Two special events are S itself, called the sure events and the impos sible event o defined to contain no elements of S, i.e. 0= the latter is defined for com- pleteness a third ingredient of 8 associated with(b) which Kolmogorov had to formal ed was the idea of uncertainty related to the outcome of any particular trial 8. This he formalized in the notion of probabilities attributed to the various events associated with 8, such as P(A1), P(A2), expressing the"likelihood"of occurrence of these events. Although attributing probabilities to the elementary events presents no particular mathematical problem, going the same for events in general is not as straightforward. The difficulty arise because if Al and A2 are ts, A1=S-Al, A2=S-A2, A10A2, A1 UA2, etc, are also events becaus the occurrence or non-occurrence of A1 and A2 implies the occurrence or not of these events. This implies that for the attribution of probabilities to make sense we have to impose some mathematical structure on the set of all events, say F, which reflects the fact that whichever way we combine these events, the end result is always an event. The temptation at this stage is to define f to be the set of all subsets of S, called the power set; Surely, this covers all possibilities In the above example, the power set of s take the form F={S,0,{(HT)},{(TH)},{(HH)},{(TT)},{(HT),(TH)},{(HT),(HH)},{(HT),(TT)} {(TH),(HH)},{(TH),(TT)},{(HH),(①T)},{(HT),(TH),(HH)},{(HT),(TH),(TT)}, {(TH),(HH),(TT)},{(HT),(TH),(HH),(TT)}} Sometimes we are not interested in all the subsets of s. we need to define a set independently of the power set by endowing it with a mathematical structure which ensures that no inconsistency arise. This is achieved by requiring that F in the following has a special mathematical structures, It is a a-field related to S Definition 3 Let f be a set of subsets of s. F is called a o-field if (a) if A E F, then A E F-closure under complementation (b)ifA1∈F,i=1,2,…,then(U1A)∈F- -closure under countable union Note that (a) and (b) taken together implying the followingTwo special events are S itself, called the sure events and the impossible event ∅ defined to contain no elements of S, i.e. ∅ = { }; the latter is defined for com￾pleteness. A third ingredient of E associated with (b) which Kolmogorov had to formal￾ized was the idea of uncertainty related to the outcome of any particular trial of E. This he formalized in the notion of probabilities attributed to the various events associated with E, such as P(A1), P(A2), expressing the ”likelihood” of occurrence of these events. Although attributing probabilities to the elementary events presents no particular mathematical problem, going the same for events in general is not as straightforward. The difficulty arise because if A1 and A2 are events, A1 = S −A1, A2 = S −A2, A1 ∩A2, A1 ∪A2, etc., are also events because the occurrence or non-occurrence of A1 and A2 implies the occurrence or not of these events. This implies that for the attribution of probabilities to make sense we have to impose some mathematical structure on the set of all events, say F, which reflects the fact that whichever way we combine these events, the end result is always an event. The temptation at this stage is to define F to be the set of all subsets of S, called the power set; Surely, this covers all possibilities ! In the above example, the power set of S take the form F = {S, ∅, {(HT)}, {(T H)}, {(HH)}, {(TT)}, {(HT),(TH)}, {(HT),(HH)}, {(HT),(TT)}, {(T H),(HH)}, {(T H),(TT)}, {(HH),(TT)}, {(HT),(T H),(HH)}, {(HT),(T H),(TT)}, {(T H),(HH),(TT)}, {(HT),(T H),(HH),(TT)}}. Sometimes we are not interested in all the subsets of S, we need to define a set independently of the power set by endowing it with a mathematical structure which ensures that no inconsistency arise. This is achieved by requiring that F in the following has a special mathematical structures, It is a σ-field related to S. Definition 3: Let F be a set of subsets of S. F is called a σ-field if: (a) if A ∈ F, then A ∈ F–closure under complementation; (b) if Ai ∈ F, i = 1, 2, ..., then (∪ ∞ i=1Ai) ∈ F–closure under countable union. Note that (a) and (b) taken together implying the following: 7