16 July 2001 I've been lulling in this journal for a bit (though I've been producing lots of book reviews, you should go check them out). So I guess I will hit you measure theory as a reward for returning: (reward?!) Infinity is NOT a number IV: Measure for Measure What's funny about mathematicians is they'll tell you they're measuring stuff, but you'll notice that the measurements all seem to have to do epsilon or delta, and your best chance at seeing a number is either e or pi. That doesn't seem quite right. Especially as many of these epsilon-wielding characters call themselves =applied= mathematicians. So what exactly =is= this thing called measure theory? Well, there are two main stories here: set stuff and integration stuff. You see, when Cantor started letting people see the amount of fun they could have with sets, people did all sorts of things with sets: they intersected them, complemented them (oh, that open cover suits you =so= well), unioned them - not just a finite number of times, not just a countably finite number of times, but also an uncountably infinite number of times (does that even make sense?) One thing people noticed is that even though one could show that the "shape" and/or "countability" of two infinite sets were the same, in some "applied" senses, one was bigger than the other. For example, I can put a one-to-one correspondence between all the real numbers and all the real numbers between -pi/2 and pi/2. Just pair any x between -pi/2 and pi/2 with the real number tan(x). If you know something about the tangent function, you will realize that every single real number will get hit this way. This kind of correspondence between infinite sets really messed with earlier heads, because they'd see things like two concentric circles of vastly different radii, notice that one can pair each point on the smaller circle with a unique point on the larger circle, and they'd end with the absurd conclusion that 2=1. As ridiculous as much medieval thought is, even =they= realized this was beyond the pale (but other conclusions as to the bodily humours and the spontaneous generation of frogs from mud seemed perfectly normal. But then, accounting isn't messed up if you think that rotten meat naturally transforms into maggots.) So obviously the kind of measuring as to what kind of infinity one had, called cardinality, wasn't of much help when one wanted to know how "big" a set was. So let's develop something called =Lebesgue measure=. It's going to seem nice and reasonable to begin with. Of course, if it stayed reasonable, it wouldn't be mathematics. We don't study things that behave the way we expect them to. We study mind-warping subjects for years, and then we explain them to other people by claiming "it's obvious". So, as to interval notation: A) [alpha,beta] means the interval of all the real numbers from alpha to beta, including alpha and beta. Another way to write this is: {X | alpha <= X <= beta}, which in mathese is "all x such that x is greater than or equal to alpha and less than or equal to beta." B) (alpha, beta) means the interval of all the real numbers from alpha to beta, NOT including alpha nor beta. Another way to say this is: {X | alpha < X < beta}, which is "all X such that X is greater than alpha and less than beta." So how long is [1,3]? Pull out your real number line if you need to. It's not a trick question. Yes, it's 2. How long is [2.3,6.78]? Think hard. It's 6.78 - 2.3 = 4.48 This measure theory stuff is really easy. So we're going to name our measure function. Like all traditional mathematicians, I'm going to call it m. so one would write m([1,3]) = 2 and m([2.3, 6.78]) = 4.48. First observation: m([alpha, beta]) = beta - alpha (that only works if beta >= alpha). A natural conclusion from this rule is that the measure of a single point is zero, because the set of a single point alpha can be written as [alpha, alpha] (hmmm, all points greater than or equal to alpha =and= less than or equal to alpha? Are there any other points other than alpha for which this can work? Right. You can't. Okay, we're all on the same page.) So let's make a few rules. Let's see - if you have two sets, and they're disjoint (meaning they don't share any points), then the measure of their union is simply the sum of their measures. This makes sense - you should be able to break up sets into smaller, non-overlapping sets and add up their measures. Let's try this out: Let's see, the interval [1,3] is the same as putting the open interval (1,3) together with the two endpoints [1,1] and [3,3]. So m([3,3]) = m([1,1]) + m([3,3]) + m( (1,3) ) = 0 + 0 + m((1,3)). Hmm, so the interval with endpoints has the same measure as the interval =without= its endpoints. This makes sense as points have no measure on their own. In fact, any =countable= set of points will have zero measure. Zero measure sets are lovely