I'm working on problem 2 (p.95) of Stein & Shakarchi Real Analysis textbook:
Prove the Cantor-Lebesgue theorem: if $$\sum_{n=0}^\infty A_n(x) = \sum_{n=0}^\infty (a_n\cos nx + b_n\sin nx)$$ converges for $x$ in a set [let's call it $E$] of positive [Lebesgue] measure (or in particular for all $x$), then $a_n\to 0$ and $b_n\to 0$ as $n\to \infty$.
There is a well-known proof of this statement for the case that $m(E) < \infty$ but it is not specified in the problem whether $E$ has finite measure. My doubt arises from the fact that if $E$ does not have finite measure, then the indicator function $\chi_E$ is not integrable, and consequently, we cannot apply the Lebesgue-Riemann theorem.
For the case $m(E) = \infty$, I was trying to apply dominated convergence theorem to a sequence $E_n \subset E$ such that $m(E_n) < \infty$ and $E_n \to E$ but I couldn't find an integrable function to dominate $E_n$.
Can you help me with this?
Thank you!