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

Question

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 $L_1.$

Now, this may be already solved in the following link:

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.

Answer 1

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

(Note that it happens to be the very link from your question yet Davide Giraudo edited his answer after PhoemueX's comment).