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One of the most fundamental concepts in Euclidean geometry is that of the measure of a solid body in one or more dimensions. In one, two, and three dimensions, we refer to this measure as the length, area, or volume of respectively. In the classical approach to geometry, the measure of a body was often computed by partitioning that body into finitely many components, moving around each component by a rigid motion (e.g. a translation or rotation), and then reassembling those components to form a simpler body which presumably has the same area. One could also obtain lower and upper bounds on the measure of a body by computing the measure of some inscribed or circumscribed body; this ancient idea goes all the way back to the work of Archimedes at least. Such arguments can be justified by an appeal to geometric intuition, or simply by postulating the existence of a measure that can be assigned to all solid bodies , and which obeys a collection of geometrically reasonable axioms. One can also justify the concept of measure on “physical” or “reductionistic” grounds, viewing the measure of a macroscopic body as the sum of the measures of its microscopic components.
With the advent of analytic geometry, however, Euclidean geometry became reinterpreted as the study of Cartesian products of the real line . Using this analytic foundation rather than the classical geometrical one, it was no longer intuitively obvious how to define the measure of a general subset of ; we will refer to this (somewhat vaguely defined) problem of writing down the “correct” definition of measure as the problem of measure. (One can also pose the problem of measure on other domains than Euclidean space, such as a Riemannian manifold, but we will focus on the Euclidean case here for simplicity.)
To see why this problem exists at all, let us try to formalise some of the intuition for measure discussed earlier. The physical intuition of defining the measure of a body to be the sum of the measure of its component “atoms” runs into an immediate problem: a typical solid body would consist of an infinite (and uncountable) number of points, each of which has a measure of zero; and the product is indeterminate. To make matters worse, two bodies that have exactly the same number of points, need not have the same measure. For instance, in one dimension, the intervals and are in one-to-one correspondence (using the bijection from to ), but of course is twice as long as . So one can disassemble into an uncountable number of points and reassemble them to form a set of twice the length.
Of course, one can point to the infinite (and uncountable) number of components in this disassembly as being the cause of this breakdown of intuition, and restrict attention to just finite partitions. But one still runs into trouble here for a number of reasons, the most striking of which is the Banach-Tarski paradox, which shows that the unit ball in three dimensions can be disassembled into a finite number of pieces (in fact, just five pieces suffice), which can then be reassembled (after translating and rotating each of the pieces) to form two disjoint copies of the ball . (The paradox only works in three dimensions and higher, for reasons having to do with the property of amenability; see this blog post for further discussion of this interesting topic, which is unfortunately too much of a digression from the current subject.)
Here, the problem is that the pieces used in this decomposition are highly pathological in nature; among other things, their construction requires use of the axiom of choice. (This is in fact necessary; there are models of set theory without the axiom of choice in which the Banach-Tarski paradox does not occur, thanks to a famous theorem of Solovay.) Such pathological sets almost never come up in practical applications of mathematics. Because of this, the standard solution to the problem of measure has been to abandon the goal of measuring every subset of , and instead to settle for only measuring a certain subclass of “non-pathological” subsets of , which are then referred to as the measurable sets. The problem of measure then divides into several subproblems:
- What does it mean for a subset of to be measurable?
- If a set is measurable, how does one define its measure?
- What nice properties or axioms does measure (or the concept of measurability) obey?
- Are “ordinary” sets such as cubes, balls, polyhedra, etc. measurable?
- Does the measure of an “ordinary” set equal the “naive geometric measure” of such sets? (e.g. is the measure of an rectangle equal to ?)
These questions are somewhat open-ended in formulation, and there is no unique answer to them; in particular, one can expand the class of measurable sets at the expense of losing one or more nice properties of measure in the process (e.g. finite or countable additivity, translation invariance, or rotation invariance). However, there are two basic answers which, between them, suffice for most applications. The first is the concept of Jordan measure of a Jordan measurable set, which is a concept closely related to that of the Riemann integral (or Darboux integral). This concept is elementary enough to be systematically studied in an undergraduate analysis course, and suffices for measuring most of the “ordinary” sets (e.g. the area under the graph of a continuous function) in many branches of mathematics. However, when one turns to the type of sets that arise in analysis, and in particular those sets that arise as limits (in various senses) of other sets, it turns out that the Jordan concept of measurability is not quite adequate, and must be extended to the more general notion of Lebesgue measurability, with the corresponding notion of Lebesgue measure that extends Jordan measure. With the Lebesgue theory (which can be viewed as a completion of the Jordan-Darboux-Riemann theory), one keeps almost all of the desirable properties of Jordan measure, but with the crucial additional property that many features of the Lebesgue theory are preserved under limits (as exemplified in the fundamental convergence theorems of the Lebesgue theory, such as the monotone convergence theorem and the dominated convergence theorem, which do not hold in the Jordan-Darboux-Riemann setting). As such, they are particularly well suited for applications in analysis, where limits of functions or sets arise all the time. (There are other ways to extend Jordan measure and the Riemann integral, but the Lebesgue approach handles limits better than the other alternatives, and so has become the standard approach in analysis.)
In the rest of the course, we will formally define Lebesgue measure and the Lebesgue integral, as well as the more general concept of an abstract measure space and the associated integration operation. In the rest of this post, we will discuss the more elementary concepts of Jordan measure and the Riemann integral. This material will eventually be superceded by the more powerful theory to be treated in the main body of the course; but it will serve as motivation for that later material, as well as providing some continuity with the treatment of measure and integration in undergraduate analysis courses.
In this supplemental note to the previous lecture notes, I would like to give an alternate proof of a (weak form of the) Carathéodory extension theorem. This argument is restricted to the -finite case, and does not extend the measure to quite as large a -algebra as is provided by the standard proof of this theorem, but I find it conceptually clearer (in particular, hewing quite closely to Littlewood’s principles, and the general Lebesgue philosophy of treating sets of small measure as negligible), and suffices for many standard applications of this theorem, in particular the construction of Lebesgue measure.
Let us first state the precise statement of the theorem:
Theorem 1. (Weak Carathéodory extension theorem) Let be a Boolean algebra of subsets of a set X, and let be a function obeying the following three properties:
- (Pre-countable additivity) If are disjoint and such that also lies in , then .
- (-finiteness) X can be covered by at most countably many sets in , each of which has finite -measure.
Let be the -algebra generated by . Then can be uniquely extended to a countably additive measure on .
We will refer to sets in as elementary sets and sets in as measurable sets. A typical example is when X=[0,1] and is the collection of all sets that are unions of finitely many intervals; in this case, are the Borel-measurable sets.