Suppose $\{f_n\}$ are Lebesgue measurable functions on $[0,1]$, such that $\int_0^1 |f_n|\,d\mu=1$ for all $n$, and $f_n\to 0$ almost everywhere.
I have proved: given $\epsilon>0$, there exists a Lebesgue meausurable $E\subseteq [0,1]$ such that $\mu(E)<\epsilon$ and $$\lim_{n\to\infty}\int_E |f_n|\,d\mu=1$$ (using Egorov's Theorem, where $E$ turns out to be $[0,1]\setminus F$ for some closed $F$ in which the convergence is uniform.)
Hence, or otherwise, how do we prove that there exists a subsequence $f_{n_k}$ of $f_n$ and sequences of measurable functions $g_k$ and $h_k$ such that
(i) $f_{n_k}=g_k+h_k$ for all $k$
(ii) $g_kg_j=0$ a.e. for $k\neq j$
(iii) $\lim_{k\to\infty}\int_0^1|h_k|\,d\mu=0$
The hardest condition in my opinion is (ii). I managed to find a candidate that satisfies both (i) and (iii), but not (ii). Let $f_{n_k}$ be a subsequence such that $\int_E |f_{n_k}|\,d\mu>1-\frac 1k$. Let $g_k=f_{n_k}\chi_E$ and $h_k=f_{n_k}\chi_F$, where $E=[0,1]\setminus F$.
Then (i) $f_{n_k}=g_k+h_k$ is satisfied.
$\lim_{k\to\infty}\int_0^1 |h_k|\,d\mu=\lim_{k\to\infty}\int_F |f_{n_k}|\,d\mu=1-\lim_{k\to\infty}\int_E |f_{n_k}|\,d\mu=1-1=0$. So (iii) is satisfied.
However, condition (ii) remains unsatisfied.
Thanks for any help.
Using Egorov, we can choose $E_1\subset [0,1]$ such that $\int_{E_1}|f_1| > 0$ and $f_n\to 0$ uniformly on $E_1.$ Now
$$1= \int_0^1|f_n| = \int_{E_1}|f_n| + \int_{[0,1]\setminus E_1}|f_n|,$$
and since $f_n\to 0$ uniformly on $E_1,$ we have $\int_{[0,1]\setminus E_1}|f_n|\to 1.$ Thus there exists $n_2>1$ such that $\int_{[0,1]\setminus E_1}|f_{n_2}| > 1/2.$ By Egorov we can choose a sequence $F_k \subset [0,1]\setminus E_1$ such that
$$\mu (F_k) \to \mu([0,1]\setminus E_1),$$
with $f_n \to 0$ uniformly on each $F_k.$ So if $k$ is large enough, we'll have both $f_n \to 0$ uniformly on $F_k$ and
$$\int_{F_k}|f_{n_2}| > 1/2.$$
Let $E_{2}$ be be any one of these $F_k.$ So we now have pairwise disjoint $E_1,E_2$ and $1= n_1 < n_2$ such that $f_n \to 0$ uniformly on $E_1\cup E_2$ and $\int_{E_k}|f_{n_k}| > 1-1/k$ for $k=1,2.$
We can continue this process by induction to obtain pairwise disjoint subsets $E_1, E_2, \dots$ and $1=n_1 < n_2 < \cdots $ such that for each $k,$ $f_n \to 0$ uniformly on $E_k$ and $\int_{E_k}|f_{n_k}| > 1-1/k.$
Now it's easy street. Define $g_k = f_{n_k}\chi_{E_k}, h_k = f_{n_k}\chi_{[0,1]\setminus E_k}.$ Then (i)-(iii) are satisfied.