A Probability Path.pdf

# M anam 0 an e c unan e c the old postulates are

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m, AnAm = 0, An E .C UnAn E .C. The old postulates are equivalent to the new ones. Here we only check that old implies new. Suppose >..;, >..;, >..) are true. Then .l..t is true. Since Q E .C, if A E .C, then A C Q and by>..;, Q \A = Ac E .C, which shows that >..2 is true. If A, B E .C are disjoint, we show that A U B E .C. Now Q \ A E .C and B C Q \ A (since (J) E B implies (J) fl. A which means (J) E Ac = Q \A) so by>..; we have (Q \A)\ B = AcBc E .C and by >..2 we have (AcBc)c =AU BE .C which is AJ for finitely many sets. Now if A j E .Care mutually disjoint for j = 1, 2, ... ,

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2.2 More on Closure 37 define Bn = U}=tAi . Then Bn e £by the prior argument for 2 sets and by >..3 we have UnBn = limn-+oo t Bn E £. Since UnBn = UnAn we have UnAn E £ which is AJ. 0 Remark. It is clear that a a-field is always a >..-system since the new postulates obviously hold. Recall that a rr - system is a class of sets closed under finite intersections; that is, 'Pis a rr-system if whenever A, B e 'P we have AB e 'P . We are now in a position to state Dynkin's theorem. Theorem 2.2.2 (Dynkin's theorem) (a) lf'P is a rr-system and£ is a >..-system such that P c £, then a ('P) C £. (b) lf'P is a rr-system a('P) = £('P), that is, the minimal a-field over P equals the minimal >.. - system over P. Note (b) follows from (a). To see this assume (a) is true. Since 'P c C('P), we have from (a) that a('P) c C('P). On the other hand, a('P), being a a-field, is a >..-system containing P and hence contains the minimal >..-system over 'P, so that a ('P) ::> £('P). Before the proof of (a), here is a significant application of Dynkin's theorem. Proposition 2.2.3 Let Pt. Pz be two probability measures on (Q, 8) . The class £ : ={A e 8: Pt(A) = Pz(A)} is a >..-system. Proof of Proposition 2.2.3. We show the new postulates hold : (J..t) n E £since Pt(fl) = Pz(n) = 1. (J..z) A e £implies Ace£, since A E £means Pt(A) = Pz(A), from which (J..3) If {A i} is a mutually disjoint sequence of events in £, then P 1 (A i) = Pz (A i) for all j, and hence P 1 (UAj) = LPt(Aj) = L:Pz(Aj) = Pz<UAj) j j j j so that 0
38 2. Probability Spaces Corollary 2.2.1 If Pt. P2 are two probability measures on (Q, B) and ifP is a ;r-system such that 'v'A e P: Pt(A) = P2(A), then 'v'B e a(P): Pt(B) = P2(B). Proof of Corollary 2.2.1. We have C ={A e B: Pt(A) = P2(A)} is a A-system. But C :::> P and hence by Dynkin's theorem C :::> a(P) . 0 Corollary 2.2.2 Let n = JR. Let Pt, P2 be two probability measures on (JR., B(IR.)) such that their distribution functions are equal: 'v'x E JR.: Ft(X) = Pt((-oo,x)) = F2(x) = P2((-oo,x]) . Then on B(IR.). So a probability measure on JR. is uniquely determined by its distribution func- tion. Proof of Corollary 2.2.2. Let P = {(-oo,x]: x e JR.}. Then P is a ;r -system since (-oo,x) n (-oo,y] = (-oo,x Ay) E P. Furthermore a (P) = B(IR.) since the Borel sets can be generated by the semi- infinite intervals (see Section 1.7). SoFt (x) = F2(x) for all x e JR., means Pt = P2 on P and hence Pt = P2 on a(P) = B(IR.). 0 2.2.2 Proof of Dynkin 's theorem Recall that we only need to prove: If Pis a ;r-system and Cis a A-system then P C C implies a (P) C C.

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