Let S be a collection of elements a, b, c,..., which we shall call elementary events, and F a set of subsets of S; the elements of F will be called random events.

I. F is a field of sets.

II. F contains the set S.

III. To each set A in F is assigned a non-negative real number P(A). This number P(A) is called the probability of the event A.

IV. P(S) = 1.

V. If A and B have no element in common, then P(AÈB) = P(A) + P(B)

VI. For a decreasing sequence of events A₁É A₂É... É A_nÉ... of F, for which ÇA_m = 0, the following equation holds: lim P(A_n) = 0, n ® ¥.

    A system of sets, F, together with a definite assignment of numbers P(A), satisfying Axioms I - VI, is called a field of probabilities. Our system of axioms is not, however, complete, for in various problems in the theory of probability different fields of probability have to be examined.

    The Construction of Fields of Probability

    The simplist fields of probability are constructed as follows. We take an arbitrary finite set
S = {a₁, a₂, a₃, ... , a_k} and an arbitrary set {p₁, p₂, p₃, ... , p_k} of non-negative real numbers with the sum p₁ + p₂ + p₃ + ... + p_k = 1. F is taken as the set of all subsets in S, and we put

    In such cases , p₁, p₂, p₃, ... , p_k are called the probabilities of the elementary events a₁, a₂, a₃, ... , a_k or simply elementary probabilities. In this way are derived all possible finite fields of probability in which F consisits of the set of all subsets of S. (The field of probability is called finite if the set S is finite).