Counting, uncounting, and dust, part five

Alright, so now we finally know that there isn’t one infinity any more. In fact, there is an infinite number of infinities, each infinitely larger than the one smaller than it, and so on upwards. Excelsior is a good motto for the infinitude of infinities (as it is for the Coat of Arms of the State of New York).

But we haven’t dealt with another problem: this way of measuring the size of a set just isn’t good enough for our purposes. We need something more refined, something that does distinguish between countable and uncountable while also capturing common-sense notions of the “length” of a line, the “area” of a sheet of paper, the “volume” of a hill, and so on. This is exactly what is studied in the branch of mathematics known as measure theory.

Measuring mountains and molehills

Now a measure is basically a rule (a function) that assigns to a set a number (this being the measure of the set), in a manner that follows certain common-sensical restrictions. The two most important of these restrictions are:

• the measure of a set has to be greater than or equal to 0. This is sort of obvious—we don't want areas and lengths to be negative! [It is also not entirely obvious—there are certain other cases where we do want “signed” areas and volumes. This comes into play when dealing with such things as differential forms, integration on manifolds, and all that. But that’s another post for another time!]
• if a set can be separated into non-overlapping subsets, then the measure of the set equals the sum of the measures of the subsets. So for instance, the measure of the interval \( (0, 1) \) must equal the sum of the  measures of the interval \( (0, \frac{1}{2}) \), the single-point set \( \{ \frac{1}{2} \} \), and the interval \( (\frac{1}{2}, 1) \).

[Technical note: strictly speaking, a measure is not defined on a set but a \( \sigma \)-algebra over the set, i.e., on a special collection of subsets of the set that satisfy a few important properties. Since this special collection of subsets always includes the set itself, it is not hugely wrong to say that the measure is defined on the set. There are some technical reasons for defining the measure on a  \( \sigma \)-algebra over the set rather than on the set itself, but these rise from certain pathological complications that sometimes make it impossible to define the measure of every single subset of a set. Using the  \( \sigma \)-algebra restricts us to dealing with the measures only of a certain collection of subsets of the set, but in practice this is virtually never a problem since pretty much every conceivably useful subset of \( \mathbb{R} \) can be assigned a consistent and meaningful measure.]

As can be imagined, there are many different measures that can be defined on one set. One example is the so-called counting measure, which assigns to a finite set \( A \) the number of elements in it, and to an infinite set (whether countable or uncountable) the symbol \( \infty \). Another extremely important example is the Lebesgue measure, which is defined over virtually all “useful subsets” of \( \mathbb{R} \) and which preserves our common-sense notion of length.

• The Lebesgue measure of isolated points is zero (and hence of any countable set of isolated points); in other words, any set for which the counting measure is not \( \infty \) has Lebesgue measure zero.
• The Lebesgue measure of a closed interval \( [a, b] \) is defined as \( b-a \). Again, this preserves our common-sense notion of length.
• The Lebesgue measure of a closed rectangle \( [a,b] \times [c,d] \) in the plane \( \mathbb{R}^2 \)  is \( (b-a) \cdot (c-d) \), which again accords with our common-sense notion of area. The same principle generalizes to higher-dimensional closed structures.

Using Lebesgue measure thus immediately allows us to distinguish mountains from molehills—even if they both have the same cardinality. 

Lebesgue measure is the foundation of the Lebesgue integral, the standard theoretical integral of integration that is taught in higher mathematics, much more powerful than the Riemann (or Cauchy) integral introduced (but seldom labeled as such) in high school calculus. The fact that it is usually cloaked behind an impenetrable veil of mathematical jargon means that the beauty of Lebesgue integration is usually not at all apparent to the casual reader. Perhaps at some point in the near future I will attempt to explain the need for more and more complicated definitions of integration.