Download PDF by Tao T.: An introduction to measure theory

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6. Give an example to show that the reverse statement m∗ (E) = sup m∗ (U ) U ⊂E,U open is false. 2. Lebesgue measurability. 2, and set out their basic properties. First, we show that there are plenty of Lebesgue measurable sets. 13 (Existence of Lebesgue measurable sets). 32 1. Measure theory (i) Every open set is Lebesgue measurable. (ii) Every closed set is Lebesgue measurable. (iii) Every set of Lebesgue outer measure zero is measurable. ) (iv) The empty set ∅ is Lebesgue measurable. (v) If E ⊂ Rd is Lebesgue measurable, then so is its complement Rd \E.

Bk ) = |B1 | + . . + |Bk | holds for almost disjoint boxes B1 , . . , Bk , and not just for disjoint boxes. 9 (Outer measure of countable unions of almost disjoint ∞ boxes). Let E = n=1 Bn be a countable union of almost disjoint boxes B1 , B2 , . .. Then ∞ m∗ (E) = |Bn |. n=1 Thus, for instance, Rd itself has an infinite outer measure. Proof. 6 we have ∞ m∗ (E) ≤ ∞ m∗ (Bn ) = n=1 |Bn |, n=1 so it suffices to show that ∞ |Bn | ≤ m∗ (E). n=1 But for each natural number N , E contains the elementary set B1 ∪ .

The arguments will be fairly simple, but the sets constructed are somewhat artificial in nature. 18. There exists a subset E ⊂ [0, 1] which is not Lebesgue measurable. Proof. We use the fact that the rationals Q are an additive subgroup of the reals R, and so partition the reals R into disjoint cosets x + Q. This creates a quotient group R/Q := {x + Q : x ∈ R}. 10]. 2. Lebesgue measure 45 with [0, 1]. Applying the axiom of choice, we may thus find an element xC ∈ C ∩ [0, 1] for each C ∈ R/Q. We then let E := {xC : C ∈ R/Q} be the collection of all these coset representatives.

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An introduction to measure theory by Tao T.


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