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 ) 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 ,µ )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.