Measure theory was established on naive set theory(Not totally sure). But after Russell discovered the paradox named by him, set theory was reconstructed in the sense of axiomatization.
My question is in the first chapter of many measure theory textbooks, there is a set theory introduction, most likely describing the naive set theory. How can I be convinced that measure theory is rigorous? Or I can just take as granted that it is, and the approach by introducing naive set theory is only because it is easier to understand?
And could we encounter such a example of the paradox when using measure theory, e.g to analyze integrability.