Talk:Null set/Archive 1

Archive 1

Untitled

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)

_ I believe the person is talking about the fact that under the heading of Definition, the word "null" appears multiple times. Also, instead of explaining the meaning or the word, that section seems to express a whole formula for using a null set. Perhaps the mathmaticians' interpretation of the word "definition" is a little different than what we're expecting the word to mean? As for myself, when I saw in the introduction that the word "null" meant "negligable," I assumed at first that it was something like a remainder of .2 when splitting a whole item amoung four people: unimportant. Then I looked at the Definition section, and saw, appearently, instructions use.

_ Anyways, can we get a definition written for those of us who just want to know generally what people who say "null set" are talking about? For instance, I wanted to know what greater and less than signs were called when used like parentheses. I found "null set," but my husband told me that the term meant was something in math. So I looked it up, to learn what it is. Instead, I'm leaving this page with the impression that a negligable amount is not something left over, but something you want to work with ... and ... I don't know why you want to work with it ... it's unimportant ... right?

_ I guess we're asking, "Dilbert ..., where's Tina?" Pvtbuddie (talk) 06:30, 8 March 2013 (UTC)Pvtbuddie

(PS, They're called angle brackets!).

First of all, a null set is a set. It's important to understand what is meant by a set: it is a collection of mathematical objects. For example, perhaps we are working with sets of geometric points.

Now, let's suppose that we have a rectangle. Actually, we're going to view the rectangle as a set of points—the set that contains every point that is in the rectangle (both the boundary of the rectangle and its interior). Let's call this set R, for rectangle. Obviously R is an infinite set, i.e., a set with infinitely many points in it. It is meaningful to talk about the area of the set R, because the points in R form a rectangle, and we can compute the area of a rectangle by multiplying its length by its width. Let's say the area of the rectangle (and hence the area of the set R) is 1 square meter.

Next, suppose that we remove a single point from the set R. Put that single point into a new set, called S. So now the set R contains every point of the rectangle, except one; it's like a rectangle with an infinitesimally small pinprick in it. And the set S contains just one point. A single point has no area, so removing it from the rectangle does not subtract any area. The area of the leftover part of the rectangle (i.e., the area of the set R after we've removed the one point from it) is still 1 square meter, and the area of the single point (i.e., the area of the set S) is 0 square meters. So in this context we can say that the set S is a null set, not because it is empty (it does contain a point, after all), but because its area (the "measure" we are working with here) is zero.

In fact, we could remove two points from the rectangle, or three points, or even a million points, and the total area of the points we'd remove would still be zero, because each of the points individually has zero area. In the context of geometric area, any set of finitely many points would be a null set. In order to remove some actual area from the rectangle, we would have to remove some "shape" from it—for example, we could remove a smaller rectangle, or a triangle, or a circle. Shapes like this that have nonzero area (and therefore are not null sets) must have infinitely many points. (This is why you can't just add up the areas of all the individual points to get zero area for the whole shape—there are too many points.)

In most situations, if you are working with area, and you have some shape that has had a null set removed from it (like our rectangle with a single point removed), you can ignore the null set and pretend that the whole shape is there. The area is the same whether that single point is included in the shape or not. That's what is meant when it is said that null sets are "negligible."

So an example of a null set is a set of finitely many points, if the problem you're working on involves geometric area. Null sets also show up in other contexts, but the essential idea is the same. —Bkell (talk) 11:19, 8 March 2013 (UTC)

By the way, it's important not to confuse this idea of a null set with the idea of an empty set, even though some authors use the phrase "null set" to mean empty set. An empty set is just a set that has nothing in it. If you imagine sets as cardboard boxes that can hold mathematical objects (like points), then an empty set is the equivalent of an empty box. It's possible that whatever you were reading used the angle-bracket notation ⟨⟩ to represent an empty set. That's unusual; it would be more common to see curly brackets, like {}, or the empty set symbol ∅. But it would be really strange, I think, to see something like ⟨⟩ being used to represent a null set in the sense that I described above. —Bkell (talk) 11:37, 8 March 2013 (UTC)

Thank you, very much. I can understand this explanation, at least well enough not to have to look up every word in it. But it still looks like what you're talking about can't really be measured, so how is it that the definition section calls N a measurable set, then says than N is null?

(As far as the < > vs { } vs ∅, I was first looking up what < >'s would be called when used in writing, such as to set off a URL.) Pvtbuddie (talk) 20:46, 12 March 2013 (UTC)Pvtbuddie

A null set can be measured—it just has measure zero. —Bkell (talk) 23:45, 19 March 2013 (UTC)

Could Bkell's explanation be used as the basis for a "Layman's explanation" section in the article, or for a sub-section of "Definition"? Pvtbuddie (talk) 20:46, 12 March 2013 (UTC)Pvtbuddie

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)

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)

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)

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)

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

I appreciate your attempt to make a serious case without emotionalism, and I'll try to reciprocate.

It's not a formal definition in the sense of formal logic. It's a definition for human readers. Excessive formalism detracts from readability and adds hardly anything (especially in cases like this, where in your text, Σ_X would be introduced and then never used). I think you'll find plenty of sources that would introduce it without the Σ_X, and as for the rest of Wikipedia mathematics, most of us try to maximize readability as long as it's not at the expense of correctness. The principle does not, of course, get applied uniformly; different editors have different styles. --Trovatore (talk) 02:59, 23 December 2012 (UTC)