creatures - they're also called nullsets. (Now I could talk about sets of first category, but I'm not going to.) So what's wild is that infinite sets like all the counting numbers, or all rational numbers, both countable sets, have zero measure. Now, that's true under Lebesgue measure -- there are other measures in which even single points have measure. But that's not what we're playing with now. That can feel really odd, because rational numbers are =dense= in the real numbers, meaning that one can get a rational number as close as you want to to any real number. So here's a question: Though a countably infinite set of points has zero measure, can an uncountably infinite set have zero measure? All the intervals that are bigger than a single point are uncountably infinite, and have non-zero measure. Of course, that doesn't mean =all= uncountably infinite sets have non-zero measure. In fact, I've already told you about an uncountably infinite set with zero measure -- it's the Cantor set. That's the set where you start with [0,1], and then you get rid of (1/3,2/3), then (1/9,2/9) and (7/9,8/9), and so on. You get more and more holes in the set, and one finds that any interval in the cantor set has at least one "hole" in it. You can prove that to yourself if you'd like. However, there's an uncountable number of points in the Cantor set. But if you think about it, there's a countable process going on here - there's a step at which each "hole" appears, and at each step the length of the entire set is multiplied by 2/3 - so on the "zeroth" step, the set has measure 1, then on the first step it has measure 2/3, on next step it's 4/9, then it's 8/27, so on the nth step the measure is (2/3)^n. And if you remember what I told you about limits, you can figure out that the limit of that as n goes to infinity is 0. So if we just had that the measure of the limit of a sequence of sets (we'd have to define "limit of a sequence of sets") is the limit of the measures. And with Lebesgue measure, under certain circumstances (like, the sets are bounded - as these are), one can do that. =That's= why we love measure theory, kiddos - because we can pass limits through other things. This comes in real handy when you're trying to do stuff like take the integral of a limit of functions, or some such nonsense. Oh yes, integration. Well, I'll explain how measure theory and integration are related. Some of you may remember the definition of a definite integral as "the area under the curve". You might have a hazy memory of a bunch of rectangles, and you do this limit thing with the areas of the rectangles and there was some hoo-doo called the fundamental theorem of calculus, blah de blah blah blah. Thing is, you learned Riemann integration, which is fine as long as your functions are well behaved (like, oh, x*sin(1/x)). But what if your function were this: f(x) = 0 for all irrational x and =1 for all rational x. What is the integral of f from 0 to 1? So we go back to the idea of integral as area, and area for rectangles is easy: base times height. We've got two heights -- 0 and 1. So let's split our base into two sets: the irrational numbers and the rational numbers. Those are disjoint sets, so we can look at our integral as adding up the areas of two separate rectangles. I hear you cry -- they don't =look= like rectangles! Well, if you took a single rectangle and split it into two rectangles with the same height, but two smaller bases which added up to the original one, it would still have the same area, right? Well, forget infinity is messing around here, and that's pretty much what's going on. So the "rational rectangle" has a height of 1, and its base, the measure of the rationals from 0 to 1.. has to be 0. It's a countably finite set of points, remember? So the area of that rectangle is 0. What about the "irrational rectangle"? It has a height of zero, so it doesn't matter what the base is. But still, the measure of the irrationals from 0 to 1 has to be 1, because the rationals and irrationals together make up [0,1], and since the rationals have measure 0, the irrationals have to make up the difference. Anyway, that's just a =little= taste of measure theory. Things one can consider as a result of mesure theory is stuff like probability, because probability is a measure (albeit one that never exceeds 1). One can do stuff like integrate over brownian motion. One can do Fourier analysis. And one can look at continuous functions that have no derivatives. Isn't life fun? Ok, so I shall let infinity slumber once more. But remember all this next time you wish to treat infinity like a number. Infinity deserves more respect than that. Sure, e, pi, epsilon, and delta all have their uses, but they pale in comparison to infinity. And now you've got one more way to compare things to infinity. Behave yourselves, hear?

Prev | Year | Next |