 
Summary: Math 501. 2nd Homework. Due Wednesday, September 26, 2007.
Homework on "Chapter 3".
Definition 1.1. Let (, F, P) be a probability space. Let C1, . . . , Cm be collections of
measurable subsets of . We say that C1, . . . , Cm are independent, if for each C1
C1, . . . , Cm Cm, C1, . . . , Cm are independent events.
1. Let (, F, P) be a probability space. Let C1, . . . , Cm be independent collections of
measurable subsets of . Suppose that Ci is a class for each 1 i m. Let
1 k < m. Show that (C1 · · · Ck) and (Ck+1 · · · Cm) are independent
collections of events.
2. Let X, Y and Z be three independent r.v.'s. Let h : R2
R be a Borel function.
Show that X and h(Y, Z) are independent r.v.'s.
3. Let (, F, P) be a probability space. Let C1, . . . , Cm be independent collections of
measurable subsets of . Suppose that Ci is a class for each 1 i m. Let
1 k < l < m. Show that (C1 · · · Ck), (Ck+1 · · · Cl) and (Cl+1 · · · Cm)
are independent collections of events.
4. Let C be a class of sets of a universal set . Suppose that C is a class and a
class. Show that C is a field.
5. Let µ1 and µ2 be two measures on (R2
, B(R2
