MA359 2009-2010 Assignment 2 PDF

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Measure Theory MA359 Assignment 2 DUE on Friday November 6th at NOON. A. Warming up questions (will be discussed in the support sessions - they should not be handed in for marking) 1. Prove Pn that if µ1 , · · · , µn are measures on (X, M), and a1 , · · · , an ∈ [0, ∞) then i=1 a1 µi is a measure on (X, M). 2. Define the outer Jordan content J∗ (E) of a set E in R by N X N Ij } J∗ (E) = inf { |Ij |, Ij intervals and E ⊂ ∪i=1 i=1

¯ denotes the closure of E. Use Prove that J∗ (E) = J∗ (E¯), for any E . Here E that result to produce an example of a bounded set such that J∗ (E) > 0 but m∗ (E) > 0. (m∗ here denotes the Lebesgue measure, i. e., the measure of chapter I) 3. Prove that if (X, M, µ) is a measure space and E, F ∈ M, then µ(E) + µ(F ) = µ(E ∪ F ) + µ(E ∩ F ). 4. Given a measure space (X, M, µ), and set E ∈ M, define µE (F ) = µ(F ∩ E ), for F ∈ M. Prove that µE is a measure.

B. Assignments for credit (should be deposited in the postbox outside the Undergraduate Office by NOON on Friday November 6th.) 1. Prove that a finitely additive measure µ is a measure if and only if it is continuous from below (i.e If B1 ⊂ B2 ⊂ · · · Bn ⊂ · · · then µ(∪∞ j=1 Bj ) = lim µ(Bn )). n→∞

If µ(X) < ∞, where X is the set where the measure is considered, then prove that µ is a measure if and only if it is continuous from above. ∞ is a sequence of 2. Prove that if µ∗ is an exterior measure on X and {A Pj∞}j=1∗ S∞ ∗ ∗ disjoint µ -measurable sets, then µ (E ∩ ( 1 Aj )) = i=1 µ (E ∩ Aj ).

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P∞ 3. Let {Ej }∞ j=1 m(Ej ) < ∞. j=1 be a sequence of measurable sets in R such that Prove that the m(lim supEk ) = 0. (limsupEk is defined in the previous problem set. m here is the Lebesgue measured from Chapter I) 4. Let E ⊂ R with m∗ (E) > 0. Prove that for each 0 < α < 1 there exists an open interval I such that m∗ (E ∩ I) ≥ αm∗ (I). (m∗ denotes the exterior measure of chapter I)

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