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.

Tutorial 11: Complex Measures 1 11. Complex Measures In the following, (Ω,F) denotes an arbitrary measurable space. Definition 90 Let (a n) n≥1 be a sequence of complex numbers. We say that (a n) n≥1 has the permutation property if and only if, for all bijections σ: N∗ → N∗,theseries k=1 a σ(k) converges in C 1 Exercise 1. Let (an)

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