Let $f$ be nonnegative measurable function from $X$ to $[0,\infty]$ and $\int f <\infty$, show $\{x:f(x)=\infty\}$ is a null set.

Question

Let $f \in L^+$ and $\int f <\infty$. First denote $\{x:f(x)=\infty\}=E$, then suppose that $E$ has positive measure, then define $\phi_n=n\chi_{E}+f\chi_{E^c}$ then $\phi_n\le \phi_{n+1}$ and $\phi_n \nearrow f$, then by MCT $$\int f = \lim \int \phi_n=\lim n\mu(E)+\int_{E^c}f=\infty$$ A contradiction. Hence, $E$ has to be a null set.

Is this correct? Thanks!