Talk:Inner measure

Cleanup

This page needed cleanup, so I rewrote the introduction a little and separated out the definition and the completion. I'm not sure if this qualifies as enough to remove the clean up template. I'm not very happy with the $\scriptstyle {\hat {\Sigma }}$ but it was better than Σ̂ which is nearly impossible to distinguish from Σ. Steve Checkoway (talk) 02:46, 28 December 2008 (UTC)

I'm going to change it to ${\hat {\Sigma }}$ as suggested by the discussion of alignment with normal text flow. This apparently is not ideal, and I tried to stack it over nothing. That didn't work though maybe some modification of the following will: ${\underset {a}{\displaystyle {\hat {\Sigma }}}}$. Wpathooper (talk) 18:58, 18 November 2014 (UTC)

To make the equality of inner and outer measure imply that a set is measurable with respect to the completed measure, the measure space must be finite. Ideally, this subsection would be improved to handle non-finite measures too. Perhaps it would be simplest to state the result for sigma-finite measure spaces. Wpathooper (talk) 19:20, 18 November 2014 (UTC)

Definition

The definition given says "An inner measure is a function ... defined on all subsets of a set X, t ..." Is that correct? Or may it be defined only on some subsets of the set X, as is the case for a measure, which may be defined only on the measurable sets? It is so many years since I studied measure theory that I can't remember for certain. I also note that the editor who put that definition there said "Added an actual definition... I don't know if this is the best one", which does not give me faith that the definition is sound. Can anyone clarify this? The editor who uses the pseudonym "JamesBWatson" (talk) 15:28, 2 April 2015 (UTC)

Removed incorrect definition

The definition added by Intangir in April 2010 is incorrect because of the condition that "[i]nfinity must be approached". That is not correct if the underlying measure is finite.

The only source I have handy is Folland, who treats inner measures very briefly in an exercise, and as derived from a given outer measure. I am not aware of any source that defines inner measures independently, as Intangir's definition attempts to do.

What to do with the article is a bit of a puzzle. Possibly it should be merged into outer measure; that would make sense if Folland is typical of sources. Or arguably we could have a single inner and outer measures article. I'm generally not that fond of that solution but it might make sense in this case. --Trovatore (talk) 05:43, 3 April 2024 (UTC)

The text you removed said clearly If $\varphi (A)=\infty$ ... and so for a finite measure, the rest of the clause does not apply, and the definition would appear to be plausible and reasonable. Saying its "incorrect" seems too strong. My reference, Klenke, Probability theory has outer measure on page 22 and nothing in the index for inner measure. 67.198.37.16 (talk) 03:10, 11 April 2024 (UTC)

I read and re-read the part you removed, and concluded the right thing to do is to revert you. I see nothing wrong in that definition, albeit unsourced. I have no desire for an edit war, so if you still think the definition is flawed, then do revert me and explain here. But right now, I just don't see anything wrong. 67.198.37.16 (talk) 03:16, 11 April 2024 (UTC)

Hmm. I did indeed miss the text that assumed there was a set of infinite measure. It's not clear to me what case this is supposed to be excluding. Suppose there's a single point that has infinite point mass, and otherwise the measure is finite; why should the inner measure derived from that not count as an inner measure? And maybe more to the point, what are we sourcing this to? --Trovatore (talk) 13:52, 12 April 2024 (UTC)