Is σ-finiteness necessary for the “in measure” version of the dominated convergence theorem to hold?

Let $(X,\Sigma,\mu)$ be a measure space, $g\in L_1$, $|f_n|\le g$ and $f_n\to f$ in measure. I want to prove that $\int f_n\to f$, and $f_n\to f$ in $L_1.$

Generalisation of Dominated Convergence Theorem

Except for, there it says the measure space is $\sigma$-finite.

So, my question is, being $\sigma$-finite is completely necessary? Or this can be solved without that?

Thank you.

As demonstrated here, the $\sigma$-finiteness assumption is not necessary for the "in measure" version of the dominated convergence theorem to hold.