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sets distance construction and properties

Denote by \Sigma the quotient space of the family of Lebesgue measurable sets of \Bbb{R}^N by the equivalence relation E_1 \sim E_2 \Leftrightarrow \chi_{E_1}=\chi_{E_2} a.e.. Denote by |X| the Lebesgue measure of the measurable set X.
1) Prove that \delta(E_1,E_2)=\arctan( |E_1 \Delta E_2|) is a distance on \Sigma.
2) Prove that given (E_n)_{n \geq 1}, E measurable sets in \Bbb{R}^N the following three properties are equivalent.

  • \delta(E_n,E) \to 0;
  • \chi_{E_n}-\chi_E \xrightarrow{\sigma(L^1,L^\infty)} 0;
  • \chi_{E_n}-\chi_E \xrightarrow{L^1} 0.

3) Prove that (\Sigma,\delta) is a complete metric space.

4) Given the sequence (f_n) of integrable real valued functions on \Bbb{R}^N, such that for any measurable set E of \Bbb{R}^N there exists \displaystyle \lim_{n \to \infty}\int_E f_n, prove that if |E| \to 0 then \displaystyle \sup_n\int_E |f_n| \to 0.

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  1. Diana
    February 21, 2011 at 10:06 pm

    Ce ai notat cu \Delta?

    • February 21, 2011 at 10:15 pm

      \Delta e diferenta simetrica a doua multimi.

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