final - Final Exam Fall 2009 Statistics 778 Answer any FIVE...

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Final Exam, Fall 2009, Statistics 778 Answer any FIVE questions. Answers must be supported by clear argu- ments. If you use any theorem, clearly indicate that. 1. Define symmetric difference A Δ B of two sets A and B . Show that ( i =1 A i )Δ( i =1 B i ) ⊂ ∪ i =1 ( A i Δ B i ). Let (Ω , A , μ ) be a measure space and E be a sub- σ -field of A . Consider F = { F A : μ ( E Δ F ) = 0 for some E E } . Show that F is also a σ -field, containing E . Construct an example to argue that F may be strictly bigger than E . If f is an F -measurable function, show that there exists an E -measurable function g such that f = g a.e. [ μ ]. (Hint: Start with indicator functions.) [2+4+4+5+5=20] 2. Let μ be a Lebesgue-Stiltjes measure with continuous and strictly in- creasing distribution function F . Show that μ ( A ) = 0 for any countable set A . If μ ( A ) = 0, is A necessarily countable? Prove or give a counterexample with all details. If A contains an open interval, show that μ ( A ) > 0.
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