Real Analysis Homework #1 Due 9/29 1. Let

Real Analysis Homework #1

Due 9/29 1. Let S be the σ-algebra generated by A, then S is the union of the σ- algebras generated by E as E ranges over all countable subsets of A.

2. Let A ⊂ 2^X be an algebra, A_σ the collection of countable unions of sets in A, and A_σδ the collection of countable intersections of sets in A_σ. Let µ : A → [0, ∞] be countably additive on A and µ^∗ the induced outer measure.

(i) For any set E ⊂ X and  > 0 there exists A ∈ A_σ with E ⊂ A such that µ^∗(A) ≤ µ^∗(E) + .

(ii) If µ^∗(E) < ∞, then E is µ^∗-measurable iff there exists B ∈ A_σδ with E ⊂ B and µ^∗(B \ E) = 0.

(iii) Generalize (ii) to the case where µ is σ-finite.

3. As in last problem, we further assume µ(X) < ∞. Let E ⊂ X, we define the inner measure of E to be µ∗(E) = µ(X) − µ^∗(E^c). Then E is µ^∗-measurable iff µ^∗(E) = µ_∗(E).

4. Let E be the collection of the sets A_a,b = [−b, −a) ∪ (a, b] for all 0 < a <

b < ∞ and A the algebra generated by E . Define µ(A_a,b) = b − a. Show that µ extends to a countably additive measure on a σ-algebra. Is [1, 2]

measurable for µ^∗?

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