Non-measurable sets and sigma-algebra definition

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?

Solution 1:

It's not that we define a $\sigma$-algebra "in order to avoid [the] problem" that non-(Lebesgue-)measurable sets exist. Rather, we define exactly what we mean by "measurable set", form the collection $\cal S$ of all such sets, and then prove that this collection is a $\sigma$-algebra.

Having done so, it's a triviality that every set in $\cal S$ is measurable.

Of course, there are still non-measurable sets! They're just not members of $\cal S$.