Wikipedia:Reference desk/Archives/Mathematics/2008 June 3

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June 3

Non measurable sets having continuous boundaries

Is there any example of a non-measurable set in the plane whose boundary is a continuous closed curve?--Pokipsy76 (talk) 07:09, 3 June 2008 (UTC)

Yes. You can choose a Vitali set which is dense in [0,1]. Cross this with the interval, and we get a non-measurable subset of [0,1]x[0,1] whose boundary is the whole square. But [0,1]x[0,1] is a continuous closed curve. Algebraist 10:25, 3 June 2008 (UTC)

Ok, let's consider a more restrictive hypothesis: the boundary must be a Jordan curve, that means it must also be non self-intersecting. Can we still find an example?--Pokipsy76 (talk) 13:31, 3 June 2008 (UTC)

Let $B$ be a set in the plane with boundary being some set $C$. Suppose we are given that $B$ and $C$ are disjoint. This implies that $B$ is open, and thus Lebesgue measurable.

If $C$ is a Jordan curve, then we know from the Jordan curve theorem that its complement in the place is two connected components, one unbounded component $A$ and one bounded component $B$, whose boundaries equal $C$. By definition $B$ and $C$ are disjoint, so the above argument applies and $B$ is Lebesgue measure. Eric. 144.32.89.104 (talk) 14:35, 3 June 2008 (UTC)

I don't follow your argument. By what definition are B and C disjoint? You've assumed they are, but you defined B to be any set in the plane... Doesn't that proof still allow for a non-measurable set which is the union of an open set and part (or all) of its Jordan curve boundary? I suspect that it isn't actually a problem (I haven't really studied any measure theory, so I'm not 100% sure), but it still needs to be addressed. --Tango (talk) 16:08, 3 June 2008 (UTC)

Suppose B is non-measurable and has boundary C, a Jordan curve. By subtracting the interior of B, we may assume that B is contained in C. Since Lebesgue measure is complete, it's enough to show that Jordan curves have zero area. Is this true? Algebraist 18:16, 3 June 2008 (UTC)

JSTOR tells me it is not. Damn. Algebraist 18:17, 3 June 2008 (UTC)

I was thinking the same thing, but didn't know enough measure theory to know either way. I know not all curves have zero area (Peano curves being the obvious example), but wasn't sure about Jordan curves. Did your investigation find a example of a such a curve? Although, we don't actually need the measure to be zero, as long as it is measurable, correct? Did JSTOR tell you they weren't necessarily measurable, or just that the measure wasn't necessarily zero? --Tango (talk) 18:42, 3 June 2008 (UTC)

A Jordan curve is measurable (it's closed, after all), but it is not obvious to me that any dense subset of one is measurable, which is what we need. Any subset of a zero area set is measurable, so I was hoping for that. Yes, Osgood's ancient paper gives a construction, though I didn't read it. Algebraist 18:45, 3 June 2008 (UTC)

Assuming property 15 in Lebesgue measure is true, we're fine (any Jordan curve is a continuous injective image of the circle, all of whose subsets are measurable). That property is neither obvious nor referenced. Algebraist 19:01, 3 June 2008 (UTC)

Well, this conclusion seems to be incompatible with the assertion that there's a Jordan curve of positive area -- any set of positive measure has a non-measurable subset, in fact a subset of inner measure zero and outer measure equal to the measure of the original set.

Are you sure that any Jordan curve is a continuous image of a circle embedded in the plane? You seem to need that to get your conclusion from the property 15 stated above. But I think the stated property is just false -- there are continuous (in fact order-preserving) injections that blow the middle-thirds Cantor set up to a set of positive measure, so if you take a non-measurable subset of that set and pull it back, you refute the claim. --Trovatore (talk) 03:28, 4 June 2008 (UTC)

Actually I seem to have misread it -- I thought it was talking about continuous functions from Rⁿ to Rⁿ. It doesn't matter; it's wrong in either case. I went ahead and took it out. --Trovatore (talk) 03:36, 4 June 2008 (UTC)

So we know that any subset of a Jordan curve is measurable and this completely solve the problem. However I'm still wondering if a Jordan curve can really have non-zero measure...--Pokipsy76 (talk) 19:24, 3 June 2008 (UTC)

