Given a totally finite measure μ defined on a [itex]\sigma[/itex]-field X, define the (pseudo)metric d(A,B)=μ(A-B)+μ(B-A), (the symmetric difference metric), it can be shown this is a valid pseudo-metric and therefore the metric space (X',d) is well defined if equivalent classes of sets [itex][A_\alpha][/itex] where [itex]d(A_{\alpha_1},A_{\alpha_2})=0[/itex] are considered.(adsbygoogle = window.adsbygoogle || []).push({});

How do I show this metric space (X',d) is complete? In other words, given a Cauchy sequence {An}, the limit seems to be given by [itex]A=\cap_{n=1}^\infty A_n[/itex], but how do I formalize the proof? d(An,A)=μ(An-A)=[itex]\mu(A_n-\cap_{n=1}^\infty A_n[/itex])=...,

How do I make use of the Cauchy sequence {An}?

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# Convergence of sequence of measurable sets

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