Talk:Almost all

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I wonder if we should distinguish between "almost everywhere" and "almost all". I don't think "almost all" is used in measure theory at all, nor is "almost everywhere" used in number theory. Also, sometimes "almost all" means "all but finitely many". AxelBoldt

Yes, I think they should be separated. I've seen "almost all" used in a measure-theoretic sense, but it's not the usual usage. --Zundark, 2002 Mar 16

The real problem is that both "almost all" and "almost everywhere" are used in all of these contexts. Each of these is a special case of the most general situation (to my knowledge) where such terms are used -- the case where one has an ideal in a power set (or other Boolean algebra). Another example, used in Baire category theory, is meagre sets in a topological space. So these eventually need to be combined, but in conjunction with a rewriting of the article that describes the general case -- and yet gives precedence to the situations where one is most likely to see the phrases (such as measure theory). I'll have to do this someday, but we're probably OK for now. -- Toby 20:09 Feb 12, 2003 (UTC)

Is the 'perverse' statement correct?

The statement

'Perversely, if we allow "almost all" to mean "all but a countable set", then it follows that almost all prime numbers are even, since the set of all prime numbers is itself countable.'

looks debatable. It relies on the assumption that the countable set referred to in 'all but a countable set' is the set of all odd prime numbers. If the countable set referred to in 'all but ...' were the set of even primes, the statement 'almost all prime numbers are odd' holds. Conversely, if one chooses the set of odd primes, the statement 'almost all primes are even' holds. The conclusions I would draw from that are i) The statement is not sound as written and ii) defining 'almost all' as 'all but a countable set' depends on some ancillary statement about _which_ countable set is excluded. This could be fixed by extending the statement somewhat to include the issue above - but only if I'm more or less right about it... SLR Ellison (talk) 12:23, 21 June 2017 (UTC)[reply]

"a", in "all but a countable set", means "some", "any"; or more wordy, "there exists some countable set so that ...". Because of this existential quantification, it is not possible to have the truth of the statement "depend on ... _which_ countable set is excluded". Still, explaining that there is a quite implicit existential quantification in "all but a countable set" would help many people reading this to understand what's going on.

Also, I would heartily suggest to replace the word "perversely" (which does not transport any meaning except the emotion some writer had here) with "unfortunately" (which implies that this is a possible source of misunderstandings). --User:Haraldmmueller 18:49, 29 July 2017 (UTC)[reply]

"could mean" -> "can mean"?

Could "could mean" be replaced with "can mean" everywhere (except in the introduction)? "Could mean" sounds like "but there are strings attached to what I'm going to tell you, and I won't tell you about them - hehe" - which is mean for a dictionary. "Can mean" means "sometimes it actually means what I'm telling you - no strings attached". --User:Haraldmmueller 06:48, 31 May 2018 (UTC)[reply]

@Haraldmmueller: sorry for that, and for not noticing your request earlier. I'm just used to write that way. Is there anything else in the article that bothers you? Please ping me when you reply, this page is on my watchlist but I (almost) never check it. Professor Proof (talk) 18:49, 10 June 2018 (UTC)[reply]

“The Cantor set is also null”

Should this be “The Cantor set is almost null”?

Not a mathematician, but it seems wrong as written (since article is about “is almost”, and based on looking at Cantor set, which seems to not be null). 24.217.225.229 (talk) 09:12, 25 May 2023 (UTC)[reply]

The Cantor set is in fact a null set, i.e., it is a Lebesgue measurable set, and its Lebesgue measure is zero. ("The Cantor set is null" is math jargon for "the Cantor set is a null set", not for "the Cantor set is the empty set". I think this could be better phrased in the article.) -- Tea2min (talk) 06:11, 26 May 2023 (UTC)[reply]