An Introduction To Measure Theory (January 2011 Draft) by Terence Tao

By Terence Tao

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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.

The first of which covered E, and the second of which covered F . From definition of Lebesgue outer measure, we have ∞ m∗ (E) ≤ |Bn | n=1 and ∞ m∗ (F ) ≤ |Bn |; n=1 summing, we obtain ∞ m∗ (E) + m∗ (F ) ≤ |Bn | n=1 and thus m∗ (E) + m∗ (F ) ≤ m∗ (E ∪ F ) + ε. Since ε was arbitrary, this gives m∗ (E) + m∗ (F ) ≤ m∗ (E ∪ F ) as required. Of course, it is quite possible for some of the boxes Bn to intersect both E and F , particularly if the boxes are big, in which case the above argument does not work because that box would be doublecounted.

If E is Lebesgue measurable, we refer to m(E) := m∗ (E) as the Lebesgue measure of E (note that this quantity may be equal to +∞). We also write m(E) as md (E) when we wish to emphasise the dimension d. 3. The intuition that measurable sets are almost open is also known as Littlewood’s first principle, this principle is a triviality with our current choice of definitions, though less so if one uses other, equivalent, definitions of Lebesgue measurability. 5 for a further discussion of Littlewood’s principles.

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