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# exercises - Mathematics 741 Advanced Probability Theory I...

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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 (Ω , 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 σ -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 = σ ( { ( −∞ , 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 Ω and S be sets and let S be a σ -algebra on S . Let X : Ω S . Define a collection σ ( X ) of subsets of Ω by σ ( X ) = { X - 1 ( B ) : B ∈ S} . Show that σ ( X ) is a σ -algebra on Ω. Show all details. b) Let H be a σ -algebra on Ω. Let D = { B S : X - 1 ( B ) ∈ H} . Show that D is a σ -algebra on S . Think of D as all subsets of S whose inverse images under X are “nice”. Notes: (i) σ ( X ) is called the σ -algebra generated by X . It is the collection of all events concerning X . (ii) The function X : (Ω , H ) ( S, S ) is measurable if and only if σ ( X ) ⊆ H . Think of σ ( X ) as the subsets of Ω that need to be measurable for the mapping X to be measurable. 4. Represent Ω by a square in the plane with sides of length 1. Divide Ω in half hor- izontally and vertically. Let X : Ω 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 Ω by shading in the appropriate squares. For example, the set { X = 1 } is the upper–left smaller square. Represent all elements of σ ( X ) by shading and in the form X - 1 ( B ) for different sets B . Check σ ( X ) to make sure it is a σ -algebra. 5. Let lim sup n →∞ A n n =1 m = n A m . Show that IP(lim sup n →∞ A n ) lim sup n →∞ IP( A n ). 6. a) Let X : Ω S . Let C be a non-empty collection of subsets of S . Show that σ ( X - 1 ( C )) = X - 1 ( σ ( C )). Here, X - 1 of a collection means the collection of inverse images of each element of the collection. For example, X - 1 ( C ) = { X - 1 ( B ) : B ∈ C} .

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