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Homework 2, Undergraduate Analysis
Refer to class notes for definitions of sets and functions in this homework assignment.
1. Prove that E , the collection of elementary sets in Rn, is not closed under countable union.
2. Verity the inequalities for the ”distance function” d(A, B) = µ∗(S(A, B)) defined in class.
3. Given A ∈ M(µ), and > 0, prove that there exists closed set F ∈ M(E ), open set G ∈ M(E ), such that F ⊂ A ⊂ G and
• µ(G) ≤ µ(A) +
• µ(F ) ≥ µ(A) − .
4. Let α : R → R be monotonic.
(a) Prove that left or right limit always exists at every point x ∈ R.
(b) Prove that the set function
µ([a, b]) = α(b+) − α(a+), µ([a, b)) = α(b−) − α(a+), µ((a, b]) = α(b+) − α(a−), µ((a, b)) = α(b−) − α(a−)
is regular. Here, α(x−) = limt→x−α(t) and α(x+) = limt→x+α(t).
5. Rudin Chapter 11, Exercise 3.
6. Rudin Chapter 11, Exercise 15.
7. Given a measure space (X, M, µ) and f : X → ¯R such that f−1((r, ∞]) ∈ M for all r ∈ Q, prove that f is measurable.
8. Given a measure space (X, M, µ) and X = AS B with A, B ∈ M, prove that a function f on X is measurable if and only if f |Aand f |B are measurable on A and B, respectively.
