version of the dominated convergence theorem where the almost-everywhere convergence is used

Question

Let $f\in \mathcal{L}^0(S,\mathcal{S},\mu)$ be a function

State and prove a version of the dominated convergence theorem where the almost-everywhere convergence is used. Is it necessary for all $\{f_n \}_{n∈\mathbb N}$ to be dominated by $g$ for all $x\in S$, or only almost everywhere?

I don't even have a direction. What do I need to show? where do I start?

Answer 1

Almost everywhere dominated by $g$ will do.

Let $E$ be the set where $|f_n|\leq g$ everywhere, then $\mu(S\setminus E)=0$.

Use the ordinary Dominated Convergence Theorem to show that $\int_E f_n\to\int_E f$.

But then $\int_S f_n=\int_E f_n$ and $\int_S f=\int_E f$, since if the domain of integration differs by a set of measure zero, the integral still remains the same.