You don't get new theorems if you remove assumptions. Rather, you get the ability to add different assumptions.
The Banach-Tarski paradox shows that classical set theory makes the wrong assumptions to intrinsically model measure theory and probability.
There are other systems which don't suffer from this paradox and hence don't need all the machinery of sigma algebras and measurable sets.
I wish there was a good accessible book/article/blog post about this, but as is you'd have to Google point-free topology or topos of probability (there are several).
The Banach-Tarski paradox shows that classical set theory makes the wrong assumptions to intrinsically model measure theory and probability.
There are other systems which don't suffer from this paradox and hence don't need all the machinery of sigma algebras and measurable sets.
I wish there was a good accessible book/article/blog post about this, but as is you'd have to Google point-free topology or topos of probability (there are several).