Failure of dominated convergence theorem but the result still holds(?).

Question

For $x>1$ let $g (x)=(x \log x)^{-1}$ & for $\{A_n\}$ a sequence of measurable subsets of $[2,\infty)$ & $\{c_n\}$ a sequence of nonnegative numbers, we put $f_n=c_n \chi _{A_n}$.

if $\{f_n\}$ converges to $ 0 $ & for every $n $ ,$|f_n|\le g$,

prove or disprove that $\int _2 ^ {\infty}f_n dx$ converges to $ 0 $.

Here the integral of $g $ is not finite ,so we can't use the dominated convergence theorem.but the result seems to hold.

Can anyone help me with this problem? Thank you very much.

Answer 1

This is correct for the simple reason that $A_n\subseteq [2,t)$ where $t\log t=1/c_n$, so $\int f_n \le c_n t$ and $t\lesssim 1/(-c_n\log c_n)$.