Real Variables Midterm
Monday, October 25, 2010
You must nish by 6:25 p.m.
Please solve all 4 problems and all parts of each problem.
Each part of each problem counts the same (4 percentage points).
Identify your answers by drawing boxes around them.
Real Variables, Fall 2014
Problem set 2
Solution suggestions
Exercise 1. (Egoros Theorem) Let (fn ) : B R be a sequence of measurable
n=1
functions that converge almost everywhere to f : B R and assume
that m(B) < . Then for every > 0 there exists a measu
Real Variables, Fall 2014
Problem set 1
Solution suggestions
Exercise 1. Let E be a given set. Prove that the following statements are equivalent.
(i) E is measurable.
(ii) Given > 0, there exists an open set O E so that m (O\E) < .
(iii) Given > 0, there
Real Variables, Fall 2014
Problem set 4
Solution suggestions
Exercise 1. Let f be of bounded variation on [a, b]. Show that for each c (a, b),
limxc f (x) and limxc f (x) exist. Prove that a monotone function (and
hence a function of bounded variation) ca
Real Variables, Fall 2014
Problem set 3
Solution suggestions
Exercise 1. Let f be a nonnegative measurable function. Show that
f = sup
,
where is taken over all simple functions with f .
Answer:
For each n N we divide [0, n) to disjoint intervals
Ik =
k1
Real Variables, Fall 2014
Problem set 5
Solution suggestions
Exercise 1. Let f be absolutely continuous on [a, b] Show that
b
b
Ta (f ) =
|f (x)| dx
a
and
b
b
Pa (f )
[f ]+ .
=
a
Conclude that if f is in AC then it is the dierence of two monotone
absolute
Real Variables, Fall 2014
Problem set 9
Solution suggestions
Exercise 1. Let C be a semi-algebra of sets and a non negative set function dened
on C with () = 0 (if C). Then has a unique extension to a
measure on the algebra A generated by C if the followi
Real Variables, Fall 2014
Problem set 11
Solution suggestions
Exercise 1. Is Hausdor measure m on Rn -nite?
Answer:
We consider the cases n = , n < and n > separately.
(i) For = n the Hausdor measure is a constant times the
dimensional Lebesgue measure on
Real Variables, Fall 2014
Problem set 10
Solution suggestions
Exercise 1. Show that if f : X Y R is measurable with respect to the -algebra
A B, then fx : Y R dened by fx (y) = f (x, y) is measurable for
all x X.
Answer:
If E X Y , then for each x X we de
Real Variables, Fall 2014
Problem set 6
Solution suggestions
Exercise 1. Show that if f Lp and g Lp then f + g Lp even for 0 < p < 1.
Answer:
Fix 0 < p < . Then note that
|f (x) + g(x)| |f (x)| + |g(x)| 2 maxcfw_|f (x)|, |g(x)|
for all x. Since the functi
Real Variables, Fall 2014
Problem set 7
Solution suggestions
Exercise 1. Let (X, B, ) be a measure space. Show that we can nd a complete
measure space (X, B0 , 0 ) so that
(a) B B0
(b) E B (E) = 0 (E)
(c) E B0 if and only if E = A B where B B and A D with
Real Variables, Fall 2014
Problem set 8
Solution suggestions
Exercise 1. Show that the Radon-Nikodym theorem for a nite measure implies
the theorem for a -nite measure.
Answer:
Assume that the Radon-Nikodym theorem holds for nite measures
and let
where