I´m starting to study about measure theory, but I have problems regarding the definition of measure space.
In my class we saw that there exists sets that are not measurable(Vitali sets in $\mathbb R$) and in order to avoid this problem we define a sigma-algebra on a set $X$ and we call the elements of $\Sigma$ "measurable sets"
But the problem I have is that how can we guarantee that with this "definition" we cannot construct a non-measurable set. Is there a theorem that says that there can´t be non-measurable sets on a sigma-algebra over $X$ with that definition?
Or I just need to assume that we cannot find such sets over these circumstances?