Due Friday in class, Oct 13. Relevant sections in Durrett’s textbook: 1.1,1,2,1.3, ap-
pendix A1; in Resnick book: Chapter 2. Justify all your answers.
1. SupposeCis a class of subsets of and suppose B satisfies that B2 (C). Show
that there exists a countable subclassCB C such that B2 (CB).
(Hint: This exercise is similar to Problem 6 on HW1. Set L to be the collection of
subsets of which satisfy the property described, use theorem to show L contains
(C) . Mind when you set up the proof thatC is not necessarily a -system.)
2. An atom in a probability space ( ;B;P) is defined as a setA2Bsuch thatP(A) > 0
and if B A and B2B, then either P(B) = 0 or P(AnB) = 0.
i) If = R and P is determined by a distribution function F(x), show that the atoms are
the discontinuity points of F, that isfx : F(x) F(x ) > 0g, where F(x ) is the left
limit of F at x.
ii) Let A;B be two atoms in ( ;B;P) . Show that either P(A4B) = 0 or P(A\B) = 0.
iii) Show that a probability space contains at most countably many atoms. In particular,
a distribution function F has at most countably many discontinuities.
3. nN =f1;2;:::g,defineF =fA N : A is finite orA containsfn;n+1;:::gfor some n2
Ng. Let : F ![0;1] be
(A) =
(
0 if A is finite
1 otherwise:
i) Show that F is a field and is a finitely additive meausure on F (meaning for a finite
disjoint collection A1;A2;:::An2F, ([ni=1Ai) = Pni=1 (Ai)).
ii) Such a finitely additive measure of total mass 1 is often called a mean, it behaves
differently from a probability measure. If An 2F is a countable sequence of subsets
such that An#;, is it true that (An)#0?
4. Let P be a probability measure on a field F0. Recall in the Carathéodary extension
theorem, we defined the outer measure P by
P : 2 ![0;1]
P (A) = inff
1X
n=1
P(Bn) : A [1n=1Bn; Bn2F0 for all ng:
Also recall the -fieldF which consists of P -measurable sets.
1
i) Denote by P the probability measure on ( ;F) which extends P on F0. Show that
P : 2 ![0;1] can be written as
P (A) = inffP(B) : A B;B2Fg:
Moreover, the infimum is always achieved.
ii) Similarly, one can define an inner measure
P : 2 ![0;1]
P (A) = supfP(B) : B A;B2Fg:
Show that A is P -measurable (that is A2F) if and only if P (A) = P (A).
5. (Exercise 1.3.5 on P16 Durrett) A function f : R!R is lower semicontinuous (l.s.c.)
if
lim infy!x f(y) f(x):
Show that f is l.s.c. if and only if fx : f(x) ag is closed for any a 2 R. Using this
characterization, show that a l.s.c. function is measurable with respect to the Borel -field
on R.