Real Analysis Ph. D. Comprehensive Exam
August 2011
Do all parts of problem 1, and work 3 of the other 4 problems.
If not explicitly specified, the measure space is R
with the Borel σ-algebra B and Lebesgue measure λ.
1. True or false? Justify your answers.
(a) If (fn ) is a sequence of measurable non-negative functions on a measure space
Z X
∞
∞ Z
X
(X, X, µ), then
fn dµ =
fn dµ.
n=1
n=1
(b) If µ1 and µ2 are σ-finite measures with µ1 µ2 , then there exists a µ2 -integrable
R
function f with µ1 (A) = A f dµ2 for all measurable sets A.
(c) For every f ∈ L2 there exists a sequence of functions φn ∈ L2 , vanishing outside
[−n, n], such that φn → f in L2 .
(d) For every f ∈ L∞ there exists a sequence of functions φn ∈ L∞ , vanishing outside
[−n, n], such that φn → f in L∞ .
2. Let (X, X, µ) be a measure space, and let (Ek )∞
k=1 be a sequence of measurable sets
such that µ(Ek ) ≥ 2011 for all k ∈ N. Let E be the set of points in X which belong to
Ek for infinitely many indices k.
(a) Show that E is measurable.
(b) Under the additional assumption that µ(X) < ∞, show that µ(E) ≥ 2011.
(c) Give an example to show that the additional assumption in (b) is necessary.
3. Let fn : R → R be a sequence of Borel-measurable functions converging to 0 uniR
formly, and satisfying R |fn | dλ ≤ 1 for all n.
(a) Show that fn → 0 in Lp for all p > 1.
(b) Give an example to show that the assumptions do not imply fn → 0 in L1 .
Z 1 α
x −1
4. For α ≥ 0 let F (α) =
dx.
ln x
0
(a) Show that F is differentiable with F 0 (α) =
Z
(b) Use this result to calculate
0
1
1
.
α+1
x−1
dx.
ln x
5. Let f beZ a non-negative
Z ∞ measurable function on a σ-finite measure space (X, X, µ).
Show that
f dµ =
µ(Ft ) dt, with Ft = {x ∈ X : f (x) > t}.
X
0