An introduction to measure theory by Tao T. PDF

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By Tao T.

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R need not be Jordan measurable, even when bounded. 2. Give an example of a sequence of uniformly bounded, Riemann 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 integrable. 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 m(B) B⊃E;B elementary of a set E ⊂ Rd (we adopt the convention that m∗,(J) (E) = +∞ if E is unbounded, thus m∗,(J) now takes values in the extended nonnegative reals [0, +∞], whose properties we will briefly review below).

Covering E ∪F have diameter at most r. In particular, we may assume that all such boxes have diameter strictly less than dist(E, F ). Once we do this, then it is no longer possible for any box to intersect both E and F , and then the previous argument now applies. g. E = [0, 1) and F = [1, 2]). But the situation improves when E, F are closed, and at least one of E, F is compact: 11The diameter of a set B is defined as sup{|x − y| : x, y ∈ B}. 2. 4. Let E, F ⊂ Rd be disjoint closed sets, with at least one of E, F being compact.

24 (Lebesgue measure as the completion of elementary measure). The purpose of the following exercise is to indicate how Lebesgue measure can be viewed as a metric completion of elementary measure in some sense. g. A = [0, 1]d ). (i) Let 2A := {E : E ⊂ A} be the power set of A. We say that two sets E, F ∈ 2A are equivalent if E∆F is a null set. Show that this is a equivalence relation. (ii) Let 2A / ∼ be the set of equivalence classes [E] := {F ∈ 2A : E ∼ F } of 2A with respect to the above equivalence relation.

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An introduction to measure theory by Tao T.


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