Quick question regarding the Lebesgue's Dominated Convergence Theorem

Question

I'm reading Rudin's Real and Complex Analysis book. In the statement where we require $$|f_n(x)| \le g(x)$$ for all $n$. Could this be relaxed to all, but finitely many $n$? I was looking at his proof and couldn't find any reason why not, but I thought I go ahead and ask to make sure.

Answer 1

Of course you can. If $|f_n(x)|\leq g(x)$ for $n> N$, you may throw away first $N$ terms since it does not affect the limit. If $f_n \in L^1$ for all $n$, you may take $$ g' = g + |f_1| +\cdots + |f_{N}|$$ as a majorant explicitly.
Note: Actually, condition of single majorant $g$ can be replaced with existence of $L^1$-sequence $\{g_n, g \}$ such that $|f_n| \leq g_n$, $|\lim_n f_n|\leq g$, and $\int g_n d\mu\to \int gd\mu.$