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 $X$.
> 
> 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 $\displaystyle
> \sup_n\int_E |f_n| \to 0$.
> (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?