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August 28, 2010

The Difference between Measure Zero and Empty Interior

Posted by Tom Leinster

This is a post in the category of “Small things I have learned”.

Close your eyes and picture a set of measure zero, where by “set” I mean subset of n\mathbb{R}^n. Now open them again so that you can carry on reading…

Close them again and picture a set with empty interior.

For me, the two mental images are about the same. Should they be?

There are a couple of easy mathematical points. One is that a set of measure zero certainly has empty interior—because if not, it would contain a nontrivial cuboid (a 1,b 1)××(a n,b n)(a_1, b_1) \times \cdots \times (a_n, b_n). The other is that the converse fails: a set can have positive measure but empty interior (e.g. the irrational numbers).

What I only just learned is that there are compact sets with positive measure but empty interior. (I’ll give an example in a moment.) That might sound like an instantly forgettable technicality, but actually it’s the gnarly fact behind an important distinction. Exaggerating only slightly:

the difference between measure zero and empty interior is the difference between Lebesgue and Riemann integrability.

Before I get to that, here’s the promised example of a compact subset of \mathbb{R} with positive measure but empty interior. It’s easy, but it’s not really the main point of this post, so skip the next two paragraphs if you’re not in the mood for details.

Consider the classic Cantor set, obtained from [0,1][0, 1] by removing the middle open 1/31/3, then removing the middle open 1/31/3 of each of the 22 resulting intervals, then removing the middle open 1/31/3 of each of the 44 resulting intervals, and so on ad infinitum. This isn’t the example, since the Cantor set has measure zero: the set produced at the nnth stage has measure (11/3) n=(2/3) n(1 - 1/3)^n = (2/3)^n, which converges to 00 as nn \to \infty.

But we can adapt the construction: at the nnth stage, instead of removing the middle open 1/31/3 from each of the 2 n2^n intervals, remove the middle open x nx_n. Here (x n)(x_n) is any sequence of numbers between 00 and 11. So, the three occurrences of “1/31/3” in the first sentence of the previous paragraph become “x 0x_0”, “x 1x_1” and “x 2x_2”, respectively. The measure of the resulting Cantor-like set is n=0 (1x n), \prod_{n = 0}^\infty (1 - x_n), which is nonzero if we choose (x n)(x_n) to converge rapidly to 00. And our set certainly has empty interior, since the nnth stage contains no intervals of length > 2 n> 2^{-n}. For a particular choice of (x n)(x_n), our set is called the fat Cantor set.

What does this have to do with Lebesgue and Riemann integrability? The answer lies in the notion of ‘Jordan measurability’. A subset of n\mathbb{R}^n is called Jordan measurable if its characteristic function (indicator function) is Riemann integrable. You might expect this to be called ‘Riemann measurability’, but apparently not. It’s sometimes called ‘Peano–Jordan measurability’ too.

The crucial fact relating Jordan and Lebesgue measurability, at least for compact sets, is this:

A compact set is Jordan-measurable if and only if its boundary has Lebesgue measure zero.

An equivalent statement is that a compact set AA is Jordan-measurable if and only if Vol(A)=Vol(IntA)Vol(A) = Vol(Int A), where VolVol means Lebesgue measure and IntInt means interior.

But the boundary of a compact set is compact and has empty interior. So if every compact set with empty interior had measure zero, Jordan measurability would be the same as Lebesgue measurability. Conversely, if Jordan measurability were the same as Lebesgue measurability then every compact set AA with empty interior would satisfy Vol(A)=Vol(IntA)=0Vol(A) = Vol(Int A) = 0.

Conclusion:  “Measure zero” is different from “empty interior”—even for compact sets—precisely because Lebesgue measurability is different from Jordan measurability.

Posted at August 28, 2010 3:52 AM UTC

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28 Comments & 0 Trackbacks

Re: The Difference between Measure Zero and Empty Interior

I’m typing from home so don’t have access to MathSciNet, but if you go there and search for “Swiss cheese” then one should get plenty of evidence that compact planar sets with empty interior and measure zero have been of interest to people who like function algebras (or, at least, whose PhD supervisors like function algebras).

