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)
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 assum
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
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
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 e
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
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
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
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
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
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)
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-N