## Homework 2

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## Homework 2

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Problems/Exams

- Pages:
- 3
- School:
- University of Louisville
- Course:
- Math 660 - Probability Theory

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1 HomeWork 2 DUE DATE OCT 19 2010 MATH 660 Instructor Dr RON SAHOO Direction This homework worths 100 points To receive full credit answer each problem correctly and must show all works 1 Let X 1 2 3 C1 1 2 3 X and C2 3 1 2 X Verify that C1 and C2 are both algebras but C1 C2 is not a algebra 2 Let X and Y be non empty sets and Y F be a measurable space For any function f X Y show that the collection f 1 A A F is a algebra in X 3 Let X C be a measurable space If the algebra C has a finite number of subsets of X in it then C is also a topology on X 4 Let X a b c d C X a b c d and let F P X the set of all subsets of X Define the functions f g X X by f x a for x X a if x a b c if x c d and g x Show that f is C F measurable but g is not C F measurable 5 a Let f X R be a constant function f x for all x X Show that f is measurable b Let f X R be a measurable function and R be any real constant Show that f is also measurable 6 Let X C be a measurable space and let f g X R be any two measurable real valued functions Show that the functions f g f g and f g are measurable 7 Define a relation on 0 1 as follows for x y 0 1 we say x is related to y written as x y if x y Q Prove the followings 2 a Show that is an equivalence relation on 0 1 b Let E I denote the set of equivalence classes of elements of 0 1 Using the axiom of choice choose exactly one element x E for every I and construct the set E x I Let r1 r2 r3 rn denote an enumeration of the rationals in 1 1 Let En rn E n 1 2 3 Show that En Em for n 6 m and En 1 2 for every n Deduce 0 1 En 1 2 n 1 c Show that E is not Lebesgue measurable 8 Let X be a set and A B C X The function A X 0 1 defined by 1 if x A 0 if x 6 A A x is called the characteristic or indicator function of A Show a A B A B where A B x A x B x b A B A B A B 9 Let X C be a measure space M X C be a simple function and c 0 be a real number Then show that Z Z c d c d 10 Let X C

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