Let $\{E_j\}$ be measurable subsets in $B_1\subset\mathbb{R}^n$ and $\exists$ $A>0$, such that $\int_{B_1}\chi_{E_j}(x)dx\geq A$ for any $j=1,2,3,...$. Can we select a subsequence of functions $\{\chi_{E_j}(x)\}$ such that $\chi_{E_j}(x)\rightarrow\chi_E(x)$ a.e. $x\in B_1$, for some measurable set $E$ in $B_1$.
Note that $\chi_{E_j}\in L^2(B_1)$, then $\exists$ subsequence (we still use $E_j$) $\chi_{E_j}$ converge weakly to $f\in L^2(B_1)$. From the weakly convergence, it not hard to see $0\leq f(x)\leq 1$ for $a.e. x\in B_1$ and $\int_{B_1}f(x)dx\geq A$.
The main question is that whether $f$ is a characteristic function for some measurable set in $B_1$. Maybe there exists a counterexample. I don't know how to solve this problem.
Thanks!