Given a sigma algebra over some set which is generated by some set , and a probability function why is it sufficient to check the conditions: for all where is a disjoint sequence, to determine that is a valid probability? Intuitively it is obvious that if we check the conditions to be true on a generating set then we should have the conditions true for all the elements of , but how do we rigorously prove this fact?
For example suppose (the Borel sigma algebra) and . Now it is easy to check that satisfies the requisite conditions. How does it follow that the conditions are now met for ? Is it because of the result known as the Carathéodory's extension theorem? If so, can anyone refer me to proof, preferably in the context of probability measure?