Talk:Negligible set
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It doesn't appear that there is an actual mathematical definition of "negligible" on this page, just examples of its usage in the literature and definitions in those specific cases. If there isn't a common definition, it might be more clear to change the wording to say that sets are defined to be negligible iff such-and-such condition holds in that setting.
Dzackgarza (talk) 20:05, 30 May 2020 (UTC)
I don't think the last sentence of the following is correct:
- Let X be a measurable space equipped with a measure m, and let a subset of X be negligible if it is m-null. Then the negligible sets form a sigma-ideal. The preceding example is a special case of this using counting measure.
With the counting measure, only the empty set is m-null (i.e. with measure zero). Moreover, a countable subset has the same counting measure as the full set, namely infinity. -- 134.95.128.246
You're right! The correct measure to use assigns 0 to any countable set but infinity to any uncountable set. I don't know a name for this measure, so I'll replace "counting measure" with "a suitable measure". (But if anybody else does know a name for this measure, then please add it in, with a link! ^_^) -- Toby Bartels 14:39, 16 Jul 2004 (UTC)
Every sigma ideal - from a measure; however...
[edit]"Let X be measurable space equipped with a measure m, and let a subset of X be negligible if it is m-null. Then the negligible sets form a sigma-ideal." This is OK with me. "Every sigma-ideal on X can be recovered in this way by placing a suitable measure on X." Well, this is right, as far as rather pathologic measures are allowed. But if only finite (or equivalently, sigma-finite) measures are allowed, then the sigma-ideal of all sets of first category, say, on [0,1] is not like that. Boris Tsirelson (talk) 15:59, 30 November 2008 (UTC)
At some point in the past several years, somebody added "although the measure may be rather pathological". —Toby Bartels (talk) 11:12, 24 August 2015 (UTC)