MTH404 Quiz 1 Set 2

MTH 404: Measure and Integration

Quiz 1 (Feb 18, 2016) Maximum Duration: 55 minutes

Note: Each problem carry 10 points. Clearly state the results used.

Q1. Let A = {A ⊆ R | either A or Ac is finite}. Determine whether

1. A is an algebra of sets.
2. A is a σ-algebra of sets.

Q2. Let E be a measurable subset of R with finite measure and c > 0 be a fixed real number. Let (En) is a sequence of measurable sets in E with m(En) > c, for all n ≥ 1. Let F = lim sup En.

1. Prove that F is a measurable set.
2. Prove that m(F ) ≥ c.
3. Give an example to show that conclusion
4. can fail if m(E) = ∞.
5. Let (fn) : R → R be a sequence of non-negative integrable functions converging a.e. to an integrable function f . Let limn→∞ R R fn = R R f . Prove that for any measurable subset E of R,

lim n→∞ Z

E fn = Z

E f.

4. Let f : [1, ∞) → R be a non-negative measurable function defined as f (x) = 1/x. Is the function f Lebesgue integrable over [1, ∞)? Justify your answer.