Borel - Cantelli Lemma 1

The notation and terminology used here have been introduced in the following papers: [17], [3], [4], [8], [13], [1], [2], [5], [15], [14], [21], [9], [12], [11], [16], [6], [20], [19], and [18]. For simplicity, we adopt the following rules: O1 is a non empty set, S1 is a σ-field of subsets of O1, P1 is a probability on S1, A is a sequence of subsets of S1, and n is an element of N. Let D be a set, let x, y be extended real numbers, and let a, b be elements of D. Then (x > y → a, b) is an element of D. We now state two propositions: (1) For every element k of N and for every element x of R such that k is odd and x > 0 and x ≤ 1 holds (−xExpSeqR)(k+1)+(−xExpSeqR)(k+2) ≥ 0. (2) For every element x of R holds 1 + x ≤ (the function exp)(x). Let s be a sequence of real numbers. The functor ExpFuncWithElementOf s yielding a sequence of real numbers is defined as follows: (Def. 1) For every natural number d holds (ExpFuncWithElementOf s)(d) = ∑ −s(d) ExpSeqR . Next we state two propositions: (3) (The partial product of ExpFuncWithElementOf(P1·A))(n) = (the function exp)(−( ∑κ α=0(P1 ·A)(α))κ∈N(n)). The author wants to thank Prof. F. Merkl for his kind support during the course of this work.