Go read A Jordan Curve of Positive Area, William F. Osgood, Transactions of the American Mathematical Society, Vol. 4, No. 1 (Jan., 1903), pp. 107-112 if you really want to. Unfortunately JSTOR doesn't have the diagrams, and I don't feel up to understanding it without them. Algebraist 19:32, 3 June 2008 (UTC)

Tango, yes for some reason I was (incorrectly) assuming that the set whose boundary was a Jordan curve was necessarily one of the two connected components produced by Jordan's theorem. But fortunately Algebraist has sorted it out. Eric. 144.32.89.104 (talk) 21:16, 3 June 2008 (UTC)

The answer to the original question is YES:

1. Any Lebesgue set of positive measure has non-measurable subsets.
2. There are Jordan curves with positive Lebesgue measure.
3. Let $\gamma$ be a Jordan curve with positive Lebesgue measure and let $A$ be a non-measurable subset of $\gamma$, then the domain surrounded by $\gamma$ union with $A$ is a non-measurable set as required.

I'll be happy to provide a reference or proof to any of these claims apon demand. Oded (talk) 05:07, 4 June 2008 (UTC)

Yes, I thought your (1) was true, but I couldn't find a reference online. Could you provide one, so I can add it to Lebesgue measure? Algebraist 09:42, 4 June 2008 (UTC)

While you are at it, you may consider adding 2 to Lebesgue measure. There are references for it at books.google.com and curve mentions it. I don't know a reference for 1, but it is easy to prove. First, we construct a measure preserving transformation from the set to an interval in R. Then argue that the pre-image of a non measurable set in the interval has to be non-measurable as well. (Alternatively, perhaps one of the proofs of the existence of non-measurable sets in R applies directly to this case as well.) I got to go now, but I'll come back to this with more details soon. Oded (talk) 10:25, 4 June 2008 (UTC)

So does that mean property 15 in Lebesgue measure isn't actually true? --Tango (talk) 12:03, 4 June 2008 (UTC)

Yes. Trovatore has removed it. Algebraist 12:07, 4 June 2008 (UTC)

I don't have a reference, but I can give you a proof: Let A be a set of positive measure, and together wellorder the open sets of measure less than m(A) and the closed sets of positive measure, in order-type $2^{\aleph _{0}}$. Now we're going to build up disjoint sets B and C by transfinite recursion. When you hit a closed set K of positive measure, pick an element of K that is not yet committed, and throw it into C (guaranteeing that K will not be a subset of B). You can do that because so far you've committed fewer than $2^{\aleph _{0}}$ points, and K has cardinality $2^{\aleph _{0}}$. Similarly, when you come to an open set U, pick an element of A\U that is not yet committed, and throw it into B, guaranteeing that B will not be a subset of U.

At the end, B is a subset of A that has no closed subsets of positive measure, so it has inner measure 0. Also, B is not contained in any open set of measure less than m(A), so B has outer measure at least (and therefore exactly) m(A). --Trovatore (talk) 17:51, 4 June 2008 (UTC)

Thanks. Algebraist 21:22, 4 June 2008 (UTC)

By the way, in answer to your question above, it is the case that for any Jordan curve in the plane, there is a homeomorphism of the plane that maps the unit circle onto your curve. This is the Jordan–Schönflies theorem. Algebraist 21:27, 4 June 2008 (UTC)

I think the following proof is easier and more transparent than the one by transfinite induction. (I mean a proof that any measurable set of positive Lebesgue measure has a non-measurable subset.) Let A be the set, and with no loss of generality assume that A is bounded (this is anyway the case in the present application). Say that two points in A are equivalent if the difference between them is a vector with rational coordinates. Let B be a subset of A which contains one element from each equivalence class. (I guess that the existence of B is proved using Zorn's lemma, the axiom of choice is used here.) We claim that B is not measurable. Fix a bounded open set U, and let U' be the set of points in U with rational coordinates. For u in U' set B_u := B+u. Then these sets are disjoint by construction. If they were measurable, the union would have measure equal to the sum of the measures, and this has to be zero or infinity, since the measure of each B_u is the same as that of B. But now we get a contradiction, since if U is sufficiently large, then on the one hand the union of all the B_u (for u in U') contains A, while on the other hand it is bounded. Thus, at least on of the B_u is non-measurable, which implies that B is non-measurable. (This proof mimicks one of the standard constructions of a non-measurable set in R.) Oded (talk) 21:57, 4 June 2008 (UTC)

