Let $(E,\mathcal{A},\mu)$ be a probability space. We have the following lemma:
Lemma : (biting lemma)
Suppose $\{\phi_n\}$ is a sequence in $\mathcal{L}^1_{\mathbb{R}}$ such that $$ \sup_{n}{\int_{E}{|\phi_n(t)|d\mu(t)}} $$ Then there exists a non-increasing sequence $\{B_p\}$ in $\mathcal{A}$ such that $\mu(\cap_{p}B_p) = 0$ and for every $p$ $$ \{\phi_n\}\text{ is uniformly integrable over } E\setminus B_p. $$
Let $A\in \mathcal{A}$ be a nonnull set. Let $\{f_n\}$ and $\{g_n\}$ are two sequences boundeds in $\mathcal{L}^1_{\mathbb{R}}$. I did not understand the following paragraph:
"Without loss of generality we may suppose that the nonnull set $A$ is such that on $A$ the two sequences $\{f_n\}$ and $\{g_n\}$ are uniformly integrable (apply the biting lemma two times: after two sufficiently small bites there is still a nonnull set left)"
My thinking
step 1: By biting lemma, let non-increasing sequence $\{B_p\}$ such that : $\mu(\cap_{p}B_p) = 0$ and for every $p$ $\{f_n\}\text{ is uniformly integrable over } E\setminus B_p$, then $\{f_n\}\text{ is uniformly integrable over } E\setminus X_1$, with $X_1=\cap_{p}B_p$.
step 2: By biting lemma, let non-increasing sequence $\{C_p\}$ such that : $\mu(\cap_{p}C_p) = 0$ and for every $p$ $\{g_n\}\text{ is uniformly integrable over } E\setminus C_p$, then $\{g_n\}\text{ is uniformly integrable over } E\setminus X_2$, with $X_2=\cap_{p}C_p$.
Set $X=X_1\cup X_2$; we have $\mu(X)=0$ and $\{f_n\}$ et $\{g_n\}$ are uniformly integrable over $E\setminus X$. In particular, $\{f_n\}$ et $\{g_n\}$ are uniformly integrable over $A\setminus X$.
What I wrote is true?