Serge Ballif
MATH 501 Homework 5
October 31, 2007
In all problems (X, M, ) is a measure space
1. Let f : X [a, b] where [a, b] is a finite interval. Assume that the function x f (x, y) is integrable for every y [a, b]. Assume also that y f (x,
1 1.1
Measure Rectangles
Almost Disjoint. A union of rectangles is said to be almost disjoint if the interiors of the rectangles are disjoint. Lemma 1.1. If a rectangle is the almost disjoint union of finitely many other rectangles, say R = N N k=1
Serge Ballif
MATH 501 Homework 9
December 12, 2007
1. Let and be measures on (X, M) such that . Set = + . Show that if f is the Radon-Nikodym f d d derivative of with respect to , f = d , then 0 f < 1 -a.e. and d = 1-f .
We first note that
Serge Ballif
MATH 501 Homework 8
November 30, 2007
1. Let (n ) be a Dirac sequence such that each n is continuous and supp n = B 1/n (0). (a) Show that (n (x) converges to 0 for almost every x d . Deduce that (n ) does not converge in L1 ( d ). (
Serge Ballif
MATH 501 Homework 7
November 16, 2007
1. Let E be a Borel subset of 2 . Prove that for every y , the slice E y is a Borel set in collection C of subsets of 2 having the property that E y is a Borel set in for every y .
R
R
R
R
Serge Ballif
MATH 501 Homework 6
November 9, 2007
1. Let (V, , ) be an inner product space. Prove that if | x, y | = x y and y = 0, then x = ay for some a
C.
We follow the proof outlined in Folland. Define a = Claim 1. x, z = z, x = | x, y |
Serge Ballif
MATH 501 Homework 4
October 22, 2007
In problems 1 and 5, (X, M, ) stands for the measure space and L1 = L1 (X, M, ).
1. Let (X, M, ) be a measure space such that (X) < . Assume that (fn ) is a sequence of integrable functions fn : X
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MATH 501 Homework 3
October 10, 2007
In problems below L stands for the -algebra of Lebesgue measurable subsets of d , m for Lebesgue measure on L, (X, M, ) stands for the measure space, and L1 = L1 (X, M, ).
1. Let E L be such that
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MATH 501 Homework 2
October 3, 2007
In problems 1-3, (X, M, ) denotes a measure space.
1. Let (Ek )kN M. Define lim inf Ej =
i=1 k=i
Ek
and
lim sup Ej =
i=1 k=i
Ek
Prove the following. (a) (lim inf Ej ) lim inf (Ej ). (b) I
Serge Ballif
MATH 501 Homework 1
September 26, 2007
In all problems below, measurable means Lebesgue measurable; m is the Lebesgue measure and m is the Lebesgue outer measure. 1. The Cantor Set. Denote by C0 = [0, 1]. Let C1 be the set obtained by