Posted by: Yemon Choi on August 28, 2010 6:42 AM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

Thanks! This amuses me, as there’s something called the Swiss cheese operad (and also a Swiss cheese multicategory), whose algebras are of course called Swiss cheese algebras.

At the time I was thinking about them most I was living in Paris, where the Swiss Cheese Board (which is about as ambiguous as “golf club”, but you know what I mean) happened to be running an advertising campaign to try to convince the French that Swiss cheese wasn’t just the holey stuff that cartoon mice eat. So there were posters all over the metro saying

Si c’est troué, c’est pas Suisse

—if it’s got holes in, it’s not Swiss.

By the way, I assume you mean “compact planar sets with empty interior and positive measure”, not “measure zero”…
right?

Posted by: Tom Leinster on August 28, 2010 8:14 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

By the way, I assume you mean “compact planar sets with empty interior and positive measure”, not “measure zero”

Oops. Yes, that is what I meant. Too much late-night typing…

Posted by: Yemon Choi on August 29, 2010 9:45 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

A compact set is Jordan-measurable if and only if its boundary has Lebesgue measure zero.

In fact, this generalizes very nicely from sets (indicator functions) to arbitrary functions:

A compactly supported function is Riemann integrable if and only if it is continuous except on a set of Lebesgue measure zero.

Posted by: Mark Meckes on August 29, 2010 6:23 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

That should be “A bounded compactly supported function”.

Posted by: Mark Meckes on August 30, 2010 4:39 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

For me, the two mental images are about the same. Should they be?

If I’m not being asked to accomodate any more topological regularity, I guess I think of these more or less as follows. I think of a set A nA \subset \mathbb{R}^n as n\mathbb{R}^n with some points removed. If AA has empty interior, that means every ball in n\mathbb{R}^n (or cuboid, if you prefer) is missing points. So the picture here is n\mathbb{R}^n with holes poked in every part of it.

If AA has Lebesgue measure 00, then every ball has holes poked in it, but furthermore the measure of those holes adds to the measure of the ball. So the subtly different picture is n\mathbb{R}^n with big holes poked in every part of it.

I think the example of the fat Cantor set fits well into this viewpoint: if the x nx_ns (which just measure the sizes of the holes we’re poking in [0,1][0,1]) are “large” we get a set of Lebesgue measure 00, and if they are “small” we get a set with positive Lebesgue measure.

I’m not sure how to relate this hand-waving to integrability, but there’s probably some nice way to do so.

Posted by: Mark Meckes on August 30, 2010 7:20 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

I think of sets of measure zero as being more analogous to meager sets than sets with empty interior. But, they’re different concepts.

Now open them again so that you can carry on reading…

How did you expect me to read this sentence with my eyes closed?

Posted by: John Baez on August 31, 2010 8:09 AM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

I thought that was obvious: use a screen reader!

Posted by: Mikael Vejdemo-Johansson on August 31, 2010 9:41 AM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

Posted by: John Baez on August 31, 2010 10:50 AM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

Justin Roberts made the following observation to me, something that sounds obvious in hindsight, and something that I’m probably misquoting:

Riemann integration suffices for geometry, whereas Lebesgue integration is necessary for probability theory.

Terry Tao made a more precisely nuanced point at his recent post The problem of measure.

I have pointed out elsewhere the fact that I have never taken a course in measure theory. This means I have very little intuition for these unusual, “non-geometric” sets like the Cantor set, fat or otherwise. However, this has not been a problem for me until recently. Now I’m coming across measure theory all over the place! It is therefore good to hear other peoples intuition, like how Mark thinks about these things.

It is perhaps worth pointing out a difference in the education of Tom and me. Tom did his undergraduate degree at Oxford, I did mine at Cambridge. At Cambridge we did Riemann integration in the first year: at Oxford they did Lebesgue integration in the first year. As a geometer/topologist I managed to avoid analysis in later years.

Posted by: Simon Willerton on September 16, 2010 11:48 AM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

A related point is that Riemann integration generally suffices for the purposes of applied mathematics. For example, (almost?) all the integrals one encounters in classical physics are Riemann integrals. Open up, say, an undergraduate text on electromagnetism and see how they derive an integral formula for the electric field at a given point resulting from a charge distribution. They chop up space into little pieces, approximate little pieces by point sources, add them all up and call the result an integral. If you make this more precise, you’re taking a limit of Riemann sums.

