1.2. Lebesgue measure 15 Exercise 1.1.26. Extend the definition of the Riemann and Darboux inte- grals to higher dimensions, in such a way that analogues of all the previous results hold. 1.2. Lebesgue measure In Section 1.1, we recalled the classical theory of Jordan measure on Eu- clidean spaces Rd. This theory proceeded in the following stages: (i) First, one defined the notion of a box B and its volume |B|. (ii) Using this, one defined the notion of an elementary set E (a finite union of boxes), and defines the elementary measure m(E) of such sets. (iii) From this, one defined the inner and Jordan outer measures m∗,(J)(E),m∗,(J)(E) of an arbitrary bounded set E ⊂ Rd. If those measures match, we say that E is Jordan measurable, and call m(E) = m∗,(J)(E) = m∗,(J)(E) the Jordan measure of E. As long as one is lucky enough to only have to deal with Jordan mea- surable sets, the theory of Jordan measure works well enough. However, as noted previously, not all sets are Jordan measurable, even if one restricts attention to bounded sets. In fact, we shall see later in these notes that there even exist bounded open sets, or compact sets, which are not Jordan mea- surable, so the Jordan theory does not cover many classes of sets of interest. Another class that it fails to cover is countable unions or intersections of sets that are already known to be measurable: Exercise 1.2.1. Show that the countable union ∞ n=1 En or countable in- tersection ∞ n=1 En of Jordan measurable sets E1,E2,... ⊂ R need not be Jordan measurable, even when bounded. This creates problems with Riemann integrability (which, as we saw in Section 1.1, was closely related to Jordan measure) and pointwise limits: Exercise 1.2.2. Give an example of a sequence of uniformly bounded, Rie- mann integrable functions fn : [0, 1] → R for n = 1, 2,... that converge pointwise to a bounded function f : [0, 1] → R that is not Riemann inte- grable. What happens if we replace pointwise convergence with uniform convergence? These issues can be rectified by using a more powerful notion of measure than Jordan measure, namely Lebesgue measure. To define this measure, we first tinker with the notion of the Jordan outer measure m∗,(J)(E) := inf B⊃E B elementary m(B)

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