Real Analysis Homework #3
Due 10/13 1. Let µ be a finite measure on (X, S) and µ∗ the outer measure induced by µ. Assume that E ⊆ X satisfies µ∗(E) = µ∗(X) (but not that E ∈ S).
Let SE = {A ∩ E : A ∈ S}, and define the set function ν on SE defined by ν(A ∩ E) = µ(A). Show that SE is a σ-algebra on E and ν is a measure on SE. You should first prove that ν is well-defined, i.e., if A, B ∈ S with A ∩ E = B ∩ E, then µ(A) = µ(B).
2. Let X be the set of rational numbers. Denote R = {(a, b] ∩ X : −∞ ≤ a ≤ b ≤ ∞}. Let A be the collection of finite unions of sets of R.
(i) Prove that A is an algebra.
(ii) Show that σ(A) = 2X.
(iii) Let µ be a set function on A such that µ(∅) = 0 and µ(A) = ∞ for A 6= ∅. Then the extension of µ to σ(A) is not unique.
3. (i) Show that N (µ) = {E ⊂ X : E ⊂ N for some N ∈ S with µ(N ) = 0}
and S ∨ N (µ) = {E ∪ F : E ∈ S and F ∈ N (µ)}.
(ii) If ¯µ is the completion of µ, show that a set E is in the σ-algebra on which
¯
µ is defined iff there exist some A and C in the domain of µ with A ⊂ E ⊂ C and µ(C \ A) = 0.
4. If (X, S, µ) is a measure space and A1, A2, · · · , are any subsets of X, let Cj be a measurable cover of Aj for j = 1, 2, · · · .
(i) Show that S
jCj is a measurable cover ofS
jAj.
(ii) Give an example to show that C1∩ C2 need not be a measurable cover of A1 ∩ A2.
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