MTH404 Midterm Past Papers

MTH 404: Measure and Integration

Mid-Semester Examination Maximum Duration: 120 minutes Feb 29, 2016 Maximum Marks: 75

Note: Clearly state the results used.

Q1. (10 points)

1. Prove that any set with zero outer measure is measurable.
2. Let E be such that m∗E < ∞. Prove that E is measurable if and only if given there exists a Gδ-set O containing E such that m∗(O \ E) = 0.

Q2. (10 points) Let f : [0, 1] → R be an integrable function.

1. For every n ∈ N, define gn(x) = xnf (x), x ∈ [0, 1]. Prove that each gn is an integrable function on [0, 1].
2. Prove that

lim n→∞ Z 1 gn = 0.

3. (10 points) Let f be a non-negative integrable function on R. Prove that for any ϵ > 0, there is a measurable set E with mE < ∞ and Z R\E f < ϵ.

(Hint: Use the definition of R R f .)

Q4. (15 points) Let f be an integrable function on E.

1. Prove that f is finite a.e. on E.
2. Prove that

R A f = 0 for every measurable subset A of E if and only if f = 0 a.e. on E. (c) If f ≥ 0, prove that R E f = 0 if and only if f = 0 a.e. on E.

Q5. (15 points)

1. Let f be a non-negative measurable function on R. Prove that

lim n→∞ Z n

−n f = Z R f.

(b) Is part (a) true for a general function f ? Prove or give a counterexample.

Q6. (15 points)

1. Let f : [a, b] → R be continuous on [a, b] and differentiable on (a,
2. with supx∈(a,b) |f ′(x)| ≤ M , for some M > 0. Prove that f is a function of bounded variation.
3. Let f : [0, 1] → R be defined as f (x) = 1, x ∈ Q and f (x) = −1, x /∈ Q. Is f a function of bounded variation? Justify your answer.
4. Is f defined in part
5. an absolutely continuous function? Justify your answer.