Kolmogorov ’ s Zero - One Law Agnes Doll

The articles [8], [19], [2], [10], [12], [18], [20], [1], [15], [5], [21], [11], [3], [9], [7], [6], [17], [4], [16], [14], and [13] provide the terminology and notation for this paper. For simplicity, we adopt the following convention: Ω, I are non empty sets, F is a σ-field of subsets of Ω, P is a probability on F , D, E, F are families of subsets of Ω, A, B, s are non empty subsets of F , b is an element of B, a is an element of F , p, q, u, v are events of F , n is an element of N, and i is a set. Next we state three propositions: (1) For every function f and for every set X such that X ⊆ dom f holds if X 6= ∅, then rng(f X) 6= ∅. (2) For every real number r such that r · r = r holds r = 0 or r = 1. (3) For every family X of subsets of Ω such that X = ∅ holds σ(X) = {∅,Ω}. Let Ω be a non empty set, let F be a σ-field of subsets of Ω, let B be a subset of F , and let P be a probability on F . The functor Indep(B,P ) yielding a subset of F is defined as follows: (Def. 1) For every element a of F holds a ∈ Indep(B,P ) iff for every element b of B holds P (a ∩ b) = P (a) · P (b). Next we state several propositions: (4) Let f be a sequence of subsets of F . Suppose for all n, b holds P (f(n)∩ b) = P (f(n)) · P (b) and f is disjoint valued. Then P (b ∩ ⋃ f) = P (b) · P ( ⋃ f).