Real Analysis Comprehensive Examination–Math 921/922
Wednesday, January 21, 2015, 12:00-6:00p.m.
• Work 6 out of 8 problems. • Each problem is worth 20 points, parts of each problem don’t necessarily carry the same weight.
• Write on one side of the paper only and hand your work in order.
• Throughout the exam, the Lebesgue measure is denoted by m, BR denotes the Borel σ-algebra on R, and (X, M, µ) denotes a general
measure space.
(1) Let E ⊂ [0, 1] be a Lebesgue measurable set with m(E) > 0, and let 0 < < 1 be given.
a) Prove that there exists an open interval I such that m(E ∩ I) > m(I).
1
b) Assume A ⊂ [0, 1] is a Lebesgue measurable set such that m(A ∩ I) ≥ 10
m(I) for every
open interval I ⊂ [0, 1]. Prove that m(A) = 1.
(2) Let g : [0, 1] × [0, 1] → R be given by: g(x, y) = xy 3 , (x, y) ∈ [0, 1]2 . Define: µ(E) =
m(g −1 (E)), E ∈ BR , where m denotes the two dimensional Lebesgue measure on R2 .
a) Show that µ is a Borel measure on R.
R
b) Show all details in evaluating
the integral R t2 dµ(t).
R
(Hint: first show µ(E) = [0,1]2 χE ◦ g dm, for all E ∈ BR ).
(3) Assume f : [0, 1] → R is a continuous function. Prove the limit:
Z
f (xn )dm
lim
n→∞
[0,1]
exists and evaluate the limit. Does the limit always exist in R if we only assume f ∈
L1 ([0, 1], m)?
(x, t) exists
(4) Let f : [0, 1] × [0, 1] → R be B[0,1]2 -measurable such
that the partial derivative ∂f
∂t
for each (x, t) ∈ [0, 1]2 , and M := sup(x,t)∈[0,1]2 ∂f
(x, t) < ∞.
∂t
R
R
Prove that ∂f
is B[0,1]2 -measurable and dtd [0,1] f (x, t)dm(x) = [0,1] ∂f
(x, t)dm(x)..
∂t
∂t
(5) Let f, fn : [0, 1] → R such that f, fn ∈ L2 ([0, 1], B[0,1] , m), for every n ∈ N. Prove or disprove
the following statements:
a) If fn → 0 a.e. [0, 1] then kfn k2 → 0.
b) If fn → 0 in L2 ([0, 1], m), then fn → 0 in measure.
c) If fn → f weakly in L2 ([0, 1], m) and kfn k2 → kf k2 , then kfn − f k2 → 0.
d) If kfn − f k2 → 0, then kfn − f kp → 0 for every p ∈ [1, 2).
(6) Let f, g : R −→ R be a Borel measurable functions such that f, g ∈ L1 (R, m).
a) Briefly explain why the
R mapping (x, y) 7→ f (x − y)g(y) is BR2 -measurable.
b) Define: (f ∗ g)(x) := R f (x − y)g(y)dm(y). Prove that f ∗ g ∈ L1 (R, m) and
kf ∗ gk1 ≤ kf k1 kgk1 .
(7) Let µ and ν be σ-finite positive finite measures on a measurable space (X, M) with ν µ.
f
dν
dν
Put λ = µ + ν. Prove that f = dλ
exists, 0 ≤ f < 1 µ-a.e., and that dµ
= 1−f
µ-a.e.
(8) Let X be a compact Hausdorff topological space with the following property:
If E ⊆ X is open then its closure E is also open. Let C(X) denotes the space of all
continuous C-valued functions on X. Let f ∈ C(X) and > 0 be given. Prove that there
exist λ1 , ..., λn ∈ C, E1 , ..., En ⊆ X such that χEj ∈ C(X) for every j and
n
X
sup f (x) −
λj χEj (x) < .
x∈X
j=1