In Antoine Henrot Michel Pierre - Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of the problem are pretty easy, and I solved them. The 4-th seems a little harder. The only indication I get is to use point 3) and the Baire theorem for $(\Sigma,\delta)$.
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 $X$.
Prove that $\delta(E_1,E_2)=\arctan( |E_1 \Delta E_2|)$ is a distance on $\Sigma$.
Prove that given $(E_n)_{n \geq 1}, E$ measurable sets in $\Bbb{R}^N$ the following three properties are equivalent.
$\delta(E_n,E) \to 0$;
$\chi_{E_n}-\chi_E \xrightarrow{\sigma(L^1,L^\infty)} 0$;
$\chi_{E_n}-\chi_E \xrightarrow{L^1} 0$.
Prove that $(\Sigma,\delta)$ is a complete metric space.
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 $\displaystyle\sup_n\int_E |f_n| \to 0$. (Hint: Use the Baire category theorem for $(\Sigma,\delta)$)
The question is: How can I apply Baire theorem to solve the 4-th point in the problem?
