Probability Axioms

Given an event  in a sample space  which is either finite with  elements or countably infinite with  elements, then we can write

and a quantity , called the probability of event , is defined such that

1. .

2. .

3. Additivity: , where  and  are mutually exclusive.

4. Countable additivity:  for , 2, ...,  where , , ... are mutually exclusive (i.e., ).

REFERENCES:

Doob, J. L. "The Development of Rigor in Mathematical Probability (1900-1950)." Amer. Math. Monthly 103, 586-595, 1996.

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 26-28, 1984.