Real Analysis Primer, part 2: Lebesgue Measure.
May 27, 2011
Review and Motivation!
Last time we constructed the notion of a measure in a pretty general setting. What was it? It was a function from a nice set of subsets (a sigma-algebra) to the non-negative reals union infinity that satisfied a relatively reasonable condition: if we were to union up a bunch of disjoint measurable sets, the measure of the union should be the sum of the measure of the sets. That’s a pretty general kind of function. But even the most unobservant of readers wouldn’t be able to help but notice my lack of creativity when talking about measures — they all related to the real line! The reader should note at this point that there are other kinds of measures out there; in fact, there are measures which are wild and crazy and have nothing to do with the intuitive notion of "length"! We may be touching on some of these measures in later posts.
The real line is comfortable. We know a lot about it. We can picture it. We can even apply it to things in the real world. Thus, if we start talking about measures, a good place to start building them would be on the real line. And if measures are supposed to, somehow, be this generalization of "length" or "size", then what better a way to make a measure on the subsets of the real line than to have our measure give us the total length of the subset!
And this is the general idea behind Lebesgue measure. So what’s the big deal? Well, remember when we defined measures before, we needed to define them on a sigma-algebra. It turns out that this kind of measure doesn’t work so well on the powerset of the reals: indeed, there are subsets of the real line which are not measurable (!) in the way that we just vaguely defined it. The first time I learned about this, it blew my mind. And it still does. How could something so simple go wrong?
Maybe we are expecting too much. Let’s start modestly. At the very least, we can talk about intervals — we know what length we’d like to assign to them.
[In this post, I have made liberal use of additional notes, which I will put in-between braces and in italics, just like the sentence you are reading right now! These additional notes can be skipped with no major loss of understanding the subject, and only provide additional rigor or verify statements which are necessary but break the flow of the post. As I’ve mentioned before in this blog, my intention is NOT to write the Great American Mathematics text book, but simply give motivation and a general argument as to how we do what we do. Last, this post, for the curious reader, follows Kolmogorov’s Introduction to Real Analysis text, section 25. My exposition is slightly different, but I don’t think there should be any problems with it.]
Outer Measure, Inner Measure, Lebesgue Measure.
Instead of working on the real line, which can get tricky with things that are infinitely long, let’s just work on the unit interval . Now, what do you usually do when you measure something? You get out a ruler. There are many different rulers of many different sizes out there, and we’re going to essentially make a whole ton of rulers on the real line.
Definition. A subset is called an elementary set if can be represented (not necessarily uniquely) as the union of an at most countably infinite number of disjoint intervals, where the intervals may be open, closed, or half-open.
For example, is elementary. So is . So is . It should be clear that if we union, intersect, or take the difference of elementary sets we will still get an elementary set. This should begin looking something like a sigma-algebra to you.
Now, elementary sets are nice because they’re just unions of disjoint intervals. At this point, let’s define something which is ALMOST a measure but not quite, which we will write with an ominous upper-star, the following way:
Definition. The outer measure of an interval (half-open, open, or closed) with endpoints is defined to be the difference of the endpoints and is denoted . We also define .
For example, , , and . The last example there emphasizes that a point IS an elementary set, and that its outer measure is 0. Notice that we have only defined this outer measure for intervals, but we can easily extend this to elementary sets.
Definition. The outer measure of an elementary set is as follows: decompose into disjoint intervals and define .
Good. So now we have something which is almost a measure which can measure the lengths of intervals and things which look approximately like finite sets of disjoint intervals. This is a good start, but there are an uncountable number of subsets of the real line that don’t look like this. We need to do a bit better.
[Another aside here: we have not shown that the outer measure of an elementary set is well-defined; in other words, we have not shown that it does not depend on how we break up our set. In fact, it does not matter, and it is not hard to show that it does not. The proof is given in Kolmorogov’s Introductory Real Analysis text, page 256. The argument goes like this: suppose we have two different pairwise disjoint interval decompositions of our set. Let’s call them and . Then we note that the set ranging over all possible values of and are disjoint intervals (and potentially empty sets) which union to our set. Thus, and so this notion is well-defined.]
