What Is Measure Theory?
A Brief History ft. Sets Of Measure Zero
Hey readers! In this post I want to go over a brief look at measure theory -- what it is, and some of the big questions that motivated its conception.
First, a motivating mystery!
If you throw a dart on an interval of the real number line, the probability of landing on a rational number is 0. This is despite the fact that rational numbers are everywhere: they include all the integers as well as all fractions of integers.
In the video above, all the red dots represent rational numbers. Between any two numbers, there are infinitely many rational numbers between them. And yet, their presence is negligible. The set is almost empty. Almost all real numbers are irrational!
Why? Because of Lebesgue Measure.
A History of Measure Theory
Measure theory was born out of a crisis in calculus. Recall from previous blogs that calculus can be broken down into differential calculus and integral calculus -- the latter of which can be characterized by solving the area problem, that is, to determine the area underneath a curve by approximating it using rectangles that get increasingly thinner.
But this level of understanding falls too short. There are lots of functions which are not integrable under this definition, like this one proposed by Dirichlet in the 1820s:
Another strange but important class of functions is the Fourier series, which is an expansion of an arbitrary function into an infinite sum of trigonometric functions. Here’s one example:
Fourier used integrals to find the coefficients of this infinite series. But we didn’t know how integration worked for an infinite series. Can you integrate an infinite series by integrating each term individually? Is the corresponding series guaranteed to converge to the original function?
To answer these questions, we needed to broaden our definition of integration by broadening our definition of area.
Some Facts About Area
Let's start by stating the obvious:
Areas can’t be negative
The area of the empty set is zero
The area of a union of sets A and B is the area of A plus the area of B, as long as A and B are disjoint (no overlap).
These 3 statements form the building blocks of what one might call a measure. In 2D, we call it area, in 1D we call it length, in 3D we call it volume, but in general, we’ll just call it a measure from here onwards.
The goal is to build a definition of measure that makes sense. It should obey the three statements we listed above, and it should feel intuitive, at least for familiar objects, like the length of the interval (0,1) should be 1, and the area of a square with side lengths 2 should be 4.
The key will be how we define the lengths and areas for more abstract sets of points, like a scattered set of disconnected points.
The Lebesgue Measure
Lebesgue built his concept of measure based on the idea of a countable cover. A countable cover of a set S is a countable collection of intervals whose union contains S.
One way to think about a countable cover is to think of open intervals like (a,b) as measuring cups.
If S can fit inside of 3 measuring cups, then the total measure of S must be less than or equal to the capacity of those 3 measuring cups.
If S is measurable, then its measure will be equal to the capacity of the smallest possible collection of measuring cups that still fit S.
Note the caveat that some sets may not be measurable. This is another cool fact that I’ll revisit in another post!
Sets of Measure Zero
We’ve stated that the empty set has a measure of zero. What other sets have measure zero? Using intervals as our measuring cups, the measure of a single point is zero, because you can cover a single point using an arbitrarily small interval.
E.g. the set {1} can be covered by the interval (0,2), which has a length of 2. But it can also be covered by the set (0.9, 1.1), which has a length of 0.2. There’s no lower bound to how small we can make this interval while still covering the point {1}, so it must have a length (or measure) of 0.
A set which contains just two points, like {1, 2} also has measure zero, because we just have to add up their individual measures. This extends for even a countably infinite number of points.
This is why the set of rational numbers has measure zero.
BTW: If you don’t know about the countability of the rationals, you can read about it here. I’ve also written about it in my book, Math In Drag, but let me know if you’d like me to make a blogpost specifically about countability!
The set of rational numbers is countably infinite, and thus has a measure of zero.
Now consider the interval (0,1), which is an uncountably infinite set of real numbers. The rational numbers within this set have measure 0, meaning that the irrational numbers within (0,1) must have measure 1. This means that almost all numbers between 0 and 1 are irrational!
You’ll see the words “almost everywhere” popping up a lot in math, and it refers to these measure-zero sets as the exceptions, hence the almost.
To give another example, the function f(x) = 1/x is continuous almost everywhere, because its set of discontinuities has measure zero: it’s the single point x=0. The analogue in probability theory would be the phrase “almost surely”, which means that an event happens with probability 1.
Mathematically, having a probability of 0 isn’t the same as something being impossible!
The Lebesgue Integral
Armed with the correct notion of measure that works for abstract sets of points, Lebesgue was able to redefine integration and find the right conditions under which we can conclude that the integral of a limit is equal to the limit of integrals. For that reason, Lebesgue’s integral is the one that’s preferred by mathematicians.
A recurring theme you’ll find everywhere in mathematics is that a single inconvenience may give birth to an entire new field of study that creates its own questions and mysteries.
If you’re interested in learning more, check out David M. Bressoud’s A Radical Approach To Lebesgue’s Theory of Integration. It’s an introduction to measure theory with an emphasis on the historical questions that motivated its development, which I find is the most fun way to learn a subject.
Until next time!