One exercise about the real analysis Let $(f_n)_{n=1}^{\infty}$ be a sequence of Lebesgue measurable functions on $[0,1]$. Assume that $\int_0^1|f_n|d\lambda=1$ for all $n$ and that $(f_n)_{n=1}^\infty$ converges to $0$ almost everywhere.
(a) Show that for any $\varepsilon>0$, there exists a Lebesgue measurable subset $E$ of $[0,1]$ such that
$$\lambda(E)<\varepsilon~~~~\mbox{and}~~~~\lim\limits_{n\rightarrow\infty}\int_E|f_n|d\lambda=1$$
(b) Show that there is a subsequence $(f_{n_k})_{k=1}^\infty$ of $(f_n)_{n=1}^\infty$ and sequences of Lebesgue measurable functions $(g_k)_{k=1}^\infty$ and $(h_k)_{k=1}^\infty$ so that (1) $f_{n_k}=g_k+h_k$ for all $k\in\mathbb{N}$; (2) $g_kg_j=0$ a.e. if $k\neq j$; (3)$\lim\limits_{k\rightarrow\infty}\int_0^1|h_k|d\lambda=0$.
For part one, by using Egoroff's Theorem, the result can be obviously deduced. For the second part, I have thought that by choosing $E_k\subseteq E_{k-1}$ so that $\lambda(E_k)<\frac{1}{k}$ and $\lim\limits_{n\rightarrow\infty}\int_{E_k}|f_n|d\lambda=1$. However, I am confused about how to choose proper $g_k$ and $h_k$. Hope you can give me some hints.
 A: I think that your idea of choosing a decreasing sequence $(E_k)$ is essentially correct.
Namely, by induction we can construct a subsequence $(n_k)$ and a sequence $(E_k)$ of measurable subsets of $[0,1]$ such that, for every $k$,
1) $E_k \supset E_{k+1}$;
2) $\int_{E_k} |f_{n_k}| \geq 1 - 1/k$
3) $\int_{E_{k+1}} |f_{n_k}| \leq 1/k$.
Indeed, in order to fulfill 3), it is enough to observe that, by the absolute continuity of the integral, there exists $\epsilon_k > 0$ such that
$$
\int_{A} |f_{n_k}| \leq \frac{1}{k}
\qquad
\forall\ \text{measurable}\ A\subset [0,1]\
\text{with}\ \lambda(A) < \epsilon_k.
$$
So, when we choose $E_{k+1}$ at the next step, it is enough to choose it of measure less than $\epsilon_k$.
Let us define, for every $k$,
$$
g_k := f_{n_k} \chi_{E_k \setminus E_{k+1}},
\qquad
h_k := f_{n_k} - g_k.
$$
Clearly $g_k g_j = 0$ if $k\neq j$.
On the other hand,
$$
\int_0^1 |g_k| = \int_{E_k} |f_{n_k}| - \int_{E_{k+1}} |f_{n_k}|
\geq 1 - \frac{1}{2k} \to 1,
$$
and $\int_0^1 |h_k| \to 0$.
