Home > Analysis, Real Analysis > sets distance construction and properties

## 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 $\displaystyle \sup_n\int_E |f_n| \to 0$.

Ce ai notat cu $\Delta$?
$\Delta$ e diferenta simetrica a doua multimi.