21-720 F09: Measure and Integration, Assignment 5
R. Pego
Due Wednesday, December 2
5.1. Let be a regular signed Borel measure on Rn , with = 1 + 2 its Lebesgue decomposition
(|1 |
m, |2 | m). Prove (a) | = |1 | + |2 |, and (b) |1 | and |2 | are regular.
21-720 F09: Measure and Integration, Assignment 2
R. Pego
Due Monday, September 28
2.1. Let (X, F) be a measurable space and suppose fn : X R is F-measurable for all n N.
Prove that the set of points x where (fn (x) converges is measurable.
2.2. Let (X, F
21-720 F09: Measure and Integration, Assignment 4
R. Pego
Due Friday, November 13
4.1. Let (X, F, ) be a measure space, and suppose g : X R is integrable and
all E F. Show g = 0 -a.e.
E
g d = 0 for
4.2. (a) Let A Rn be Lebesgue measurable with m(A) < and
21-720 F09: Measure and Integration, Assignment 3
R. Pego
Due Friday, October 23
3.1. (A generalized Hlder inequality) Suppose p, q, r [1, ] and
o
if f Lp and g Lq then f g Lr and f g
r
f
p
1 1
1
+ = . Show that
p q
r
g q.
3.2. Let (X, F, ) be a nite mea