Construction of a Probability Measure on R
This chapter is an important special case of what we dealt with in Chapter 6. We assume that Ω=R.
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This chapter is an important special case of what we dealt with in Chapter 6. We assume that Ω = R. Let B be the Borel σ-algebra of R. (That is, B = σ(O), where O are the open subsets of R.) Definition 7.1. The distribution function induced by a probability P on (R, B) is the function F (x) = P ((−∞, x]). (7.1) Theorem 7.1. The distribution function F characterizes the probability. Proof. We want to show that knowledge of F defined by (7.1) uniquely determines P . That is, if there is another probability Q such that G(x) = Q((−∞, x]) for x ∈ R, and if F = G, then also P = Q. We begin by letting B0 be the set of finite disjoint unions of intervals of the form (x, y], with −∞ ≤ x ≤ y ≤ +∞ (with the convention that (x, ∞] = (x, ∞); observe also that (x, y] = ∅ if x = y). It is easy to see that B0 is an 1 algebra. Moreover if (a, b) is an open interval, then (a, b) = ∪∞ n=N (a, b − n ], for some N large enough, so σ(B0 ) contains all open intervals. But all open sets on the line can be expressed as countable unions of open intervals, and since the Borel sets (= B) are generated by the open sets, σ(B0 ) ⊃ B (note 1 also that ∩∞ n=1 (a, b + n ) = (a, b], so B0 ⊂ B and thus B = σ(B0 )). The relation (7.1) implies that P ((x, y]) = F (y) − F (x), and if A ∈ B0 is of the form A = ∪1≤i≤n (xi , yi ] with yi < xi+1 ,
then P (A) = 1≤i≤n {F (yi ) − F (xi )}. If Q is another probability measure such that F (x) = Q((−∞, x]), J. Jacod et al., Probability Essentials © Springer-Verlag Berlin Heidelberg 2004
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7 Construction of a Probability Measure on R
then the preceding shows that P = Q on B0 . Theorem 6.1 then implies that P = Q on all of B, so they are the same Probability measure. The significance of Theorem 7.1 is that we know, in principle, the complete probability measure P if we know its distribution function F : that is, we can in principle determine from F the probability P (A) for any given Borel set A. (Determining these probabilities in practice is another matter.) It is thus important to characterize all functions F which are distribution functions, and also to construct them easily. (Recall that a function F is right continuous if limy↓x F (y) = F (x), for all x ∈ R.) Theorem 7.2. A function F is the distribution function of a (unique) probability on (R, B) if and only if one has: (i) F is non-decreasing; (ii) F is right continuous; (iii) limx→−∞ F (x) = 0 and limx→+∞ F (x) = 1. Proof. Assume that F is a distribution function. If y > x, then (−∞, x] ⊂ (−∞, y], so P ((−∞, x]) ≤ P ((−∞, y]) and thus F (x) ≤ F (y). Thus we have (i). Next let xn decrease to x. Then ∩∞ n=1 (−∞, xn ] = (−∞, x], and the sequence of events {(−∞, xn ]; n ≥ 1} is a decreasing sequence. Therefore P (∩∞ n=1 (−∞, xn ]) = limn→∞ P ((−∞, xn ]) = P ((−∞, x]) by Theorem 2.3, and we have (ii). Similarly, Theorem 2.3 gives us (iii) as well. Next we assume that we have (i), (ii), and (iii) and we wish to show F is a distribution function. In accordance with (iii), let us set F (−∞) = 0 and F (+∞) = 1. As in the proof of Theorem 7.1, let B0 be the s
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