hw8 - Mathematics 105 Spring 2004 M Christ Problem Set...

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Mathematics 105 — Spring 2004 — M. Christ Problem Set 8 (corrected 1 ) For Friday April 9: Study § 3.3 of our text. Solve the following problems from Stroock § 3.3: 3.3.16,17,19,20,23. In problem 23, simplify the statement by assuming that μ ( E ) and ν ( E ) are both finite. Thus it is given that ϕ dμ = ϕ dν for all bounded, ρ -uniformly continuous functions ϕ , and you are asked to prove that μ = ν on B E . (Recall that the Borel sigma-algebra B E is the smallest sigma-algebra of subsets of E which includes all open subsets of E . Recall also that we have discussed the universal method of proving statements about Borel sets; that discussion applies to the metric space ( E, ρ ) just as well as to R n .) In problem 3.3.22, in the phrase “whenever ϕ is a bounded ρ -uniformly continuous ϕ ”, the second “ ϕ ” should be replaced by the word “function”. There’s a typo in the second-to-last line of exercise 3.3.16; it should read “any sequence of B -measurable sets”. Our author just wants to say that Γ n ∈ B .
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