from the closed half-lines, consider the following example Example Let S be the real line R=a:-o0< a< oo and the set of events of interest {B2:x∈R} where B2=2: 2<a=(o0, 1. How can we construct a a-field, a(J)on R from the events B By definition BE a(J), then (1). Taking complements of B2: B2=2: 2ER,2>c)=(a,ooE a(J) (2). Taking countable unions of e:U∞=1(-∞,x-(1/m)]=(-∞,x)∈a(J) (3). Taking complements of(2):(=∞,m)=[x,∞)∈o(); (4).From(1),fory>x,lv,∞)∈a() ]U[v,∞)=(x,y)∈o(J) (6).∩=1(x-(1/mn),x]={x}∈(J) This shows not only that a()is a a-field but it includes almost every con- ceivable subset of R, that is, it coincides with the a-field generated by any set of subsets of R, which we denote by B, i.e. o()=B, or the Borel Field on R Having solved the technical problem in attributing probabilities to events by postulating the existence of a a-field F associated with the sample space S Kolmogorov went on to formalize the concept of probability itself A mapping p: F-0, 1]is a probability measures on S, F) provided that (a)P(∞)=0. (b)For any AE F, P(A)=l-P( (c)For any disjoint sequence (Ai) of sets in F(i.e, A; A,=0 for all i#j), P(U≌1A)=∑1P(A Examplefrom the closed half-lines, consider the following example. Example: Let S be the real line R = {x : −∞ < x < ∞} and the set of events of interest be J = {Bx : x ∈ R}, where Bx = {z : z ≤ x} = (−∞, x]. How can we construct a σ-field, σ(J) on R from the events Bx? By definition Bx ∈ σ(J), then (1). Taking complements of Bx: B¯ x = {z : z ∈ R, z > x} = (x, ∞) ∈ σ(J); (2). Taking countable unions of Bx: ∪ ∞ n=1(−∞, x − (1/n)] = (−∞, x) ∈ σ(J); (3). Taking complements of (2): (−∞, x) = [x, ∞) ∈ σ(J); (4). From (1), for y > x, [y, ∞) ∈ σ(J); (5). From (4), (−∞, x] ∪ [y, ∞) = (x, y) ∈ σ(J); (6). ∩ ∞ n=1(x − (1/n), x] = {x} ∈ σ(J). This shows not only that σ(J) is a σ-field but it includes almost every con￾ceivable subset of R, that is, it coincides with the σ-field generated by any set of subsets of R, which we denote by B, i.e. σ(J) = B, or the Borel Field on R. Having solved the technical problem in attributing probabilities to events by postulating the existence of a σ- field F associated with the sample space S, Kolmogorov went on to formalize the concept of probability itself. Definition 5: A mapping P : F → [0, 1] is a probability measures on {S, F} provided that (a) P(∅) = 0. (b) For any A ∈ F, P(A) = 1 − P(A). (c) For any disjoint sequence {Ai} of sets in F (i.e., Ai ∩ Aj = ∅ for all i 6= j), P(∪ ∞ i=1Ai) = P∞ i=1 P(Ai). Example: 9