Qualifying Exam In Analysis August 2010
Pick three of the following. Identify which ones you are doing.
1. Let f∈L1(Ω, µ) where (Ω,F, µ) is a probability space (P(Ω) = 1). Also let Gbe a σalgebra of sets, G ⊆ F.
Show there exists a unique function, denoted by E(f|G) which is Gmeasurable, and for all A∈ G,
ZA
fdµ =ZA
E(f|G)dµ
2. In the situation of Problem 1, show that if f, g are functions in L1(Ω, µ) and f≤g, then E(f|G)≤E(g|G)
a.e. and that the map f→E(f|G) is linear.
3. Show the Vitali Convergence theorem implies the Dominated Convergence theorem for finite measure spaces,
but there exist examples where the Vitali convergence theorem applies but the dominated convergence theorem
cannot be applied.
4. A sequence of functions {fn}on a measure space (Ω,F, µ) is said to converge to 0 in measure if
lim
n→∞
µ([x∈Ω : |f(x)−fn(x)| ≥ ε]) = 0
for each fixed ε > 0. Show that there exists a subsequence {fnk}such that fnkconverges pointwise to 0 off a
set of measure zero.
5. A random variable is a measurable function X: (Ω,F, P )→Rnwhere P(Ω) = 1. Show there exists a unique
Radon measure called the distribution measure, λXdefined on a σalgebra of sets of Rncontaining the Borel
sets such that whenever Eis a Borel set in Rn,
λX(E) = P([X∈E]) .
Part of this is to show that [X∈E] is in Fwhenever Eis Borel.
6. Let {fn}be a sequence of functions defined on a finite measure space (Ω, µ) meaning (P(Ω) <∞).Also suppose
there exists a constant Csuch that for some p > 1,||fn||Lp< C. Show this collection of functions is uniformly
integrable.
Pick 7 of the following Identify which ones you are doing
1. Let f∈Lp(Rn), p > 1,with respect to standard Lebesgue measure. Show that
lim
y→0µZRn
|f(x)−f(x−y)|pdx¶1/p
= 0
Your proof should be based on standard facts about Lebesgue measure and advanced calculus.
2. Let {φn}be a mollifier. Recall that this means Rφn= 1, φnis infinitely differentiable, and the support of φn
is contained in B(0,an) where limn→∞ an= 0.Show that if f∈Lp(Rn),then
lim
n→∞
||f−f∗φn||Lp(Rn)= 0
(The measure is ordinary Lebesgue measure). Also show that f∗φnis infinitely differentiable. Your argument
should be based on standard theorems about the Lebesgue integral and Lebesgue measure.
3. Suppose |f(t)| ≤ Cer t for some r. and that f∈C0([0,∞)) ,the space of continuous functions which vanishes
outside a compact set. Show that if
Z∞
0
e−stf(t)dt = 0
for each ssufficiently large, then f(t) = 0 for all t.