The way we can extend this notion from elementary sets to any set is essentially the same way we use rulers to measure things in real life. Let’s do the following thought experiment: suppose you have a ton of rulers, but instead of being marked at intervals you only knew the length of the entire ruler. One way that you could measure something is to take out a ruler and see if your object was smaller than it. If so, you know that the measure of the object is less than that ruler. Continue doing this. Maybe you don’t have a ruler that measures exactly the size of your object, but you can get arbitrarily close (with respect to your rulers). You ought to just take an estimate that says your object is a little bit smaller than all the rulers that are bigger than it: in mathematical terms, this is called the infimum of a set. Using the same idea as just described, we make the following definition:
Definition. Define the outer measure of an arbitrary set to be where is an elementary set.
If we didn’t already define a point to be the trivial interval, we could prove that a point has outer measure zero by the following argument. Without loss of generality, let’s prove the point has outer measure zero. Clearly, is covered by for every , and each of these intervals has outer measure . But the infimum of this set is 0, and since measures must be nonnegative, it must be the case that the outer measure of a point is 0.
Outer measure is nice because every subset of the unit interval has an outer measure. But why do I keep saying it is not quite a measure? Well, good readers, because it turns out that it is not additive in the way that we have defined last post. It is unfortunate, but true. In particular, there are disjoint sets and such that . This is a particularly sad state of affairs, but I have not introduced this almost-measure in vain. Indeed, we are working our way towards something much greater.
A point a few paragraphs ago when we tried to motivate the outer measure, an angry reader may have shouted: "Why take rulers BIGGER than the object? Why not take them smaller?" Indeed, we can do this as well. Since we have already defined outer measure, we note that if we have some set in , then measuring it with elementary sets smaller than it is the same as measuring the compliment of the set in with elementary sets bigger than it. Think about this for a second.
Definition. We define the inner measure of a set to be .
It is an elementary exercise to show that the inner measure is always less than or equal to the outer measure; indeed, using the rulers thought experiment above, you can intuitively see why. We simply state this.
Fact. For all sets , it is the case that .
It is NOT always the case that the reverse equality is true, though the reader might not be able to see why using examples from the real world. Shouldn’t we always be able to "hone in" on what the actual measurement should be? In fact, it is not always the case on the real line: there are weird sets (called "non-measurable sets") which have outer measure which is strictly greater than their inner measure! It is these same non-measurable sets that mess up the outer and inner measure’s additive properties so that they cannot be considered measures in the sense of the last post. That’s irritating! Why can’t we just get rid of non-measurable sets? And, indeed, the Lebesgue measure does exactly this; let’s define it now.
Definition. A set is Lebesgue measurable if . We denote the Lebesgue measure by .
One important thing to note is that while every subset of the unit interval has an inner and outer measure, it is not the case that every subset of the unit interval is defined to have a Lebesgue measure. To make up for this, the Lebesgue measure is additive in the sense which actually makes it a measure in the sense of the last post. Finally! This should hopefully give the reader some insight as to why we did not simply stop at the outer measure; we needed a bit more refining of subsets to generate a sigma-algebra for which our notion of measurement matched up with an actual measure in the sense of the last post.
Next post we will talk about some of the properties of the Lebesgue measure and discuss why we should care about it. We will also talk about measurable functions, which will be useful in discussing integrals.
This section is optional reading, but I just wanted to mention a few things. First, I have completely done away with the "finite" part of elementary sets and simply have defined them to be countable unions of disjoint intervals. Since this is on the unit interval, I don’t think I run into any problems. Also, we may extend our measures to any other compact interval in a relatively obvious way. To extend the measures to the real line is a bit trickier, and there are apparently a number of ways to do it; I do not understand enough about this yet to be able to post about it.
Another thing which I find unfortunate is the terms "outer measure" and "inner measure", since these fail to be actual measures in the sense of the previous post. This is an unfortunate fact that I found out after I had written up this post when I was wondering why we do not simply stop at outer measure when we define the Lebesgue measure. It is difficult to change terms (especially as standard as these) and so I just left it.
Last, on a personal note, these series of posts were mainly to help me study for my analysis qual. They have taken significantly more time to type up than I had thought, but it is kind of making me excited about analysis which is arguably more important (at least in the grand scheme of things!). I am unsure of how much detail I will leave out, especially because I have a significant amount of material to cover (measurable functions, convergence theorems, Egorov’s, Luzin’s, and all those theorems, all the Lebesgue integral stuff, spaces, tons of inequalities, Fourier stuff, product measures, and some of functional analysis). It is also not enough to post about the material, but also understand how to solve related problems, so it is important that I have enough time to work on these as well. We’ll cross these bridges when we get to them.