematics
Jump to: navigation, search
(→‎Comments: our "Baire set" article)
 
(2 intermediate revisions by the same user not shown)
Line 15: Line 15:
  
 
"'''(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  {{Cite|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 "$B\subset A$"... --[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 14:05, 23 September 2012 (CEST)
 
"'''(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  {{Cite|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 "$B\subset A$"... --[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 14:05, 23 September 2012 (CEST)
 +
  
 
=== Comments ===
 
=== Comments ===
Line 20: Line 21:
 
"for any open set $U$ there is a continuous function $f$ with
 
"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<sub>σ</sub> set"; the latter holds in all metrizable spaces, but not all topological spaces. --[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 14:11, 23 September 2012 (CEST)
 
$f^{-1} (]0, \infty[) = U$ (cp. with [[Separation axiom]])" — Related a bit to separation axioms, but more strongly related to "every open set is an F<sub>σ</sub> set"; the latter holds in all metrizable spaces, but not all topological spaces. --[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 14:11, 23 September 2012 (CEST)
 +
 +
About Baire sets: compare this with [http://en.wikipedia.org/wiki/Baire_set Wikipedia article] (written by me to a large extent). Also with ''our'' [[Baire set]]. --[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 15:20, 23 September 2012 (CEST)
 +
 +
== Different authors ==
 +
 +
It would be nice to know who among the sources in "References" uses (A), and (B), and (C). --[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 14:15, 23 September 2012 (CEST)

Latest revision as of 13:22, 23 September 2012

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 Fσ set"; the latter holds in all metrizable spaces, but not all topological spaces. --Boris Tsirelson (talk) 14:11, 23 September 2012 (CEST)

About Baire sets: compare this with Wikipedia article (written by me to a large extent). Also with our Baire set. --Boris Tsirelson (talk) 15:20, 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)

How to Cite This Entry:
Borel measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_measure&oldid=28128