Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. Does there exist a subsequence $n_j$ such that $$\int_A f_{n_j}(x)\,dx$$ converges for each Borel subset $A$ of $[0, 1]$?
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Note that your sequence $f_n$ is a bounded sequence in $L^\infty$ and $L^\infty$ is the dual space of $L^1$, thus the Banach–Alaoglu theorem implies that there is a subsequence $f_{n_j}$ and $f\in L^\infty$ so that
$$ \int g f_{n_j} dx \to \int gf dx,\ \ \ \forall g\in L^1$$
(Note the identification $(L^1)^* \cong L^\infty$ is given by $L(g) = \int g f dx$). Thus setting $g = \chi_A$, you have
$$\int_A f_{n_j} dx \to \int_A f dx$$
for all Borel set $A$.
