## sets distance construction and properties

Denote by the quotient space of the family of Lebesgue measurable sets of by the equivalence relation . Denote by the Lebesgue measure of the measurable set .

1) Prove that is a distance on .

2) Prove that given measurable sets in the following three properties are equivalent.

- ;
- ;
- .

3) Prove that is a complete metric space.

4) Given the sequence of integrable real valued functions on , such that for any measurable set of there exists , prove that if then .

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Categories: Analysis, Real Analysis
measurable

Ce ai notat cu ?

e diferenta simetrica a doua multimi.