Download May 2015

Real Analysis Comprehensive Examination–Math 921/922
Friday, May 29, 2015, 12:00-6:00 p.m.
• Work 6 out of 8 problems.
• Each problem is worth 20 points, the 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 m denotes the Lebesgue measure, L stands for the Lebesgue σ-algebra on R, BR denotes the Borel
σ-algebra on R and (X, M, µ) indicates a general measure space unless otherwise indicated.
(1) Let µ∗ be an outer measure induced by a finite pre-measure µ0 on an algebra A ⊂ 2X and let M∗ be the
associated Caratheodory
σ-algebra.
(a) Prove that µ∗ M∗ is complete.
(b) Prove that for any set E there is B ⊂ Aσδ such that µ∗ (E) = µ∗ (B).
(2) The parts of this problem are not directly related.
(a) Suppose f : R → R is monotone increasing. Prove that f is Borel-measurable and deduce that any function
g : R → R of bounded variation is also Borel-measurable.
(b) Let (X, M, µ) be a measure space with positive measure µ. Prove that the set {χE : E ∈ M, µ(E) < ∞}
is closed in L1 (X, µ).
(3) Suppose f : (0, 1] → R is Riemann integrable on [a, 1] for every 0 < a < 1 and the improper Riemann integral
R1
R 0 f (x)dx is convergent (“R” indicates “Riemann”).
(a) Prove that if f is bounded on (0, 1), then f ∈ L1 ((0, 1)).
Z 1
(b) Give an example of an R-valued function f on (0, 1] (it need not be continuous) such that R
f (x)dx is
0
convergent, but f is not in L1 ((0, 1)).
(c) Suppose f is not necessarily bounded, but changes sign finitely many times in (0, 1]. Prove f ∈ L1 ((0, 1)).
(4) Let (X, M, µ) be a measure space with positive finite measure µ. Suppose fn : X → [0, ∞), n ∈ N, are Mmeasurable and converge point-wise µ-a.e.R to a function f ∈ L1 (X, µ). Assume for every ε > 0 there exists δ > 0
such that E ∈ M with µ(E) < δ implies E fn dµ < ε for n ≥ N , for some N ∈ N. Prove
Z
Z
fn dµ = f dµ .
lim
n→∞
(5) For i = 1, 2, let µi , νi be finite measures on measurable spaces (Xi , Mi ) such that νi µi . (Remark: in this
problem the finiteness assumption is not essential and can be replaced with σ-finiteness, but you need not consider
this general case.) Prove the following (you may use without a proof the fact that ν1 × ν2 µ1 × µ2 ):
d(ν1 × ν2 )
dν1
dν2
(x1 , x2 ) =
(x1 ) ·
(x2 )
d(µ1 × µ2 )
dµ1
dµ2
µ1 × µ2 -a.e.
(6) Let G : R → R be continuous strictly increasing and µG denote the associated Lebesgue-Stieltjes measure such
that µG ((x, y]) = G(y) − G(x). Let G(a) = c, G(b) = d. In this problem you may not cite any change of variable
theorems.
(a) Assume f : R → R is bounded and Borel-measurable. Prove that
Z
Z
f dm =
f (G(x))dµG (x)
[c,d]
[a,b]
(b) Extend the result to the case when f is Borel-measurable and in L1 ([c, d]) (instead of being bounded).
(7) Let X = [0, 2015].
(a) Prove that there is C > 0 such that for every absolutely continuous function f : X → R with f (0) = 0
kf kL1 (X,m) ≤ Ckf 0 kL1 (X,m)
(b) Show that the statement (a) is false if the condition “absolutely continuous” is replaced by “m-a.e. differentiable on X with f 0 ∈ L1 (X, m)”.
(8) Let (X, M, µ) be a measure space where µ is positive and σ-finite.
(a) Suppose 1 ≤ p < ∞. Prove that weak limits in Lp (X, µ) are unique. In other words, show that if a sequence
converges weakly to two functions f, g ∈ Lp (X, µ), then these functions must be equal µ-a.e.
(b) Assume p, q > 1 and p1 + 1q = 1. Suppose fk ∈ Lp (X, µ) for k ∈ N and fk → f in Lp (X, µ) as k → ∞. Next,
assume functions {gk }k∈N form a bounded subset of Lq (X, µ) and gk → g weakly in Lq (X, µ) as k → ∞.
Prove that fk gk → f g weakly in L1 (X, µ).