Real Analysis Comprehensive Examination–Math 921/922
Thursday, June 5, 2014, 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 X be a nonempty set and S ⊂ P(X) be an algebra. For a fixed x0 ∈ X, define
(
µ0 : S −→ {0, 1} by:
1, if x0 ∈ E;
µ0 (E) =
0, if x0 ∈ E c .
a) Show that µ0 is a pre-measure on S.
b) Let µ∗ : P(X) −→ [0, ∞] be the outer measure induced by µ0 . Given E ⊆ X, prove that:
if x0 ∈ E, then µ∗ (E) = 1.
c) Show that the sets in S are µ∗ -measurable.
(2) Let (X, M, µ) be a measure space, 1 ≤ p < r ≤ ∞. Let X = Lp (µ) ∩ Lr (µ) be endowed
with the norm kf k = kf kp + kf kr . Prove that (X, k·k) is a complete normed space. Don’t
bother proving the fact that (X, k·k) is a normed space.
πy n
o
(3) Let f (x, y) = χE (x, y) x−3/2 cos
, where E = (x, y) ∈ (0, 1)2 : 0 < y < x, 0 < x < 1 .
2x
Use
Fubini-Tonelli’s
Theorems
to
evaluate (with all justifications) the single integral:
R
f
(x,
y)
d(m
×
m).
(0,1)×(0,1)
(4) Let {fn }∞
n=1 be a sequence of R-valued continuous functions on a complete metric space
X. Use a theorem of Baire to prove the following special case of the Uniform Boundedness
Principle: If for every x ∈ X, Mx := supn∈N |fn (x)| < ∞, then there exist a nonempty open
set U ⊆ X and a constant C > 0 such that |fn (x)| ≤ C for all x ∈ U and all n ∈ N.
1
(5) For each fixed 1 < p < ∞, let fn (x) = n p e−nx , where x ∈ (0, ∞), n ∈ N.
a) Prove that: fn −→ 0 point-wise on (0, ∞).
b) Prove that: {fn } does not converge to 0 strongly in Lp ((0, ∞), m).
c) Prove that: fn −→ 0 weakly in Lp ((0, ∞), m). Does the same statement hold when p = 1?
(6) Let ν : LR −→ [−∞, ∞] be the signed measure defined by: ν(E) = m(E) − 3 m(E ∩ [0, 1]),
where LR denotes the Lebesgue measurable sets in R.
a) Provide, with justification, a Hahn decomposition of R with respect to ν, and specifically
describe all sets E ∈ LR such that ν(E) = 0.
b) Provide, with justification, a Jordan decomposition of ν.
dν
c) Compute
.
dm
(7) a) LetR f : R → R be a non-decreasing function. Prove that: whenever −∞ < a < b < ∞,
then [a,b] f 0 (t) dm(t) ≤ f (b) − f (a). (Hint: Use Fatou’s Lemma).
b) Let f : R → R be a non-decreasing function and such that: f (−∞) = 0 and f (∞) = 1.
Prove
R 0 that: f is absolutely continuous on every closed bounded segment if and only if
f (t) dm(t) = 1.
R
(8) Let fn , f ∈ Lp ((0, 1), B(0,1) , m); n ∈ N, be R-valued functions, where 1 < p < ∞, {kfn kp }
is a bounded sequence and fn −→ f in measure. Prove that:
fn → f weakly in Lp ((0, 1), B(0,1) , m).
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