Suggestions for Problem Set 11
3. (27.11; alternative suggestion; original suggestion below):
8
8
W8 " ! \5 . Define W 8 " ! ^85 where
=8
=8
5"
Let =8 8 ln 8)"# , and
5"
\
^85 5
!
if l\5 l 8"# ln 8
.
otherwise
(note that unlike the ] variables in Billings
Suggestions for Problem Set 1
1 (Billingsley, 1.2). Recall that a set K is open if for every point B K, there is an
interval B % B % around B which is also contained in K. The definition of a dense set of
numbers is in Appendix 15: a set F ! " is dense if
Suggestions for Problem Set 8
1. (22.1) Show that T ] + is 0 or 1 for all +, so that ] must be concentrated on a single
point +.
3. (22.2) (Let \" \# be independent.). For sufficiency, you can also note that
#



!Z \8 !I l\8 l# Il\8 l Show that both s
Suggestions for Problem Set 6
2. (18.10) (a) Both . and / are the same measure on the natural numbers " # $ .
If E then .E / E number of elements in E. Show that for any function 0 B
(with B ),
( . . 0 B " 0 3
_
3_
is the sum of all values of 0 , and simi
Suggestions for Problem Set 4
4. (13.7) Following the suggestion, if 0 is a fixed function, and is 18 a sequence in k , with
0 18 k show that 0 lim 18 k , and thus that k0 1 0 1 k (with 0 still
8_
fixed) is closed under pointwise limits. Thus show that if
Suggestions for Problem Set 4
1. Consider the 5 field Z 5 E" E# . Try using Theorem 13.4 to show that the event I
of interest is measureable with respect to Z . First show that the set of points where the limit
8
exists is measureable and that the limit
Suggestions for Problem Set 9
2. (14.5) To prove .J K ! iff J K , show that if J B! KB! and B! is a
continuity point of both functions, then .J K cannot be !. To show the triangle inequality,
show first that for any %3 such that for all B J B %" %" KB J B
The HahnBanach and RadonNikodym Theorems
1. Signed measures
Definition 1. Given a set H and a 5 field of sets Y , we define a set function
. Y to be a signed measure if it has all the properties of a measure
but can be negative; in particular we requir
Notes on the Unique Extension Theorem
1. More on measures:
Recall that we were interested in defining a general measure of a size of a set
on ! ".
Defined this measure T . Defined
T (+ , ) , +
Question: how large a collection of sets Y can we extend this
Measurable Functions and their Integrals
1 General measures: Section 10 in Billingsley
Recall: a probability measure T on a 5field Y on a space H is a realvalued function on Y with the properties:
(a) T 9 !
(b) T H "
(c) T E3 ! T E3 if E3 are disjoint.
_
Product Measures and Fubini's Theorem
1. Product Measures
Recall: Borel sets U 5 in 5 are generated by open sets. They are also
generated by rectangles V N" N5 which are products of intervals
N3
Let V be the collection of all rectangles  we have shown t
Suggestions for Problem Set 13
"
1. (34.4) (a) Show first that ' 0 .T! T F 'F 0 .T for any measurable 0 . To show that
T! EllZ T FllZ T E FllZ , integrate the left side over a set K Z , obtaining
( I! ME llZ IMF llZ .T! ( ME IMF llZ.T!
K
K
where I! is the