A follow-up question from here Explanation of the Bounded Convergence Theorem:
What if $f_n$'s themselves are something of the form $\int g_n$, where $g_n$'s are uniformly bounded? Can we "push" the limit of $n$ inside again?
If not, what other changes are required for this to work?
Thanks in advance.
No, let $E= \cup_{m=1}^\infty[m,m+1/m^2].$ Then $m(E)<\infty.$ Define $f_n(x) = \int_0^x (1/n)\,dt = x/n.$ Obviously the functions $1/n, n=1,2,\dots $ are uniformly bounded. We have $f_n(x) \to 0$ pointwise. Verify that $\int_E f_n = \infty$ for each $n.$ Thus $\int_E f_n \to \int_E 0$ fails in a big way.
