Real Analysis Comprehensive Examination–Math 921/922
Tuesday, May 28, 2013, 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 g : [0, 1] × [0, 1] → R be given by: g(x, y) = x2 y, (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 µRis a Borel measure
on R.
R
b) Prove that: R χE dµ = [0,1]2 χE ◦ g dm, for all E ∈ BR .
R
c) Show all details in evaluating the integral R t2 dµ(t).
(2) Let (X, M, µ) be a measure space and f ∈ L1 (X, M, µ) is an R-valued function. Prove the
following statements:
R
a) For every > 0 there is E ∈ M such that µ(E) < ∞ and X\E |f | dµ < .
b)
R For every > 0 there exists a δ > 0 such that, whenever E ∈ M with µ(E) < δ, then
|f | dµ < .
E
(3) Let (X, M, µ) and (Y, N , ν) be arbitrary measure spaces. Assume that f : X → C is Mmeasurable and g : Y → C is N -measurable. Put F (x, y) = f (x)g(y).
a) Show that F is M ⊗ N -measurable.
R
b) Prove that: If f ∈ L1 (µ) and g ∈ L1 (ν), then F ∈ L1 (µ × ν) and X×Y F d(µ × ν) =
R
R
f
dµ
g
dν
.
X
Y
(4) Let µ and ν be positive finite measures on the measurable space (X, M). Prove the following statements:
R
R
R
a) If f ∈ L1 (µ) ∩ L1 (ν) is an R-valued function, then X f d(µ + ν) = X f dµ
+
f dν.
X
R
Rb) There exists a nonnegative M-measurable function g on X, such that E (1 − g) dµ =
g dν, for all E ∈ M.
E
Z
sin(xk )
dm = 1.
(5) Prove: lim
k→∞ [0,∞)
xk
(6) Let (X, M, µ) be a measure space, fn ∈ Lp (µ) such that supn∈N kfn kp < ∞, and fn → f ,
µ-a.e. Prove the following statements:
a) If 1 < p < ∞, then fn → f weakly in Lp (µ). (You may use without a proof the results
of question (2)).
b) The result of part a) is false in general when p = 1.
(7) Let f : [0, 1] → R be a bounded Lebesgue measurable function such that
for k = 0, 1, 2, · · · . Prove that f = 0 m-a.e. [0, 1].
R
[0,1]
xk f (x) dm = 0,
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(8) Let (X, M, µ) be a measure space with µ(X) = 1. A sequence {fn }∞
n=1 ⊂ L (µ) is said to
be
R uniformly integrable if ∀ > 0, ∃ δ > 0 such that, whenever E ∈ M with µ(E) < δ, then
|fn |dµ < , for all n ∈ N. Prove the following convergence theorem of Vitali:
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1
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If {fn }∞
n=1 is uniformly integrable and fn → f µ-a.e., then f ∈ L (µ) and fn → f in L (µ).
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