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By Terence Tao

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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, 18 1. Measure theory and call m(E) = m∗,(J) (E) = m∗,(J) (E) the Jordan measure of E.

Properties of Lebesgue outer measure. We begin by studying the Lebesgue outer measure m∗ , which was defined earlier, and takes values in the extended non-negative real axis [0, +∞]. 3 (The outer measure axioms). (i) (Empty set) m∗ (∅) = 0. (ii) (Monotonicity) If E ⊂ F ⊂ Rd , then m∗ (E) ≤ m∗ (F ). (iii) (Countable subadditivity) If E1 , E2 , . . ⊂ Rd is a count∞ ∞ able sequence of sets, then m∗ ( n=1 En ) ≤ n=1 m∗ (En ). (Hint: Use the axiom of countable choice, Tonelli’s theorem 22 1. ) Note that countable subadditivity, when combined with the empty set axiom, gives as a corollary the finite subadditivity property m∗ (E1 ∪ .

N. We abbreviate xi −xi−1 as δxi . The quantity ∆(P) := sup1≤i≤n δxi will be called the norm of the tagged partition. The Riemann sum R(f, P) of f with respect to the tagged partition P is defined as n f (x∗i )δxi . R(f, P) := i=1 We say that f is Riemann integrable on [a, b] if there exists a real b number, denoted a f (x) dx and referred to as the Riemann integral of f on [a, b], for which we have b f (x) dx = a lim R(f, P) ∆(P)→0 by which we mean that for every ε > 0 there exists δ > 0 such b that |R(f, P) − a f (x) dx| ≤ ε for every tagged partition P with ∆(P) ≤ δ.

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