Axiometic definition of probability ( Kolmogorov, 1933)

We have earlier defined an event as a subset of the sample space. Does each subset of the sample space forms an event?
The answer is yes for a finite sample space. However, we may not be able to assign probability meaningfully to all the subsets of a continuous sample space. We have to eliminate those subsets. The concept of the sigma algebra is meaningful now.

Definition Let be a sample space and a sigma field defined over it. Let be a mapping from the sigma-algebra into the real line such that for each , there exists a unique . Clearly is a set function and is called probability, if it satisfies the following three axioms.