Variant of dominated convergence theorem

Question

There are several variants of dominated convergence theorem.

The standard one requires $f_n \to f$ a.e. and $|f_n|\leq g$ a.e. where $g$ is integrable. It can be weakened to only convergent in measure, if we impose $\sigma$-finiteness to the measure, c.f. Generalisation of Dominated Convergence Theorem.

However, we know if $g$ is integrable, $N(g) = \{g\neq 0\}$ has a $\sigma$- finite measure. Does it imply that converging in measure is alway sufficient, if given a dominating integrable function $g$.

If all statements made above are correct, I want to ask the following question, why most of textbooks only introduce the one with strong assumptions rather than the more general one?

Answer 1

Yes, convergence in measure plus integrability of $\sup_n|f_n|$ give convergence in $\mathbb L^1$, as dicussed here.

Maybe it's not stated in this way in the textbooks because in a lot (but not all) examples of application of this theorem, we have almost everywhere convergence.