Understanding the monotonous convergence theorem

Question

I'm stuck at the next point in the proof of this theorem

Monotone Convergence Theorem: If $(f_n)$ is a monotone increasing sequence of nonnegative measurable functions which converges to $f$, then $$\int f d\mu=\lim\int f_n d\mu$$

How the integral preserves order in functions we have $$\int f_n \leq \int f_{n+1} \leq \int f d\mu$$ Then the succession of integrals is monotone increasing and in a certain sense, it is bounded by $\int f d\mu$. But, what happens if the integral of some $f_n$ is infinite? would have $$+\infty\leq+\infty\leq\cdots+\infty\leq+\infty$$ Is this correct? I know it has to do with the integral taking values ​​in the extended reals, but I don't see the point in the inequality of infinities.

Answer 1

Yes, that is correct.

The point is that once the integral is infinite, it stays infinite. So if $\int f_n\,\mathrm{d}\mu=\infty$ for some $n$, then $\int \lim_n f_n\,\mathrm{d}\mu=\infty$.

In the extended reals, $+\infty$ is the maximum, so if something is greater or equal than $+\infty$, it must be equal to it.

Answer 2

If $\int f_n=\infty$ for some $n$ then $\int f_m=\infty$ for all $m \geq n$ and $\int f =\infty$ so $\int f=\lim \int f_n$