By Terence Tao
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Extra info for An Introduction To Measure Theory (January 2011 Draft)
X∗n )) of [a, b] is a finite sequence of real numbers a = x0 < x1 < . . < xn = b, together with additional numbers xi−1 ≤ x∗i ≤ xi for each i = 1, . . , 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.
Letting Fn be the complement of Un , we conclude that the complement Rd \E of E contains all of the Fn , and ∞ that m∗ ((Rd \E)\Fn ) ≤ 1/n. If we let F := n=1 Fn , then Rd \E contains F , and from monotonicity m∗ ((Rd \E)\F ) = 0, thus Rd \E is the union of F and a set of Lebesgue outer measure zero. But F is in turn the union of countably many closed sets Fn . The claim now follows from (ii), (iii), (iv). Finally, Claim (vii) follows from (v), (vi), and de Morgan’s laws ( α∈A Eα )c = α∈A Eαc , ( α∈A Eα )c = α∈A Eαc , (which work for infinite unions and intersections without any difficulty).
Id . To avoid some technical issues we shall restrict attention here to “small” cubes of sidelength at most 1, thus we restrict n to the non-negative integers, and we will completely ignore “large” cubes of sidelength greater than one. Observe that the closed dyadic cubes of a fixed sidelength 2−n are almost disjoint, and cover all of Rd . Also observe that each dyadic cube of sidelength 2−n is contained in exactly one “parent” cube of sidelength 2−n+1 (which, conversely, has 2d “children” of sidelength 2−n ), giving the dyadic cubes a structure analogous to that of a binary tree (or more precisely, an infinite forest of 2d -ary trees).