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Mathematics 2224: Lebesgue integral Homework exercise sheet 2 Due 3:50pm, Wednesday 16th February 2011 Recall that m∗ denotes Lebesgue outer-measure, L is the collection of Lebesgue measurable subsets of R, and µ is Lebesgue measure on R. 1. If E, F ⊆ R, show that m∗ (E ∪ F ) ≤ m∗ (E) + m∗ (F ). 2. If N ⊆ R and m∗ (N ) = 0, prove each of the following statements. (a) (b) (c) (d) m∗ (A ∩ N ) = 0 for any A ⊆ R m∗ (A ∩ N c ) = m∗ (A) for any A ⊆ R N is Lebesgue measurable m∗ (A ∪ N ) = m∗ (A) for any A ⊆ R 3. If a, b ∈ R with a ≤ b, show that m∗ ([a, b]) = m∗ ((a, b]) = m∗ ([a, b)) = m∗ ((a, b)) = b − a. 4. Compute m∗ (R \ Q) and m∗ ([0, 1] \ Q). 5. If α ∈ R and E ⊆ R, recall that α + E = {α + x : x ∈ E}. Show that if E ∈ L, then α + E ∈ L. [Hint: show that A ∩ (α + E) = α + (−α + A) ∩ E and (α + E)c = α + E c for any A ⊆ R.] 6. If α ∈ R and E ⊆ R, let αE = {αx : x ∈ E}. (a) (b) (c) (d) Show that if α > 0, then m∗ (αE) = αm∗ (E). Show that m∗ (−E) = m∗ (E), where −E = {−x : x ∈ E}. Deduce that m∗ (αE) = |α|m∗ (E) for any α ∈ R. Show that if E ∈ L, then αE ∈ L. [Treat the case α = 0 separately.] 7. Let E0 = [0, 1] and for n ∈ N, let En = 0.1En−1 ∪ (0.5 + 0.1En−1 ) where 0.1En−1 = {0.1x : x ∈ En−1 }. (a) Compute the sets E1 and E2 . (b) Prove by induction on n that for every n ∈ N0 : (i). En is the disjoint union of 2n closed subintervals of [0, 1], each of Lebesgue measure 10−n , and hence En ∈ L and µ(En ) = 5−n . P −j (ii). Writing 0.d1 d2 d3 . . . = ∞ j=1 dj · 10 , we have En = {0.d1 d2 d3 . . . : dj ∈ {0, 5} if 1 ≤ j ≤ n, and dj ∈ {0, 1, 2, . . . , 9} for j > n}. ∞ T (c) Let E = Ej . j=0 (i). Prove that E ∈ L and µ(E) = 0. (ii). Explain why E = {0.d1 d2 d3 . . . : dj ∈ {0, 5} for all j ≥ 1}, and use Cantor’s diagonal argument to show that E is uncountable.