I have taken both courses on Riemann integration (single and multiple cases) and one course in measure and Lebesgue integration that didn't progress far. We covered a chapter on measure, measurable spaces, and their morphisms, as well as a chapter on the Lebesgue integral of complex-valued functions, including Fatou's lemma, the monotone convergence theorem, and two theorems about limits and differentiation of integrals with parameters.
At the moment, I'm more oriented toward abstract algebra, and I either have begun to forget some of these topics or didn't fully grasp them the first time. I wish to restudy them, hopefully in an optimal way, mainly for their applications in complex analysis and differential geometry: specifically, to gain a better understanding of the integration of differential forms and Stokes' theorem in differential geometry, as well as their behavior with sequence convergence.
In this post, the user Pete L. Clark points out that the Chasles relation, the linearity, and the positivity of the integral of a positive function are sufficient to prove the fundamental theorem of calculus. I found this to be an elegant and great approach, as I remember that the explicit definitions confused me a little and made the subject seem more complex, especially in the case of multiple integrals.
So, my question is about book recommendations and comments on resources that don’t dwell too much on construction details and instead illustrate how the integral should behave, deducing all the important theorems of calculus from it (behavior with sequence limits, Fubini's theorem, changes of variables), similar to how we treat the field of real numbers in either Riemann or Lebesgue integrals.