I’ve never seen an integral in classical physics that is more naturally derived or interpreted as a Lebesgue integral (i.e., starting by chopping up the codomain instead of the domain), but I’d be interested if anyone knows an example.

Posted by: Mark Meckes on September 16, 2010 3:43 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

Mark wrote:

Lebesgue integral (i.e., starting by chopping up the codomain instead of the domain)

to contrast it with the Riemann integral.

Could you expand on that?

I can certainly see that Riemann integration involves chopping up the domain. But why do you say that Lebesgue starts by chopping up the codomain?

Posted by: Tom Leinster on September 16, 2010 7:38 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

I don’t understand that either. Lebesgue integration chops up both the domain and the codomain. Indeed the integral of a positive function f:Xf : X \rightarrow \mathbb{R} is just the measure of {(x,y):0yf(x)}\{ (x,y) : 0 \le y \le f(x) \} in the product measure on X×X \times \mathbb{R}.

Posted by: Tom E on September 16, 2010 8:24 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

Once upon a time I convinced myself that what when you define Lebesgue integration for \mathbb{R}-valued functions, what really matters about \mathbb{R} is that it’s equipped with the operation of mean (or more generally, weighted mean). This is a form of chopping up. So I might have an idea of what Mark means; but I’ll wait and see what he says.

Posted by: Tom Leinster on September 16, 2010 8:44 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

I’ll start by addressing Tom E’s point. Both Riemann and Lebesgue integration involve chopping up both the domain and the range. The difference is which comes first. I’ll keep the rest very heuristic, since it’s easy enough to look up the technical details anyway. I’ll assume that the range is a bounded subset of \mathbb{R}.

In Riemann integration, you partition the domain into nice small sets, typically intervals (in one dimension) or rectangular prisms of (roughly) equal size. You then approximate the function by a constant on each of these small sets. So you’ve approximated the range by only finitely many values, which amounts to partitioning the range into bounded intervals. The sizes of these intervals vary in general, but not too wildly if the function is nice. The function itself is now approximated by a step function, and it’s clear what the integral of a step function should be.

In Lebesgue integration, you begin by partitioning the range into nice small sets, typically intervals of equal size. You then approximate the values of the function by one fixed value in each of those intervals. This gives you a partition of the domain into sets on which the function is constant. The sizes of these subsets of the domain may vary a lot, but if the function is nice (measurable) you can at least talk about their sizes sensibly. The function itself is again approximated by a step function.

Posted by: Mark Meckes on September 16, 2010 10:36 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

That’s interesting, I’ve never thought about it that way. In the Lebesgue case you take a nice partition of the codomain, and see what partition this gives you of the domain. In the Riemann case you take a nice partition of the domain (which doesn’t automatically give you a partition of the range).

Posted by: Tom E on September 17, 2010 10:01 AM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

Thanks a lot, Mark.

Posted by: Tom Leinster on September 18, 2010 2:08 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

Indeed. The first course I ever lectured was an applied maths course which involved lots of calculations of things like volumes of rotation and moments of inertia in this chopping-up fashion by writing “Now let δxdx\delta x \to dx”. Lots of fun.

Posted by: Simon Willerton on September 16, 2010 6:16 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

I’ve been waiting to see if any of the comments elaborated on what the demands on integrals that probability has that geometry doesn’t, but there’s not been anything directly referencing it. So I’ll just see if I’ve got the right end of the stick:

Geometry, and to a certain extent physics, is concerned with taking local properties (“quantity of stuff” [volume, including “accumulating over time”], moments of inertia/charge distributions, etc) which for which one wants integrals which, in a metaphorical sense, grow local summaries into global summaries. The Riemann integral suffices for this. In probability one quickly becomes interested in the “relative volume” satisfying conditions which give rise to intricate structure. The Riemann integral process doesn’t capture these sort of details. (Presumably the same thing arises in studying integrals on non-linear dynamical systems systems.)

Is this what lies behind the thought, or have I missed something?

Posted by: bane on September 19, 2010 1:46 AM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

I don’t fully understand what you’re trying to say, bane, so I’ll present what I understand about the difference.

