18.103 Fall 2013
Problem Set 1 Solutions
1.1/1 If the collection of Bernoulli sequences were countably innite, then it would be
possible to enumerate all of them in a list indexed by the positive integers: B1 , B2 , . . . .
Dene the sequence B by saying t
18.103 Fall 2013
Problem Set 6 Solutions
1. Let E [0, 1] be a nonmeasurable set. F = E cfw_0 is contained in [0, 1] cfw_0, a rectangle
with zero measure. Thus F has outer measure 0 and hence is measurable (F M(R2 ). If
F M(R) M(R), then by Proposition 3 (
18.103 Fall 2013
Problem Set 5 Solutions
2.5/11 Denote the Borel sets of [0, 1] by B and
S = cfw_E B : (E ) = (E )
where is Lebesgue measure and is another measure satisfying (I ) = (I ) for all intervals.
The intersection of intervals is again an interva
18.103 Fall 2013
Problem Set 4 Solutions
2.3/2 Since 0 fn 1/n on R, fn 0 uniformly. On the other hand, fn = (1/n)1(0,n) is
a simple function so
fn dL = (1/n)(0, n) = 1
In order to apply the dominated convergence theorem to this sequence, we would need a
m
18.103 Fall 2013
Problem Set 3 Solutions
1.4/10
a) Let
Ac
1
Ai
Bi =
i=1
otherwise.
We must show that if
1 i1 < i2 < < ik n,
then
k
k
(
Bij ) =
j =1
(Bij )
j =1
We may assume i1 = 1, because otherwise the result comes from independence of the Ai .
But then
18.103 Fall 2013
Problem Set 2 Solutions
1.1/21
Suppose that
[a, b] =
Ij
j =1
with Ij intervals. We will show that
ba
(Ij )
j =1
in which
denotes length.
Fix > 0 and choose open intervals Ij so that Ij Ij and (Ij ) (Ij ) + 2j . Then the
Ij are an open cov
First test, solutions
You are permitted to bring the book `Adams and Guillemin: Measure Theory And Probability' with
you - just the book, nothing else is permitted (and no notes in your book!) You may use theorems,
lemmas and propositions from the book.
1
Test 2 with solutions
This test is closed book. You are not permitted to bring any books, notes or such material with you.
You may use theorems, lemmas and propositions from the book or from class.
Note that most of the solutions are relatively short - th
The final
This exam is closed book, no books, papers or recording devices permitted. You may use theorems
from class, or the book, provided you can recall them correctly.
With short solutions and comments. I hope I did not damage too many delicate egos