Thanks again. You don't need Zorn there, by the way: it's just a direct application of the axiom of choice. Algebraist 22:05, 4 June 2008 (UTC)

Hmm -- it's a nice proof, and I suppose it is a little easier. But more transparent? With the transfinite recursion proof, you just figure out what you need to do (refute all possible sets witnessing positive inner measure, or outer measure less than m(A)), and straightforwardly deal with them one at a time. It also gets you more information. There's a diffidence about transfinite recursion in a lot of circles that I don't really understand. I think it's a basic technique that every mathematician (well, at least every mathematician who deals with infinite structures) should know. --Trovatore (talk) 22:52, 4 June 2008 (UTC)

Agreed. I also agree with your sentiment regarding transfinite recursion and with your observation that some mathematicians disdain it. I guess it all depends on one's background and taste. Oded (talk) 04:23, 5 June 2008 (UTC)

Set Notation, bis

In a previous question on this reference desk, someone asked about set notation. One thing that ended up being mentioned was this:

$\sum _{d\in D,\;dBc}\ldots$

I've seen things that look like that on various Wikipedia articles, but my question is, what exactly does notation like that mean? Digger3000 (talk) 14:03, 3 June 2008 (UTC)

It means "sum over all the elements of D that are related to c by the relation B". --Tango (talk) 14:09, 3 June 2008 (UTC)

Perhaps slightly more generally, "sum over all the elements of D satisfying dBc". Of course, in our case dBc means "d is related to c by the relation B". -- Meni Rosenfeld (talk) 14:36, 3 June 2008 (UTC)

And what is a relation, exactly? Digger3000 (talk) 20:37, 3 June 2008 (UTC)

Well, we do have an article about everything. The short version: Things like "=" and "<" are relations. -- Meni Rosenfeld (talk) 20:41, 3 June 2008 (UTC)

question about equations and expressions

What is the difference between an equation and an expression?Could I solve for a varible in an expression?Could I solve for a varible in an equation? —Preceding unsigned comment added by Lighteyes22003 (talk • contribs) 14:46, 3 June 2008 (UTC)

Take a look at equation and expression. It may be helpful to think of an "equation" as a complete sentence, and an "expression" as a noun: for example, the equation "8 = 3 + 5" is like a complete sentence asserting that 8 is the sum of 3 and 5, whereas the expression "3 + 5" is like a noun, representing the sum of 3 and 5. After all, in English the phrase "the sum of three and five" is a noun, and the phrase "the sum of three and five is equal to eight" is a sentence. Eric. 144.32.89.104 (talk) 15:06, 3 June 2008 (UTC)

Put simply, an equation has a equals sign in it (hence the name). An expression is a more general term, equations are examples expressions, but so is pretty much any other meaningful sequence of mathematical symbols. "Solving" doesn't make sense for general expressions. You can solve an equation, a system of equations, an inequality or system of inequalities, and probably a few other types of expression, but not all types (and even then, they need to contain a variable, 1+1=2 is an equation, but it would be meaningless to try and "solve" it). Of course, even if you have an equation with a variable, that doesn't mean it actually has a solution. There are no real (or even complex) solutions to "e^x=0", for example, but that's a perfectly valid equation. --Tango (talk) 16:00, 3 June 2008 (UTC)

Also, an equation too might not contain any variables, so you won't be able to solve 1+1=2. -- Xedi (talk) 21:59, 3 June 2008 (UTC)

I said that... I even used that example... --Tango (talk) 23:49, 3 June 2008 (UTC)

Haha, sorry, I must've read your answer very absent-mindedly. --Xedi^Talk 05:49, 4 June 2008 (UTC)

Expressions are phrases made up of combinations of variables. Equations represent relationships that expressions obey.--Fangz (talk) 22:43, 3 June 2008 (UTC)

Combinations of variables, constants, operators, etc. --Tango (talk) 23:49, 3 June 2008 (UTC)