A problem in Folland's Real Analysis...

A problem in Folland's Real Analysis...

pThis is problem 36 in Chapter 2 of the Folland's book./p pIf
$\mu(E_n)lt;\infty$ for all $n\in\mathbb N$ and $\chi_{E_n}\to f$ in
$L^1$, then $f$ is a.e. equal to the characteristic function on a
measurable set./p pMy proof:/p pConvergence in $L^1$ implies convergence
a.e. of a subsequence. So say $\chi_{E_{n_k}} \to f$ except on a set $A$
with $\mu(A)=0$/p pThere exists $N\in \mathbb N$ such that
$|\chi_{E_{n_k}}(x)-f(x)|lt;1/2 $ for all $k\geq N$, for $x\in A^c$. /p
pNow I think it is clear that if $k\geq N$ then ${E_{n_k}}$ and
${E_{n_{k+1}}}$ can differ at most by a null set. So taking $E=\bigcap
_{k\geq N} {E_{n_k}}$, we are removing at most a null set $B$. Then I
think $f=\chi_E$ on the complement of $A\cup B$./p pThat's my best attempt
so far, but I did not use the assumption $\mu(E_n)lt;\infty$. I may have
needed to use completeness of the Lebesgue measure to guarantee $B$ is
measurable./p