TITLE: Baire Category Theorem Application
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: By the way, in a posted solution we used only following properties of functionals $f(E)=\int\limits_{E} f$:

Continuous respect to $|E|$ (Lebesgue measure)
Additive (on disjoint sets)
Finite

Therefore, our problem can be reformulated in a following way:
Suppose, that $\mu_n$ is a sequence of finite measures, absolutely continuous w.r.t. the Lebesgue measure. Then $\sup\limits_{n} \ |\mu_n|(E) \to 0$ if $|E|\to 0$.
So we have proved Vitali-Hahn-Saks theorem :-)