Department of Mathematics, University of Michigan
Analysis Qualifying Exam, May 4, 2022
Morning Session, 9.00 AM-12.00
Problem 1: Suppose f: (0,1) →Ris integrable and define a function g: (0,1) →
Rby
g(x) = Z1
x
f(t)
tdt , 0<x<1.
Prove that gis also integrable.
Problem 2: Let f: [0,1] →Rbe a positive function such that fand 1/f are
integrable. Prove that log fis integrable and
lim
q→∞ q·Z1
0
f(x)1/q dx −1=Z1
0
log f(x)dx .
Problem 3: Let (Ω,A, µ) be a finite measure space. Let C ⊂ A be a sub-sigma
algebra of A. Prove that for any f∈L1(µ) there exists a C− measurable integrable
function gsuch that
ZE
g dµ =ZE
f dµ for any E∈ C .
Problem 4: Let fn: [0,1] →R, n = 1,2,..., be a sequence of non-negative
Lebesgue measurable functions such that limn→∞ fn(x) = 0 for almost every x∈
[0,1]. Prove there exists an infinite subsequence fnk, k = 1,2,..., such that the
series ∞
X
k=1
fnk(x) converges for almost every x∈[0,1] .
Hint: Use Egorov’s theorem.
Problem 5: Suppose for n= 1,2,..., the functions Fn: [a, b]→Rare increasing
and nonnegative, and that the function Fwith domain [a, b] defined by
F(x) =
∞
X
n=1
Fn(x),
is finite for all x∈[a, b]. Prove that the derivative F0(x) exists a.e. and
F0(x) =
∞
X
n=1
F0
n(x) for almost every x∈[a, b].