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- TL;DR Summary
- I am stuck at the very first sentence in the proof of Folland's version of the dominated convergence theorem. The wording confuses me and I'm not sure if he assumes the measure to be complete and the limiting function to be measurable.

The Dominated Convergence Theorem. Let ##\{f_n\}## be a sequence in ##L^1## such that (a) ##f_n\to f## a.e., and (b) there exists a nonnegative ##g\in L^1## such that ##|f_n|\leq g## a.e. for all ##n##. Then ##f\in L^1## and ##\int f=\lim\int f_n##.

Proof. ##f## is measurable (perhaps after redefinition on a null set) by Prop. 2.11 and 2.12, and since ##|f|\leq g## a.e., we have ##f\in L^1##. ...

That's the first sentence in the proof. Prior to this Folland mentions the spaces ##L^1(\overline{\mu})## and ##L^1(\mu)## and how "we can (and shall) identify these spaces." (here ##\overline{\mu}## is the completion of ##\mu##). The propositions mentioned in the proof read as follows:

Proposition. 2.11. The following implications are valid iff the measure ##\mu## is complete:

a) If ##f## is measurable and ##f=g## ##\mu##-a.e., then ##g## is measurable.

b) If ##f_n## is measurable for ##n\in\mathbb N## and ##f_n\to f## ##\mu##-a.e., then ##f## is measurable.

Proposition. 2.12. Let ##(X, \mathcal M,m)## be a measure space and let ##(X,\overline{\mathcal M} ,\overline \mu)## be its completion. If ##f## is an ##\overline{\mathcal M}##-measurable function on ##X##, there is an ##\mathcal M##-measurable function ##g## such that ##f=g## ##\overline{\mu}##-almost everywhere.

I'm really confused by Folland's first sentence in the proof of the dominated convergence theorem. My interpretation of Folland's theorem and first sentence is that he assumes ##\{f_n\}\subset L^1(\overline{\mu})##, so by Prop. 2.11b ##f## is measurable. Now by Prop. 2.12, i.e. by redefinition on a null set, we can find ##f_0\in L^1(\mu)## such that $f=f_0$ a.e. Does this make sense to you? My interpretation assumes the measure to be complete; I don't see otherwise why he'd refer to Proposition. 2.11.

Grateful for any thoughts or comments.