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Unformatted text preview: Mathematics 741  Advanced Probability Theory I  Fall 2008  Zirbel Exercises to accompany the lecture notes These exercises may or may not be assigned as homework for Math 741. Some of them are referred to in the lecture notes by number. Let ( , H , IP) be a probability space. 1. a) Let A 1 A 2 be sets in H . Let A = i =1 A i . Show that IP( A i ) increases to IP( A ) as i . b) Let A 1 A 2 be sets in H . Let A = i =1 A i . Show that IP( A i ) decreases to IP( A ) as i . 2. Let F be the smallest algebra on R containing all intervals of the form ( ,b ], b R . Show that F = B R . We say that B R is generated by the open subsets of R or by intervals of the form ( ,b ]. We write B R = ( { ( ,b ] : b R } ). Hint: Show that F B R and then that B R F . Use the fact that B R contains all sets of the form ( ,b ]. Show that F contains all open intervals in R and thus all open subsets of R (each open subset is a countable union of open intervals). 3. a) Let and S be sets and let S be a algebra on S . Let X : S . Define a collection ( X ) of subsets of by ( X ) = { X 1 ( B ) : B S} . Show that ( X ) is a algebra on . Show all details. b) Let H be a algebra on . Let D = { B S : X 1 ( B ) H} . Show that D is a algebra on S . Think of D as all subsets of S whose inverse images under X are nice. Notes: (i) ( X ) is called the algebra generated by X . It is the collection of all events concerning X . (ii) The function X : ( , H ) ( S, S ) is measurable if and only if ( X ) H . Think of ( X ) as the subsets of that need to be measurable for the mapping X to be measurable. 4. Represent by a square in the plane with sides of length 1. Divide in half hor izontally and vertically. Let X : R take on the values 1, 2, 3, 4 in the four smaller squares. Represent X by showing the four values in the four smaller squares, and represent subsets of by shading in the appropriate squares. For example, the set { X = 1 } is the upperleft smaller square. Represent all elements of ( X ) by shading and in the form X 1 ( B ) for different sets B . Check ( X ) to make sure it is a algebra. 5. Let limsup n A n n =1 m = n A m . Show that IP(limsup n A n ) limsup n IP( A n )....
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This note was uploaded on 01/01/2011 for the course MATHSTAT 741 taught by Professor Zirbel during the Spring '10 term at Bowling Green.
 Spring '10
 Zirbel

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