Talk:Null set

This definition doesn't apply to signed measures; I'll correct this.Perhaps more controversially, I'll let not require null sets to be measurable. Does anybody prefer a reference where null sets must be measurable? Something with more than just a quick definition for possibly incomplete measures that is then applied only to complete measures (where null sets are measurable anyway). -- Toby 01:37 Mar 12, 2003 (UTC)

I don't know what you're talking about...

I don't like that one of the words in the term null set is used in the definition of null set. After reading the page I still don't understand what a null set is. Isn't there a simpler way of explaining it to people who don't speak math? —Preceding unsigned comment added by 24.185.151.226 (talk) 20:45, 6 June 2007

I think the best definition for a layperson is something like the first phrase of the article: "In measure theory, a null set is a set that is negligible for the purposes of the measure in question." I guess it would be possible to expand on it a bit, but there's a difficulty. A null set is a set which is negligible "in some sense". There are many ways in which sets can be negligible (the Baire category theorem considers category 1 sets to be negligible, measure theory considers measure 0 sets to be negligible, etc...) These nuances are way beyond the scope of this article, and it would ultimately be inappropriate to do the measure theory article in this article, just to explain what a null set is.

You say that there is a cyclic definition somewhere, but I cannot find it. Can you please point to it more precisely? Loisel 21:47, 6 June 2007 (UTC)[reply]

Null set/zero measure

Is there a difference between the two?

From Null set: "any set of measure 0 is called a null set", which sounds like this is the definition of a null set.
From Measure zero: "Any set of measure zero is a null set. The opposite is not true, because a null set is not required to be measurable."

Which is correct? --Zvika 07:09, 3 September 2007 (UTC)[reply]

Hmm -- this sounds like a "some authors say this, some other authors say that" situation. In practice it's rare to consider measures that aren't complete (in the sense that every subset of a measure-0 set is measurable). For complete measures there's no difference between "measure-zero set" and "subset of a measure-zero set", so "null" could be defined either way. In the rather pathological case of non-complete measures, I don't know whether there's a uniform standard definition or not, nor which of the two it would be. --Trovatore 07:40, 3 September 2007 (UTC)[reply]

Still, we ought to maintain consistency within Wikipedia. I think the articles should be merged, and all commonly used definitions should be mentioned. --Zvika 09:54, 3 September 2007 (UTC)[reply]

Merge is a good idea. The difference of the two definitions of null set is so small that it can be explained easily in the article.wshun 01:38, 1 October 2007 (UTC)[reply]

I performed the merge per above. --Zvika 08:05, 5 October 2007 (UTC)[reply]

null sets that are not measurable! what?

I'm going to take issue with this:

"A nonmeasurable set is considered null if it is a subset of a null measurable set. Some references require a null set to be measurable; however, subsets of null sets are still negligible for measure-theoretic purposes."

What they are really saying is that were going to be lazy and use the completion of the measure without explicitly saying so. I say this is silly. It is much more sound to define null as a measurable set of measure 0, note that if a measure is complete that any subset of a null set is thus also null, and further that every measure has a completion. --Ray andrew (talk) 00:53, 13 March 2008 (UTC)[reply]

Note that it says "for measure-theoretic purposes". I think in general measure theory doesn't really bother much with measures that aren't complete, or more precisely, doesn't bother to distinguish them from their completions. What would be the point? --Trovatore (talk) 01:03, 13 March 2008 (UTC)[reply]

These distinctions are good for technical reasons. With the Lebesgue measure, composition of measurable functions with continuous functions is not necessarily measurable. [1] This is chiefly a property of the fact that some "bad" Borel sets are included by the completion process. By contrast, the composition of Borel measurable functions is Borel measurable. [2] For these reasons, it is common to work with Borel measures and not their completions. But these null sets are also useful, and in particular in Stochastic calculus, there is often a need to have lots of different sigma algebras, but a common notion of null set. So that's that. Loisel (talk) 21:49, 13 March 2008 (UTC)[reply]

But that's all stuff about the sigma-algebras. For sure we don't want to elide the distinction between, say, Borel sets and Lebesgue measurable sets. But we don't code that distinction into the measures, we talk about it directly. There is no useful reason that I can see to make a distinction between Lebesgue measure on the one hand, and the restriction of Lebesgue measure to the Borel sets, on the other. --Trovatore (talk) 02:19, 14 March 2008 (UTC)[reply]

I'm not sure what you're saying, but I think the literature admits that null sets can be non-measurable. Loisel (talk) 03:07, 14 March 2008 (UTC)[reply]

Well, it seems to be a language issue, mostly. I would always call a subset of a null set "measurable", even if it's not in the domain of the measure currently being considered, because (e.g.) there's only one value you could assign for its measure. That's different from the case, say, that

2^{\aleph _{0}}

happens to be a real-valued measurable cardinal, in which case there is an extension of Lebesgue measure whose domain is the whole of P(R), but presumably there are different extensions giving different measures to non-Lebesgue-measurable sets.

So what I'm saying is that if I want to address the question of whether a set is in a certain sigma-algebra, I wouldn't use the word "measurable" to do it (unless the sigma-algebra happens to be the domain of a complete measure).

But yes, I would always call a subset of a null set "null", if that's what you're asking. --Trovatore (talk) 07:53, 14 March 2008 (UTC)[reply]

Measurable space

The user http://en.wikipedia.org/wiki/User:Slawekb has taken it upon himself to "monitor" my contributions to Wikipedia Math. However, the definition I am using is consistent with the well documented page http://en.wikipedia.org/wiki/Measure_(mathematics), as well as just about any textbook I've ever seen on the subject. LoveOfFate (talk) 01:10, 22 December 2012 (UTC)[reply]

There is nothing wrong with saying "Let X be a measurable space." This is like saying "Let G be a group" or let "X be a topological space". To insist on emphasizing the sigma algebra in expository mathematics is needlessly pedantic. The appropriate place to emphasize such details is at the measurable space article, not here where the symbol Σ isn't even used. It's a basic principle of all mathematics writing to avoid introducing symbols unnecessarily. Sławomir Biały (talk) 02:08, 22 December 2012 (UTC)[reply]

Sławomir is correct here. It is standard to refer to a structure by the name of its underlying set, at least in contexts when there is no other structure around having the same underlying set to confuse it with. --Trovatore (talk) 02:32, 22 December 2012 (UTC)[reply]

Referring to a measurable space as

\textstyle (X,\Sigma _{X})

is more befitting an encyclopedia somehow, I think. It might be a bit pedantic, but this is a formal definition after all. It should be presented as verifiable sources do. It's also consistent with the rest of Wikipedia Math. LoveOfFate (talk) 17:00, 22 December 2012 (UTC)[reply]