HW203 - Analysis 2 Homework 03 1. Let (, F ) be a...

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Analysis 2 Homework 03 1. Let (Ω , F ) be a measurable space on which the measures μ and ν satisfy μ (Ω) = ν (Ω) < . Must the collection E = { F ∈ F : μ ( F ) = ν ( F ) } be a σ -algebra? Proof or counterexample. 2. Let Ω = N = { 1 , 2 ,... } and on F = P (Ω) define the (probability) measure μ by μ ( { n } ) = 1 n ( n + 1) . For F n = { n,n + 1 ,... } calculate n =1 μ ( F n ) and μ (lim sup n →∞ F n ). Any comments? Solutions 1. Answer: ‘No; it need not be’. Notice that E is closed under complementa- tion and countable pairwise-disjoint unions. The problem is that σ -algebras must be closed under all countable unions, but the σ -additivity of a measure only controls pairwise-disjoint countable unions; a counterexample comes by arranging nonempty overlap. Thus, let Ω =
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