The most basic property that fails for Riemann integration but that we need for probability theory is the monotone convergence theorem. Roughly speaking we need to be able to “integrate pointwise limits of integrable functions”. This seems to be something that is never necessary in geometry without some additional structure (uniform convergence say).

Posted by: Tom E on September 19, 2010 11:20 AM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

Tom E points out one technical difference between what geometry and probability want from integrals. Here’s a closely related point in broader philosophical terms. In geometry the domain of an integrand is the object of interest; in probability it’s the domain. In particular, in geometry you can, and usually do, restrict attention to functions which have suitably regular local behavior (where “local” refers to points in the domain). In probability you only care about the distribution of a random variable; its local behavior is outside the scope of probability theory, and in any case may make no sense to talk about at all because there is typically no metric or topological structure on the domain. Thus you need a method of integration that puts primary attention on the range instead.

Posted by: Mark Meckes on September 19, 2010 2:06 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

Thanks to both of you for the insight. What I was saying in a garbled way was that people ask “What’s the probability that a point in 3-D unit cube lies within the solid part of the Sierpinski gasket?” whereas they don’t ask “What’s the moment of inertia of the Sierpinski gasket?” (or even, integrate the equation of motion of a particle moving under a weird gravity that only exerts a force when the current time lies in the cantor set”) But what you seem to be saying is that it’s not just that probability naturally throws up a desire to integrate over complicated sets, but also that there are technical criteria that are needed to compute various probabilistic entities by interchanging limits and integrals, etc.

(When I deal with probability it’s mostly in the context of proability on combinatorial/discrete structures so this stuff never comes up for me so I was curious. And for the same reason as Simon, I never learned this stuff originally.)

Posted by: bane on September 19, 2010 6:37 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

bane wrote:

they don’t ask “What’s the moment of inertia of the Sierpinski gasket?”

But fractals—or at least, things very like fractals—do appear in some parts of physics, e.g. Brownian motion. I wonder whether the physical study of Brownian motion has ever needed the Lebesgue theory.

Posted by: Tom Leinster on September 19, 2010 7:09 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

More than that! It’s interesting to consider Brownian motion on fractals. Brownian motion is generally bound up with a Laplacian, so this is basically solving the heat equation on something like the Sierpinski gasket. There does at least seem to be, then, a wish to think of physical geometric things being done to these non-physical objects.

See for example: http://www-an.acs.i.kyoto-u.ac.jp/~kigami/AOF.pdf

Posted by: Tom E on September 19, 2010 7:23 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

It’s mathematically interesting, but what I was trying to get at was: is it physically interesting? Mark wrote:

Riemann integration generally suffices for the purposes of applied mathematics. For example,
(almost?) all the integrals one encounters in classical physics are Riemann integrals.

So what I wanted to know was whether physicists might ever need Lebesgue integration in order to handle Brownian motion.

Posted by: Tom Leinster on September 19, 2010 8:18 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

I’d rephrase that last question: is Lebesgue integration necessary to make rigorous some calculations about Brownian motion of interest to physicists? I suspect for instance that most physicists don’t know or care that when they use L 2L^2 theory in quantum mechanics they’re implicitly using Lebesgue integrals instead of Riemann integrals.

Posted by: Mark Meckes on September 19, 2010 8:48 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

I think I’ll have to think harder about what this question actually means … but one immediate thing to think about is whether you can establish the Feynman-Kac formula without Lebesgue integration. It relates the heat equation to diffusions, so is certainly very physical.

I’m not even sure you can do continuous random processes at all with Riemann integration, but you may be able to “simulate” them via their finite dimensional distributions.

Posted by: Tom E on September 19, 2010 8:49 PM | Permalink | Reply to this

Re: The Difference between Measure Zero and Empty Interior

So what I wanted to know was whether physicists might ever need Lebesgue integration in order to handle Brownian motion.

Sure. Under physicists’s Wick rotation aka analytic continuation quantum mechanics transmutes into stochastical mechanics and the transition is used all the time.

Heat kernel techniques and all the functional analysis that goes with it is probably applied even more often in pyhsics to stochastic processes than to genuine quantum ones – for the simple reason that there the math is better under control.

Posted by: Urs Schreiber on September 19, 2010 11:45 PM | Permalink | Reply to this

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