Talk:Lebesgue measure
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Many of the properties of te Lebesgue measure as stated on this page are simply properties of measures in general. Items 3 and 4 for example, and perhaps others (I don't know, since I'm just learning about the subject as I browse here!)
Wouldn't it be better to list only those properties that are specific to the Lebesgue measure, or at least to indicate which ones are? The standard definition is already there at measure
Also, a measure is defined on its page as a function on a sigma algebra; if that's really part of the definition, then saying "the Lebesgue measurable sets therefore form a sigma algebra" seems a little redundant and/or confusing.
- Stuart
- When the article says, "the Lebesgue measurable sets form a sigma algebra", this is said first without knowing that Lebesgue measure IS a measure. In other words, we're verifying the axioms for the particular case of Lebesgue measure on Euclidean space. This is much the same thing that happens when we verify the group axioms for a set of elements and operation -- it's not redundant, we just have to check that it really is a group. Revolver
Is it clearer now? --AxelBoldt
Thanks. I'm still a little unsure as to which properties listed are common to all measures, and which are special to Lebesgue measure, but that's mainly due to my near-total ignorance of the whole subject. - Stuart
Can any of you give an example of a non-Lebesgue measurable set? JeffBobFrank 03:37, 4 Mar 2004 (UTC)
- Sure, it's not too difficult; I'll either put it up here or on a separate article. You basically take a set of representatives of the cosets of Q (the rationals) in R (the reals), (this is where choice comes in) and consider all the translates of this set by rationals. The translates form a countable partition of R, and you use this to get a contradiction. Of course, there are some details I'm leaving out. A proof is given in Royden and a slightly more general result in papa Rudin (that every subset of R of positive Lebesgue measure contains a non-measurable subset). Revolver 17:22, 4 Mar 2004 (UTC)