Talk:Negligible set

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)[reply]

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)[reply]

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)[reply]

Indeed. Somebody anonymous. Boris Tsirelson (talk) 12:11, 24 August 2015 (UTC)[reply]