By Terence Tao
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Extra resources for An Introduction To Measure Theory (January 2011 Draft)
Then we could divide this family into two subfamilies B1 , B2 , . . 10Recall from the preface that we use the usual Euclidean metric |(x , . . , x )| := 1 d x21 + . . + x2d on Rd . 24 1. Measure theory and B1 , B2 , B3 , . , 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.
Iii) Give a counterexample to show that the hypothesis that at least one of the m(En ) is finite in the downward monotone convergence theorem cannot be dropped. 12. 3, when restricted to Lebesgue measurable sets of course. 13. We say that a sequence En of sets in Rd converges pointwise to another set E in Rd if the indicator functions 1En converge pointwise to 1E . (i) Show that if the En are all Lebesgue measurable, and converge pointwise to E, then E is Lebesgue measurable also. ) (ii) (Dominated convergence theorem) Suppose that the En are all contained in another Lebesgue measurable set F of finite measure.
Show that dist(E, F ) > 0. Give a counterexample to show that this claim fails when the compactness hypothesis is dropped. We already know that countable sets have Lebesgue outer measure zero. Now we start computing the outer measure of some other sets. 6 (Outer measure of elementary sets). Let E be an elementary set. Then the Lebesgue outer measure m∗ (E) of E is equal to the elementary measure m(E) of E: m∗ (E) = m(E). 7. Since countable sets have zero outer measure, we note that we have managed to give a proof of Cantor’s theorem that Rd is uncountable.
An Introduction To Measure Theory (January 2011 Draft) by Terence Tao