Transcription of 11. Complex Measures - Probability
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11. Complex Measures - Probability
Tutorial 11: Complex Measures111. Complex MeasuresIn the following, ( ,F) denotes an arbitrary measurable 90 Let(an)n 1be a sequence of Complex numbers. Wesay that(an)n 1has thepermutation propertyif and only if, forall bijections :N N ,theseries + k=1a (k)converges inC1 Exercise (an)n 1be a sequence of Complex Show that if (an)n 1has the permutation property, then thesame is true of (Re(an))n 1and (Im(an))n Supposean Rfor alln 1. Show that if + k=1akconverges:+ k=1|ak|=+ + k=1a+k=+ k=1a k=+ 1which excludes as 11: Complex Measures2 Exercise (an)n 1be a sequence inR, such that the series + k=1akconverges, and + k=1|ak|=+ .LetA>0. We define:N+ ={k 1:ak 0},N ={k 1:ak<0}1. Show thatN+andN are Let +:N N+and :N N be two bijections. Showthe existence ofk1 1 such that:k1 k=1a +(k) A3. Show the existence of an increasing sequence (kp)p 1such that:kp k=kp 1+1a +(k) 11: Complex Measures3for allp 1, wherek0= Consider the permutation :N N defined informally by:( (1), +(1).)
Tutorial 11: Complex Measures 5 Definition 91 Let (Ω,F) be a measurable space and E ∈F.We call measurable partition of E, any sequence (E n) n≥1 of pairwise disjoint elements of F, such that E = n≥1E n. Definition 92 We call complex measure on a measurable space (Ω,F) any map μ: F→C, such that for all E ∈Fand (E n) n≥1 measurable partition of E,theseries
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