No1(2004) - MATH 587 Assignment 1 Due October 5 2004(1 Let be a countable set F = P Let p be non-negative numbers Show that P(A = A p denes a-nite

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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 be non-negative numbers. Show that P ( A )= ω A p ( ω ) defnes a σ -fnite measure on F . (2) Billingsley (page 41) defnes 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 fnite disjoint unions). Show that this defnition 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 )bea σ -fnite measure space. A set E ∈F is an atom oF µ iF µ ( E ) > 0 and iF µ ( F )=0or 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 4 B ]=0. (b) iF A is an atom, and iF A C D where C D = , then either
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This note was uploaded on 01/25/2011 for the course STAT 235a at Stanford.

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