Baire Category Theorem Application 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?
 A: Choose $\varepsilon > 0$ and consider sets defined by
$$\Sigma_k = \{E:\ \left|\int\limits_{E} (f_n-f_m) \right| \leqslant \varepsilon, \textrm{ if } n,m \geqslant k  \}$$
Since for any measurable set a limit of integrals exists, we have
$\Sigma = \bigcup\limits_{k} \Sigma_k$. Note, that given an integrable function $f$, the functional $f(E):= \int\limits_{E}f$ is continuous respect to $E$ in metric $\delta$. Indeed, $f(E)-f(F) = \int f \cdot  ( 1_E - 1_F )$, hence $|f(E)-f(F)| \leqslant \int |f| \cdot  | 1_E - 1_F | = \int |f| \cdot  1_{E \Delta F} = \int\limits_{E\Delta F} |f|$. 
The last expression tends to $0$ if $|E\Delta F|$ tends to $0$ because of integrability of $f$, equivalently if $d(E,F)\to 0$. This remark shows that sets $\Sigma_k$ are closed as an intersection of closed subsets of the space $(\Sigma,d)$. 
From Baire theorem we obtain that one of sets $\Sigma_k$ has an interior point. Therefore, there exists a measurable set $E_0$ and integer $k$ such that the inequality 
$ |f_n(E)-f_m(E) | \leqslant \varepsilon$ holds, whenever $|E\Delta E_0| \leqslant \delta$ and $m,n\geqslant k$. We will show, that this inequality holds in fact for any set $E$, provided that its measure is sufficiency small. 
By identities $\mathbf{1}_{E\cup E_0} - \mathbf{1}_{E_0} = \mathbf{1}_{E\cap E_0^{c}}$ and $\mathbf{1}_{E_0}-\mathbf{1}_{E_0\setminus E} = \mathbf{1}_{E\cap E_0}$ we obtain for an arbitrary integrable $f$ 
$$f(E) = f(E \cap E_0^{c}) +  f(E\cap E_0) = f(E\cup E_0) - f(E_0) + f(E_0) - f(E_0\setminus E)$$
If $|E| < \delta$, then all of sets $E_0,E_0\cup E, E_0\setminus E$, belong to the ball $\{E:\ |E\Delta E_0| < \delta \}$. Applying the last inequality to $f_n-f_m$ and $|E| < \delta$ we get
$$ |f_n(E)-f_m(E)| \leqslant 2\varepsilon \quad \textrm { if } |E|<\delta, \ n,m\geqslant k$$
Finally, observe that the finite family of integrable functions $f_1,\ldots, f_k$ is obviously locally uniformly integrable, i.e. $\sup\limits_{i\leqslant k} |f_i(E)| \to 0$ if $|E|\to 0$. Therefore, for sufficiency small $\delta'$ we have
$$|f_i(E)|\leqslant \varepsilon \quad \textrm{ if } |E| < \delta', i\leqslant k$$ 
Gluing together two estimates that have been derived, we see that for some positive $\delta$ 
$$ \left|\int\limits_{E} f_n\right| \leqslant 3\varepsilon, \quad \textrm{ if } |E| \leqslant \delta$$
We have estimated integrals for functions $f_n$, instead their modulus. It doesn't matter, however. Namely, applying the last estimate to the set $E\cap \{f_n > 0\}$ and $E\cap \{f_n < 0\}$ respectively (both contained in $E$ hence with a smaller measure), gives finally
$$ \int\limits_{E} |f_n| \leqslant 6\varepsilon, \quad \textrm{ if } |E| \leqslant \delta$$
What finished the proof.
A: 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 :-) 
