STAT 154 UNC

Homework # 12 1. Let , , be - nite measures on (, F ). (a) Show that if < and < , we have that < and d d d = , a.e. []. d d d Hint. Show that A l.h.s d = d d (b) Show that if then = A d 1 ( d ) , r.h.s d for all A F . a.e. (or ). (c)

Homework # 11 R 1. Let T , I T and RT , RT be as in class. Show that 0 (a) RT is a Field. 0 (b) (RT ) = RT 0 Hint. Try to write the cylinder sets in terms of the canonical coordinate process. 2. Construct a probability space on which is dened an i.i

Homework # 10 1. Let 1 = 2 = {1, 2, } and F1 = F2 = 21 . Let i be the counting measure on (i , Fi ), i = 1, 2. Dene (, F, )=(1 2 , F1 F2 , 1 2 ). Let f : IR be dened as if x = y 2 2x 2 + 2x if x = y + 1 (1) f (x, y) = 0 otherwise Show tha

Homework # 9 Let (, F) be a measure space. A set function : F (, ), is said to be a signed measure i it has the countable additivity property. I.e. ( i=1 i=1 Ai ) = (Ai ), if Ai F and Ai Aj = , i = j. 1. Let (, F, ) be a measure space. Let f

Homework # 8 In all the problems below (, F, ) is a measure space and {fn }, f, g are measurable functions. 1. Let be a nite measure. Let {f , } be a family of integrable functions. Show that the family is uniformly integrable if and only if: (i)

Hint for Question 3 Let > 0 be arbitrary. since fn is uniformly integrable, we can nd a L such that supn fn 1fn >L d < . Write fn d = fn 1fn >L d + + fn 1fn L d (fn 1fn L L)d + L(). Now take lim sup both sides and try to apply Fatous lemma. 1

Homework # 7 In all the problems below (, F, ) is a measure space and {fn }, f, g are measurable functions. We say that a measurable function f is integrable if |f |d < . 1. Suppose that f 0 a.e. and { : f () > 0} > 0. Show that f d > 0. 2. Suppose

Homework # 6 1. Let (, A, P ) be a probability space. Let Xn be a sequence of measurable maps from (, A) to R, R) such that Xn converges to X in probability (i.e. in measure). Let f : I I be continuous. R (I B(I R Show that f (Xn ) f (X) in probabi

Homework # 5 1. Let (, A, ) be a measure space. Dene A ={A N |A A and N B for some B A s.t. (B) = 0}. (i) Show that A is a eld. [Hint: The main diculty is showing closed under complements. Try to write the complement of A in terms of a union o