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?