# MATH 507/420 - Assignment #2 Due on Friday October 7, 2011 ```MATH 507/420 - Assignment #2
Due on Friday October 7, 2011
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Student number —————————
1
Problem 1:
Let &micro;∗ be the Lebesgue outer measure. Give an example
P of∗a countable
∗
collection of disjoint sets An such that &micro; (∪n An ) &lt; n &micro; (An ). Next
give an example of a decreasing sequence of sets A1 ⊃ A2 ⊃ &middot; &middot; &middot; such
that &micro;∗ (A1 ) &lt; ∞ and &micro;∗ (∩n An ) &lt; limn→∞ &micro;∗ (An ).
2
Problem 2:
Consider the collection S = {∅, [0, 1], [0, 3], [2, 3]} of subsets of R and
define &micro;(∅) = 0, &micro;([0, 1]) = 1, &micro;([0, 3]) = 1, &micro;([2, 3]) = 1. Show that
&micro; : S → [0, ∞] is a premeasure. Can &micro; be extended to a measure?
What are the subsets of R that are measurable with respect to the
outer measure &micro;∗ induced by &micro;?
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 &micro; on S by
&micro;(∅) = 0 and &micro;(A) = ∞ if A 6= ∅. Show that S is a semiring and &micro; is
a premeasure on S. Then prove that the extension of &micro; to the smallest
σ-algebra containing S is not unique.
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