Can we say that $\{f_n\}\text{ is uniformly integrable over }E\setminus (\cap_p B_p)$?

Question

Let $(E,\mathcal{A},\mu)$ be probability space and $\{f_n\}$ be sequence of functions such that $$ \sup_n\int_{E}|f_n|d\mu<+\infty. $$ Let $\{B_p\}$ be a sequence non-increasing in $\mathcal{A}$ such that $\mu(\cap_p B_p) =0$ and for every $p$ $$ \{f_n\}\text{ is uniformly integrable over }E\setminus B_p $$ Can we say that $\{f_n\}\text{ is uniformly integrable over }E\setminus (\cap_p B_p)$?

Answer 1

In general no: let $E$ be the unit interval endowed with the Lebesgue measure, $f_n=n\mathbf 1_{(0,1/n)}$ and $B_p=(0,1/p)$. For each $p$, $0\leqslant f_n\mathbf 1_{E\setminus B_p}\leqslant p$ hence $\left(f_n\mathbf 1_{E\setminus B_p}\right)_{n\geqslant 1}$ is uniformly integrable but $\left(f_n\right)_{n\geqslant 1}$ is not.