Let $(E,\mathcal{A},\mu)$ be probability space.
Lemma. Suppose $\{f_n\}$ is a sequence in $\mathcal{L}^1_\mathbb{R}$ such that $$ \sup_n\int_{E}|f_n|d\mu<+\infty. $$ Then there exists a nonincreasing sequence $\{B_p\}$ in $\mathcal{A}$ such that $\mu(\cap_p B_p) =0$ and for every $p$ $$ \{f_n\}\text{ is uniformly integrable over }E\setminus B_p $$
Let $\{f_n\}$, $\{g_n\}$ and $\{h_n\}$ are bounded sequence in $\mathcal{L}^1_\mathbb{R}$. Can we say that there exists a nonnull subset $A\in \mathcal{A}$ such that: $$ \{f_n\}, \{g_n\}\text{ and }\{h_n\} \text{ are uniformly integrable on }A $$
There are sequences of sets $B_p, B_p', B_p''$ satisfying the conclusion of the lemma for the sequences $\{f_n\}, \{g_n\}, \{h_n\}$ respectively. Since the intersection of each such sequence has zero measure, continuity from above implies $\mu(B_p) \to 0$ and similarly for $B_p', B_p''$, so given $\epsilon$ we can find $q$ sufficiently large that $\mu(B_q) < \epsilon$, $\mu(B_q') < \epsilon$, $\mu(B_q'') < \epsilon$. Set $A = (B_q \cup B_q' \cup B_q'')^c$. By union bound $\mu(A) > 1-3\epsilon$, which is positive as soon as $\epsilon < 1/3$, and by construction all three sequences are uniformly integrable on $A$.