Lebesgue's monotone convergence theorem, - boundedness

Question

Here they say the functions should be bounded. Does not the $\leq \infty$ at the end of the condition of the functions violate the boundedness? : ${\displaystyle 0\leq f_{k}(x)\leq f_{k+1}(x)\leq \infty .}$

Answer 1

Monotone Convergence Theorem does not require the boundedness of the nonnegative increasing functions $(f_{n})$, of course, it could be $\lim_{n}\displaystyle\int f_{n}d\mu=\displaystyle\int\lim_{n}f_{n}d\mu=\infty$.

The boundedness is crucial for Lebesgue Dominated Convergence Theorem: $|f_{n}|\leq g$, $g$ is $L^{1}$, then $\lim_{n}\displaystyle\int f_{n}d\mu=\int\lim_{n}f_{n}d\mu$.