Real Analysis Comprehensive Exam
August 22, 2007
Do all problems. On this exam, λ denotes Lebesgue measure.
1. True or false? Justify your answers.
R
f 2 dλ is a Borel measure.
2
R
(b) If f : R → R is non-negative and Borel measurable, then µ(A) = A f dλ
is a Borel measure.
(a) If f : R → R is Borel measurable, then µ(A) =
A
(c) If fn → f in L2 ([0, 1]), then this convergence also holds in L1 ([0, 1]).
(d) If fn → f in L2 ([0, 1]), then this convergence also holds in L∞ ([0, 1]).
(e) There exists a Borel charge µ on R with µ((a, b]) = sin b−sin a for all a, b ∈ R,
a < b.
2. Find (with justification)
Z
lim
n→∞
0
1
1
cos nx
√ dx.
x
3. Let f, g : [0, 1] → [0, ∞) be Borel measurable functions with f (x)g(x) ≥ 1 for
(Lebesgue) almost every x ∈ [0, 1]. Show that
Z
Z
f dλ ·
g dλ ≥ 1.
[0,1]
[0,1]
(Hint: Cauchy-Schwarz inequality.)
Z
4. Let f, g : R → [0, ∞) be Borel measurable functions, and let µ(A) =
Z
ν(B) =
g dλ for Borel sets A, B ⊆ R.
f dλ,
A
B
(a) Show that µ and ν are σ-finite Borel measures absolutely continuous with
respect to Lebesgue measure.
(b) Let π be the product measure of µ and ν on R2 . Find the Radon-Nikodym
derivative of π with respect to two-dimensional Lebesgue measure λ2 . Justify your
answer.