221 Analysis 2, 200809
Suggested solutions to exercise sheet 4
Let (X, M, m) be a measure space.
1. Fix a measurable set B M. Show that the function mB : M [0, ]
given by mB (A) = m(A B) for A M, is a measure.
Solution. We have mB () = m( B) = m() = 0. If
221 Analysis 2, 200809
Suggested solutions to exercise sheet 1
1. Show that if A is a ring of subsets of a set X, then A B A whenever
A A and B A.
Solution. We have A B = (A B) \ (B \ A) (A \ B) (to see this, draw
a Venn diagram). Since A is closed under
221 Analysis 2, 200809
Suggested solutions to exercise sheet 3
1. For each of the following collections S of subsets of R, determine whether
or not (S) is equal to the collection of Borel subsets of R. Justify your
answers.
(a) S = P(R)
(b) S = cfw_the cl
221 Analysis 2, 200809
Summary of theorems and denitions
Measure theory
Denition. The extended real line is the set
[, ] = R cfw_,
where and are two symbols which do not belong to R. We extend
order, addition and product on R to [, ] in the natural way b
221 Analysis 2, 200809
Suggested solutions to exercise sheet 2
1. Show that if m1 and m2 are measures on a ring A and 1 , 2 [0, ] then
m(A) = 1 m1 (A) + 2 m2 (A) denes a measure m on A.
Solution. Since m1 (A), m2 (A) [0, ] and 1 , 2 [0, ] we have m(A)
[0
221 Analysis 2, 200809
Suggested solutions to exercise sheet 5
In these questions, the words measurable and integrable should be taken
to mean Lebesgue measurable and Lebesgue integrable.
1. Either give an example, or explain why no such example exists, o
A xed proof that m extends m
February 2, 2009
Let A be a ring of subsets of a set X and let m : A [0, ] be a measure
on A. We know:
1. m is increasing: if A, B A with B A then m(B) m(A);
2. A is closed under nite intersections; and
i=1
3. m is countably s
The Cantor set is uncountable
February 13, 2009
Every x [0, 1] has at most two ternary expansions with a leading zero;
that is, there are at most two sequences (dn )n1 taking values in cfw_0, 1, 2
with
def
dn 3n .
x = 0.d1 d2 d3 =
n=1
1
3
For example, = 0
Abbreviated summary of the second half of
221 Analysis 2, 2008-09
Disclaimer: This is intended to give you an overview of the second half of the course, and
many of the statements are rather vague, being intended to evoke the precise statements
given in t