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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.

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up vote 12 down vote accepted

The only practical difference (as long as one doesn't work near to the foundations of mathematics at least) between naive set theory and axiomatic set theory is that the axioms say that there are certain things you just can't do - such as unrestricted comprehension and sets have to be wellfounded, and neither crops up in measure theory since the most complicated set theoretic "stuff" you do starts with a set $X$, considers $\sigma$-algebras on $X$, i.e. certain subsets of $\mathcal{P}(X)$, the power set of $X$, so the collection of all $\sigma$-algebras on $X$ is a subset of $\mathcal{P}(\mathcal{P}(X))$. In particular you never run into unrestricted comprehension because you're always guaranteed to have a set to do comprehension in, and these "nested structures" that lead to non-wellfoundedness don't happen either for essentially the same reason.

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So, how about non-measurable cardinals? :P. –  Jonas Teuwen Mar 8 '11 at 19:09
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