So I am taking an intro to measure theoretic probability course and am a bit confused by how we defined sigma-algebras and measurable sets.
In particular if you have a sigma-algebra is that sufficient to guarantee that a corresponding probability measure is well defined on those sets? Or do we need more structure?
Moreover, in a sigma-algebra every element must have a well defined measure is that correct? I read here that the power set of the interval [0,1] is a sigma-algebra, but we know that the Vitali set is non-measurable, so I'm not sure how to reconcile that.
Even more confusing is the fact that if the power set of [0,1] is not a sigma-algebra then it is clearly not the case that you can define probabilities on any sigma-algebra, but it seems like in lectures we can.
So much confusion!
Any help would be appreciated.