Download June 2012

Real Analysis Comprehensive Examination–Math 921/922
Thursday, May 31, 2012, 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, L[a,b] denotes the Lebesgue
measurable subsets of [a, b], and (X, M, µ) denotes a general measure space.
(1) Let X = {1, 2, 3} and define µ∗ : P(X) −→ [0, ∞] by: µ∗ (∅) = 0, µ∗ (X) = 2, and µ∗ (A) = 1,
for any other set A ⊂ X.
a) Show that µ∗ is an outer measure on X. Is µ∗ a pre-measure on P(X)?
b) Find M, the σ-algebra of µ∗ -measurable subsets of X.
c) Define (the inner measure) µ∗ : P(X) −→ [0, ∞] by: µ∗ (A) = µ∗ (X)−µ∗ (Ac ), for A ⊂ X.
A subset A of X is called reasonable, if µ∗ (A) = µ∗ (A). Put E = {A ⊂ X : A is reasonable}
and define µ : E −→ [0, ∞] by µ(A) = µ∗ (A), A ∈ E. Find E and determine whether or not
µ is a measure on E.
(2) In a general measure space (X, M, µ), state the Monotone Convergence Theorem and Fatou’s Lemma, and prove they are equivalent.
(3) Let (X, M, µ) be a measure space and fn , f ∈ Lp (µ), n ∈ N, 1 < p < ∞. Prove
that: fn −→ f weakly in Lp (µ) if and only if {kfn kp }∞
n=1 is a bounded sequence and
R
R
f
dµ
−→
f
dµ,
for
every
E
∈
M
of
finite
measure.
E n
E
(4) Prove (with details) the following (unrelated) statements:
R
a) If f ∈ L1 (R, m) and g ∈ L∞ (R, m), then limx→0 R |g(y)||f (x + y) − f (y)|dm(y) = 0.
R
x)
b) limn→∞ [0,∞) cos(e
dm(x) = 0.
1+nx2
(5) Let f : [a, b] −→ R be an absolutely continuous function. Prove the following statements:
a) If f is non-decreasing on [a, b] and E ∈ L[a,b] with m(E) = 0, then m(f (E)) = 0.
b) If 1 < p < ∞, f 0 ∈ Lp ([a, b], m), α = 1q , where p1 + 1q = 1, then there exists a constant
M > 0, such that |f (x) − f (y)| ≤ M |x − y|α , for all x, y ∈ [a, b].
(6) Let f, g : R −→ R be a Borel measurable functions
such that f ∈ L1 (R, m), g ∈ Lp (R, m),
R
where 1 < p < ∞, and define: (f ∗ g)(x) := R f (x − y)g(y)dm(y). Prove that (f ∗ g)(x)
is finite a.e. R, f ∗ g ∈ Lp (R), and kf ∗ gkp ≤ kf k1 kgkp . (Hint: If h ∈ Lq (R, m), where
R
1
+ 1q = 1, consider the linear functional Φ(h) = h(x)(f ∗ g)(x)dm(x). Address measurap
bility, use Fubini-Tonelli and Riesz).
(7) Let C([0, 1]) denote the space of all continuous R-valued functions on [0, 1] endowed with
the norm kf k := supx∈[0,1] |f (x)|.
a) For any 0 ≤ a < b ≤ 1, put E := {f ∈ C([0, 1]) : f is non-decreasing on [a, b]}. Prove
that E is nowhere dense in C([0, 1]).
b) Use a theorem of Baire to prove that there exists a function f ∈ C([0, 1]) such that f is
neither non-decreasing nor non-increasing on any sub-interval of [0, 1].
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(8) Let (X, M, µ) be a finite measure space, fn , f : X −→ R, where {fn }∞
n=1 ⊂ L (µ), and f
∞
1
is M-measurable function. A sequence {fn }n=1 ⊂ L (µ) is said to be Runiformly integrable
if ∀ > 0, ∃ δ > 0 such that, whenever E ∈ M with µ(E) < δ, then E |fn |dµ < , for all
n ∈ N. Prove that: f ∈ L1 (µ) and fn −→ f in L1 (µ) if and only if fn −→ f in measure
1
and {fn }∞
n=1 is uniformly integrable. (Hint: To show f ∈ L (µ) use Egoroff).
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