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MATH 507/420 - Assignment #2 Due on Friday October 7, 2011 Name —————————————– Student number ————————— 1 Problem 1: Let µ∗ be the Lebesgue outer measure. Give an example P of∗a countable ∗ collection of disjoint sets An such that µ (∪n An ) < n µ (An ). Next give an example of a decreasing sequence of sets A1 ⊃ A2 ⊃ · · · such that µ∗ (A1 ) < ∞ and µ∗ (∩n An ) < limn→∞ µ∗ (An ). 2 Problem 2: Consider the collection S = {∅, [0, 1], [0, 3], [2, 3]} of subsets of R and define µ(∅) = 0, µ([0, 1]) = 1, µ([0, 3]) = 1, µ([2, 3]) = 1. Show that µ : S → [0, ∞] is a premeasure. Can µ be extended to a measure? What are the subsets of R that are measurable with respect to the outer measure µ∗ induced by µ? 3 Problem 3: Given a semiring S ⊆ 2X , assume a set A ⊆ X can be written as a countable union of An ∈ S . Prove that A can be written as a countable disjoint union of some Bn ∈ S. 4 Problem 4: Let Q be the set of rationals and S the collection of all sets of the form (a, b] ∩ Q where a, b ∈ R and a ≤ b. Define the set function µ on S by µ(∅) = 0 and µ(A) = ∞ if A 6= ∅. Show that S is a semiring and µ is a premeasure on S. Then prove that the extension of µ to the smallest σ-algebra containing S is not unique.