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 ).
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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?
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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.
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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.