2 1. Measure theory 1.1. Prologue: The problem of measure One of the most fundamental concepts in Euclidean geometry is that of the measure m(E) of a solid body E in one or more dimensions. In one, two, and three dimensions, we refer to this measure as the length, area, or volume of E, 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 m(E) that can be assigned to all solid bodies E, 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 be- came reinterpreted as the study of Cartesian products Rd of the real line R. Using this analytic foundation rather than the classical geometrical one, it was no longer intuitively obvious how to define the measure m(E) of a general1 subset E of Rd we will refer to this (somewhat vaguely defined) problem of writing down the “correct” definition of measure as the problem of measure. 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 E 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 ∞ · 0 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 A := [0, 1] and B := [0, 2] are in one-to-one correspondence (using the bijection x → 2x from A to B), but of course B is twice as long as A. So one can disassemble A into an uncountable number of points and reassemble them to form a set of twice the length. 1One can also pose the problem of measure on domains other than Euclidean space, such as a Riemannian manifold, but we will focus on the Euclidean case here for simplicity, and refer to any text on Riemannian geometry for a treatment of integration on manifolds.

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