Let {$f_n$} be a sequence of real-valued functions on [0,1] that is uniformly bounded.
Show that there exists a subsequence $n_j$ such that $ \int_A f_{n_j} dx $ converge for each borel subset of A contained in [0,1]
I prove for sequence of Borel set $A_i$ there exists such subsequence which converges for each set.
But for each Borel subset case, i think for each borel subset, find countable collection of half-open
interval that cover A and difference of measure of A and union of those collection less than $\epsilon$
and for each such collection i can find subsequence but... i think it's wrong :(
You approach contains good ideas. If we manage to find a subsequence which works for each $C_i$ and the collections $\mathcal C:=\{C_i,i\geqslant 1\}$ is an algebra which generates the Borel $\sigma$-algebra, we would be done.
We can use a diagonal argument once we find the family $\{C_i,i\geqslant 1\}$ (which can be done considering finite unions of interval with rational endpoints).