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No1(2004) - MATH 587 Assignment 1 Due October 5 2004(1 Let...

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MATH 587 Assignment 1 Due October 5, 2004 (1) Let Ω be a countable set, F = P (Ω). Let p ( ω ) , ω Ω be non-negative numbers. Show that P ( A ) = ω A p ( ω ) defines a σ -finite measure on F . (2) Billingsley (page 41) defines a class C of subsets of Ω to be a λ -system if 1’. Ω ∈ C , 2’. C is closed under complementation, 3’. C is closed under countable unions of pairwise disjoint members of C (and so is also closed under finite disjoint unions). Show that this definition coincides with that of a d -system. (3) Prove the Monotone Class Theorem: Let A be an algebra. Then σ ( A ) = M ( A ). (i.e. the σ -algebra generated by A coincides with the monotone class generated by A . (4) Let (Ω , F , µ ) be a σ -finite measure space. A set E ∈ F is an atom of µ if µ ( E ) > 0 and if µ ( F ) = 0 or µ ( F ) = µ ( E ) whenever F ∈ F and F E . Show that (a) two atoms A and B are either a.s. disjoint, i.e. µ ( A B ) = 0, or a.s. coincide, i.e. µ [ A B ] = 0. (b) if A is an atom, and if A C D where C D = , then either
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