4 1. Measure theory concept is elementary enough to be systematically studied in an undergrad- uate 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 ex- tends 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 ad- ditional 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 (Theorem 1.4.43) and the dominated convergence theorem (Theorem 1.4.48), which do not hold in the Jordan-Darboux-Riemann setting). As such, they are par- ticularly well suited3 for applications in analysis, where limits of functions or sets arise all the time. In later sections, 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 the current section, 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 This material will eventually be superceded by the more powerful theory to be treated in later sections but it will serve as motivation for that later material, as well as providing some continuity with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. 1.1.1. Elementary measure. Before we discuss Jordan measure, we dis- cuss the even simpler notion of elementary measure, which allows one to measure a very simple class of sets, namely the elementary sets (finite unions of boxes). Definition 1.1.1 (Intervals, boxes, elementary sets). An interval is a subset of R of the form [a, b] := {x R : a x b}, [a, b) := {x R : a x b}, (a, b] := {x R : a x b}, or (a, b) := {x R : a x b}, where a b are real numbers. We define the length4 |I| of an interval I = [a, b], [a, b), (a, b], (a, b) to be |I| := b a. A box in Rd is a Cartesian product B := I1 ×...×Id of d intervals I1,...,Id (not necessarily of the same 3There are other ways to extend Jordan measure and the Riemann integral (see for instance Exercise 1.6.54 or Section 1.7.3), but the Lebesgue approach handles limits and rearrangement better than the other alternatives, and so has become the standard approach in analysis it is also particularly well suited for providing the rigorous foundations of probability theory, as discussed in Section 2.3. 4Note we allow degenerate intervals of zero length.
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