Real Analysis Comprehensive Examination–Math 921/922
May 31, 2011, 12:00-6:00 p.m.
• Work 6 out of 8 problems. • Each problem is worth 20 points. • Write on one side of the paper
only and hand your work in order.
(1) Let h be
R a bounded Lebesgue measurable function on R having the property that
limn→∞ E h(nx)dx = 0, for every Lebesgue measurable
subset E of finite measure.
R
1
Show that for every f ∈ L (R), we have limn→∞ R f (x)h(nx)dx = 0.
(2) Let E be a Lebesgue measurable subset of R and consider the function defined by
fE (x) := |E ∩ (−x, x)|, for x ∈ [0, ∞), where |A| denotes the Lebesgue measure of A.
Show that
(a) f is a uniformly continuous map taking [0, ∞) onto [0, |E|);
(b) limx→∞ fE (x) = |E|.
(3) Let (X, M, µ) be a finite positive
space. Suppose f ∈ L∞ (X) is such that
R measure
n
kf k∞ > 0. For n ∈ N, put αn := X |f | dµ = kf knn . Show that
(a) kf kn → kf k∞ , as n → ∞ and
(b) αn+1 /αn → kf k∞ as n → ∞.
(4) Let A be a σ−algebra on a set X and assume µ : A → [0, ∞] has the following
properties:
(a) If A1 , A2 ∈ A with A1 ∩ A2 = ∅, then µ(A1 ∪ A2 ) = µ(A1 ) + µ(A2 ).
∞
(b) If {An }∞
n=1 is a sequence in A such that An+1 ⊂ An , for all n ∈ N, and ∩n=1 An = ∅,
then limn→∞ µ(An ) = 0.
Prove that µ is a positive measure on (X, A).
(5) Let µ and ν be two regular Borel measures
onRRn . Show that µ ≥ ν (i.e. µ(A) ≥ ν(A)
R
for every Borel set A) if and only if f dµ ≥ f dν for each nonnegative f ∈ Cc (Rn ).
(You may assume that for every compact set K and open set U with K ⊂ U there
exists f ∈ Cc (Rn ) such that χK ≤ f ≤ χU .)
2
(6) Let A ⊆ P(X) be an algebra. Define
(
Aσ :=
A ∈ P(X) : A =
∞
[
)
Ak with
{Ak }∞
k=1
⊆A ,
k=1
and
(
Aσδ :=
A ∈ P(X) : A =
∞
\
)
Ak with {Ak }∞
k=1 ⊆ Aσ
.
k=1
Let µ
e be a premeasure on A and µ∗ the induced outer measure.
(a) Prove that for any E ⊆ X and ε > 0, there exists an A ∈ Aσ such that E ⊆ A
and µ∗ (A) ≤ µ∗ (E) + ε.
(b) Let E ⊆ X satisfying µ∗ (E) < +∞ be given. Prove that E is µ∗ -measurable if
and only if there exists a B ∈ Aσδ such that E ⊆ B and µ∗ (B\E) = 0.
(c) Suppose that µ
e is σ-finite. Prove (b) without the hypothesis that µ∗ (E) < +∞.
(7) (a) Let m be the Lebesgue measure on [0, ∞), and let µ be a positive measure that is
differentiable with respect to m. If µ has a continuous Radon-Nikodym derivative
dµ
, show that the function x → µ([0, x]) is differentiable, and
dm
dµ
d
µ([0, x]) =
(x) for all x.
dx
dm
(b) Let f : X → [0,
R ∞] be measurable and let µ be a measure. Show that mf defined
by mf (E)R := X χE fRdµ is a measure and that for any measurable g : X → [0, ∞]
we have X gdmf = X gf dm. (Hint: Consider using the Monotone Convergence
Theorem.)
(8) Let ν : BR → (−∞, ∞) be the finite signed measure given by
Z
Z
x−2
ν(E) :=
dx +
e−x dx + aδ−2 (E),
x4 + 1
E∩(−∞,0]
E∩[0,∞)
for all E ∈ BR . Here a ≥ 0.
(a) Provide a Hahn Decomposition of R with respect to ν assuming that a > 0.
(b) Provide the Jordan Decomposition of ν assuming that a > 0.
(c) Identify |ν| and provide its Lebesgue Decomposition with respect to the Lebesgue
measure m assuming that a > 0.
dν
.
(d) Assume that a = 0, compute
dm