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Showing a quotient space is a complete metric space; Finite measurable space; Symmetric difference; Equivalence relations

Let (X,B,mu) be a complete, finite measuable space. For each C,D in B, set

d(C,D) = mu (C / D)

where C / D is the symmetric difference of C and D. We say that two measurable sets C,D are equivalent if d(C,D)=0 (this is an equivalence relation).

Let E be the set of equivalence classes, and show that d introduces a metric on E and that (E,d) is a complete metric space.

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Showing a quotient space is a complete metric space; Finite measurable space; Symmetric difference; Equivalence relations are investigated. The solution is detailed and well presented.

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