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- 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. Letbe a sequence in such that (a) a.e., and (b) there exists a nonnegative such that a.e. for all . Then and .
Proof.is measurable (perhaps after redefinition on a null set) by Prop. 2.11 and 2.12, and since a.e., we have . ...
That's the first sentence in the proof. Prior to this Folland mentions the spaces
Proposition. 2.11. The following implications are valid iff the measureis complete:
a) Ifis measurable and -a.e., then is measurable.
b) Ifis measurable for and -a.e., then is measurable.
Proposition. 2.12. Letbe a measure space and let be its completion. If is an -measurable function on , there is an -measurable function such that -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
Grateful for any thoughts or comments.