Quote by mathwonk
Lang's Analysis II (maybe now Real analysis), has a good strong statement of Fubini. (And the functions in Lang have values in any Banach space.)

Thanks for this tip. I just checked it out. It looks really good. The actual title is "Real and functional analysis". Lang is using the same definition as Friedman, but starts with complexvalued functions right away (and doesn't use any properties of ℂ other than the ones shared by all Banach algebras) This is how he explains his choice to use the limit definition in the introduction to the chapter:
A posteriori, one notices that the monotone convergence theorem and the "Fautou lemma" of other treatments become immediate corollaries of the basic approximation lemmas derived from Lemma 3.1. Thus it turns out that it is easier to work immediately with complex valued functions than to go through the sequence of many other treatments, via positive functions, real functions, and only then complex functions decomposed into real and imaginary parts. The proofs become shorter, more direct, and to me much more natural. One also observes that with this approach nothing but linearity and completeness in the space of values is used. Thus one obtains at once integration with Banach valued functions.
I'm going to have to read more of it. It looks like a version of what Friedman did, that's just better organized and with proofs that are easier to follow.