Real Analysis Primer, part 1: Measures and Sigma-Algebras.
May 26, 2011
Every so often we do things not because we want to, but because we must. It is not a secret that I do not like Analysis. It may be because, as some of my peers suggest, I don’t truly understand it. I don’t deny this claim. It may be that I don’t see "the point" to much of it. I’m not sure. Analysis, to me, is like a rigid building where everything must be just so — I contrast this with topology, where things seem somehow more fluid and less strict…
But enough whining. These posts will go through the various topics in analysis, perhaps stopping along the way to offer solutions to various related problems. To be perfectly honest with the reader, the reason I am doing this is because I need to take a Qualifying exam in Analysis in the Fall and I need to review a lot of this material. Therefore, if you find an error — and there may be some! — do not hesitate to tell me.
Measures: The Size of Things.
To begin talking about analysis, we must talk about measures.
In Topology, there is not usually a notion of size, in that even very "large" looking things can be homeomorphic to very "small" looking things — the real line and the unit open interval, for example, are homeomorphic but the real line is "much bigger looking" than the unit interval. If topology ruled the world then things might get messy: if some fabric costs $5 for one yard, then via some homeomorphism you could potentially get an infinite number of yards of fabric for $5. This is not a great way to run a business. But notice that even in this example there is some notion of measurement: a yard is an interval of a specific size which (relativity and such aside) which does not change from place to place; indeed, a yard of fabric is the same "length" as a yard of fish.
On the real line it is not complicated to make a measure that corresponds with what our expectation of a measure should be. What is the measure of the closed interval between 0 and 1? Well, that is 1 unit long. What about between 0 and 2? Well, that’s 2 units long. And so forth. But perhaps something not so obvious: what is the measure of ? What’s the measure of ? Can these measures be the same, even if there is just a "little bit more" on the second one? We may be tempted to say that it’s such a small amount that it really doesn’t matter, and just say that every point has length 0. But then isn’t just an uncountable number of points? So isn’t this also of length 0? What’s the deal? We need to be more careful.
To begin constructing a measure, we need to think about what it should be. We would really like the measure of some set to be a real number — in fact, for now, we’d really like it to be a non-negative real number (though we may extend this later). So our measure should take a set and spit out which is the "measure of ". Note that we will also allow to have infinite measure and we will write in this case. But what is the domain of this function ? To be absolutely general here we must mention -algebras.
If we had a measure on some set, what would we want? If we could measure some set and some other set , then we’d want to be able to measure both of them together (in other words, ) and if we knew the measure of the entire set that was sitting inside, we’d like to be able to measure without ; in other words, we’d like to be able to measure the compliment of . We need to be careful, though, since taking uncountable unions (say, of points on the real line) will give us some not-so-nice properties in our measure, so we’ll stop at countable unions. Using these rules, let’s take some subsets of that would be satisfy these things. This kind of structure actually has a name, too: it’s called a -algebra.
Definition. A subset of the powerset of the set is called a -algebra if it satisfies the following properties:
- It is nonempty (this is just so that we don’t wind up with anything trivial).
- is closed under taking compliments. If then so is .
- is closed under countable unions. If are in , then so is their union.
Notice that we’ve said nothing about intersections. In fact, countable intersections ARE in the -algebra, but it is redundant to require them to be: if for then so are their compliments, and hence but by de Morgan’s laws this is the same as the compliment of the intersection, and the compliment of this is just which gives us countable intersections.
Notice also that to make a sigma algebra, the "best" we can do is to take the powerset of the set. This will give us the "biggest" -algebra over our set.
The -algebra we will most often take will be the following: let be a topological space (think about the reals if you’d like) and take the set of open sets in . There is a smallest such -algebra that contains these open sets: this is the one we will often refer to.
Back to Measures.
Since we’d like the measure to be defined on a whole bunch of subsets, it makes sense that we’d take our set and then take an associated -algebra over to be the domain of the measure. For example, in the reals we could take the smallest -algebra generated by the open sets in with the standard topology (and we will use this example for most of the rest of Lebesgue theory —).
Thus, so far we have for is a set and is a -algebra over , a measure must be a function where the latter space is the non-negative reals along with . But we’d like our measure to have a bit more structure: as is it, we could make it whatever we wanted so long as it was a function. A measure ought to have certain properties that are nice and help us differentiate the sizes of sets.
The first property is non-negativity. We’ve already talked about this. Also, we need to say that the empty-set has measure 0 (since what other measure would we want to give it?). Last, if we have a countable number of disjoint sets for , then we’d want their measures to sum up nicely, right? In other words, the measure of the union should be the sum of their measures. (If you have a 3 yard piece of string and a 5 yard piece of string, together you’d want to have 8 yards.)
At this point, we can finally define a measure in some legit way.
Definition. Let be a set and is a -algebra over . A measure is a function such that the following properties hold:
- for every and .
- If are a countable collection of (pairwise) disjoint sets in then .
Note that we do not say that if the measure of some set is equal to 0 then that set must be the empty set. The converse of the first property isn’t true, in other words.
Also, one last definition so I don’t have to keep writing . These sets in our -algebra are exactly the sets we can define a measure on, and so we call them measurable sets.
Definition. If is a set and is a -algebra over , then we call a measurable set.
Before we finish up this post, there are a few properties we can derive just from what we have here. Let’s do these.
Proposition. is monotonic; that is, if are measurable, then .
Proof. Note that . These sets are disjoint, so we have that
We don’t even need to figure out what the value of the last term here is, since we know that . Then we have plus a non-negative term is equal to ; this implies directly that .
Proposition. is a countably subadditive function. That is, if is a countable not-necessarily-disjoint collection of measurable sets, then
Proof. The idea of the proof is really just to make these sets disjoint sets. Do this in the following way (and remember the process, since it comes up a bunch):
Then if without loss of generality,
so we have that the are pairwise disjoint and moreover,
So the point of that construction was to make a pairwise disjoint collection of measurable sets that union to the same set as the not-pairwise-disjoint collection. Notice that, also, for every and hence . Now we may apply this and our countable union measure property:
which completes the proof.