The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, and taking the sequence of maxima of the functions with index up to $i$, should converge to the pointwise suprema of the functions in the set. By the MCT or DCT, that suprema is integrable. And it should be independent of the indexing.
This is a relatively straightforward result, but I can't seem to see it proven or mentioned anywhere? Is there a reference for this result? Assuming I haven't made some stupid error, it looks like a fairly simple generalization of the lattice of indicator functions of measurable sets.
