Let $(X,\Omega,\mu)$ be a measure space and let $f$ be an extended real valued measurable function defined on $X$. I have already seen that $$ \mu\left(\{x\in X : |f(x)|\geq t\}\right)\leq \frac{1}{t}\int_X |f|~d\mu $$ for any $t\in (0,\infty)$.
CONCLUDE that the measure of the set $$\{x\in X : |f(x)|\geq t\}$$
is finite for every $t$.
We are going to suppose that $f\in L'$
Then we know that $f$ is integrble if and only if $\int_X|f|d\mu$ is finite.
Therefore, the integral (finite) divided by a finite value is always going to be finite