Math 501. 2nd Homework. Due Wednesday, September 26, 2007. Homework on "Chapter 3". 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 Collections: Mathematics