8 1. Measure theory 1.1.2. Jordan measure. We now have a satisfactory notion of measure for elementary sets. But of course, the elementary sets are a very restrictive class of sets, far too small for most applications. For instance, a solid triangle or disk in the plane will not be elementary, or even a rotated box. On the other hand, as essentially observed long ago by Archimedes, such sets E can be approximated from within and without by elementary sets A ⊂ E ⊂ B, and the inscribing elementary set A and the circumscribing elementary set B can be used to give lower and upper bounds on the putative measure of E. As one makes the approximating sets A, B increasingly fine, one can hope that these two bounds eventually match. This gives rise to the following definitions. Definition 1.1.4 (Jordan measure). Let E ⊂ Rd be a bounded set. • The Jordan inner measure m∗,(J)(E) of E is defined as m∗,(J)(E) := sup A⊂E,A elementary m(A). • The Jordan outer measure m∗,(J)(E) of E is defined as m∗,(J)(E) := inf B⊃E,B elementary m(B). • If m∗,(J)(E) = m∗,(J)(E), then we say that E is Jordan measurable, and call m(E) := m∗,(J)(E) = m∗,(J)(E) the Jordan measure of E. As before, we write m(E) as md(E) when we wish to emphasise the dimension d. By convention, we do not consider unbounded sets to be Jordan measurable (they will be deemed to have infinite Jordan outer measure). Jordan measurable sets are those sets which are “almost elementary” with respect to Jordan outer measure. More precisely, we have Exercise 1.1.5 (Characterisation of Jordan measurability). Let E ⊂ Rd be bounded. Show that the following are equivalent: (1) E is Jordan measurable. (2) For every ε 0, there exist elementary sets A ⊂ E ⊂ B such that m(B\A) ≤ ε. (3) For every ε 0, there exists an elementary set A such that m∗,(J)(AΔE) ≤ ε. As one corollary of this exercise, we see that every elementary set E is Jordan measurable, and that Jordan measure and elementary measure coincide for such sets this justifies the use of m(E) to denote both. In particular, we still have m(∅) = 0.

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