I am currently learning about Lebesgue integration, and I am doing so from two source: Tao's analysis II and Lang's real and functional analysis.
The former because it's what the lecturer is using and the latter for curiosity on how what I am seeing in class generalizes.
I have just reached the monotone convergence theorem in both. In Tao's it is clear how powerful it is (relative to what we know when we introduce it). Whereas relative to how integration is developed with Lang, I don't feel like Monotone convergence really shows anything.
In Lang the statement of the theorem is that for an increasing sequence of L1 function, with bounded integrals, the sequence is $f_n$ is Cauchy (and thus convergent in L1 and almost everywhere).
Whereas in Tao the statement is simpler, if $f_n$ is an increasing sequence of functions, then we can interchange limit and integral.
These feel like related but different statements (I think Lang is strictly stronger). It also feels as though in Lang's the theorem is far less useful, because we already know so much (for example in Tao, we use the monotone convergence theorem to prove linearity of the integral, which is super important).
All of this rambling to just ask: is this just a consequence of Lang being more abstract? Or is there a more fundamental difference in between the exposition of Tao and Lang? Or am I missing the power of Monotone convergence in the case of Lang?