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Dynkin (Λ-) and Π-Systems; Monotone Classes of Sets, and of Functions – with Some Examples of Application (Mainly of a Probabilistic flavor)

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Dynkin (Λ-) and Π-Systems; Monotone Classes of Sets, and of Functions – with Some Examples of Application (Mainly of a Probabilistic flavor) Dynkin (λ-) and π-systems; monotone classes of sets, and of functions { with some examples of application (mainly of a probabilistic flavor) Matija Vidmar February 7, 2018 1 Dynkin and π-systems Some basic notation: Throughout, for measurable spaces (A; A) and (B; B), (i) A=B will denote the class of A=B-measurable maps, and (ii) when A = B, A_B := σA(A[B) will be the smallest σ-field on A containing both A and B (this notation has obvious extensions to arbitrary families of σ-fields on a given space). Furthermore, for a measure µ on F, µf := µ(f) := R fdµ will signify + − the integral of an f 2 F=B[−∞;1] against µ (assuming µf ^ µf < 1). Finally, for a probability space (Ω; F; P) and a sub-σ-field G of F, PGf := PG(f) := EP[fjG] will denote the conditional + − expectation of an f 2 F=B[−∞;1] under P w.r.t. G (assuming Pf ^ Pf < 1; in particular, for F 2 F, PG(F ) := P(F jG) = EP[1F jG] will be the conditional probability of F under P given G). We consider first Dynkin and π-systems. Definition 1. Let Ω be a set, D ⊂ 2Ω a collection of its subsets. Then D is called a Dynkin system, or a λ-system, on Ω, if (i) Ω 2 D, (ii) fA; Bg ⊂ D and A ⊂ B, implies BnA 2 D, and (iii) whenever (Ai)i2N is a sequence in D, and Ai ⊂ Ai+1 for all i 2 N, then [i2NAi 2 D. Remark 2. Let D ⊂ 2Ω. Then D is a Dynkin system on Ω if and only if (1) Ω 2 D, (2) A 2 D implies ΩnA 2 D, and (3) whenever (Ai)i2N is a sequence in D, and Ai \ Aj = ; for all i 6= j from N, then [i2NAi 2 D. [If D is a Dynkin system, then for fA; Bg ⊂ D with A \ B = ;, A [ B = Ωn((ΩnA)nB) 2 D. Conversely, if (1)-(2)-(3) hold, then for fA; Bg ⊂ D with A ⊂ B, BnA = Ωn(A [ (ΩnB)) 2 D. The rest is (even more) trivial.] Remark 3. Given L ⊂ 2Ω, there is a smallest (with respect to inclusion) λ-system on Ω containing L. It is the intersection of all the λ-systems on Ω containing L (of which 2Ω is one) and will be denoted λΩ(L). (Similarly σΩ(L) will denote the smallest (with respect to inclusion) σ-field on Ω containing L.) Definition 4. Let L ⊂ 2Ω. Then L is called a π-system on Ω, if A\B 2 L, whenever fA; Bg ⊂ L. Remark 5. Let D ⊂ 2Ω be both a π-system and a λ-system on Ω. Then D is a σ-algebra on Ω. Any λ-system contains, besides Ω, also the empty set. 1 Here is now the key result of this section. Theorem 6. Let L be a π-sytem on Ω. Then σΩ(L) = λΩ(L). Proof. Since every σ-algebra on Ω is a λ-system on Ω, the inclusion σΩ(L) ⊃ λΩ(L) is manifest. For the reverse inclusion, it will be sufficient (and, indeed, necessary) to show that λΩ(L) is a σ-algebra on Ω. Then it will be sufficient to check that λΩ(L) is a π-system. Define U := fA 2 λΩ(L): A \ B 2 λΩ(L) for all B 2 Lg: U is a λ-system, containing L, so λΩ(L) ⊂ U. Now define V := fA 2 λΩ(L): A \ B 2 λΩ(L) for all B 2 λΩ(L)g. Again V is a λ-system, containing, by what we have just shown, L. But then λΩ(L) ⊂ V, and we conclude. Corollary 7 (π-λ theorem/Dynkin's lemma/Sierpinski class theorem). Let L be a π-system, D a Dynkin system on Ω, L ⊂ D. Then σΩ(L) ⊂ D. Remark 8. The significance of the above result stems mainly from the following observation. Let A be a σ-field, and L a π-system on Ω. Suppose A = σΩ(L). Often in measure theory (and in probability theory, in particular) we wish to show that some property (typically involving measures), P (A), in A 2 A, holds true of all the members A of A, and we wish to do so, having already been given, or having already established beforehand, that P (A) holds true for all A 2 L. Now, it is usually easy (because measures behave nicely under disjoint/increasing countable unions, and because finite1 measures also behave nicely under differences/complements) to check directly that the collection D := fA 2 σΩ(L): P (A)g is a λ-system, but it is usually not so easy (because measures do not behave so nicely under arbitrary countable unions) to check directly that D is a σ-field. The π/λ lemma establishes an extremely useful shortcut of doing so indirectly. Its applicability is further strengthened by the fact that it is typically not difficult to find a generating π-system on which the property P \obviously" holds true. Some (mainly probabilistic) applications exemplifying this now follow. \It suffices to check densities on a π-system": Proposition 9. Let (Ω; F; P) be a probabilty space, (D; Σ; µ) a σ-finite measure space, X 2 F=Σ, f 2 Σ=B[0;1]. Suppose L is π-system on D, such that σD(L) = Σ, and suppose that the following holds for all L 2 L [ fDg: Z P(X 2 L) = fdµ. L −1 −1 d(P◦X ) 2 Then f is a density for X on (D; Σ; µ) under P, i.e. P ◦ X µ and f = dµ a.e.-µ. Proof. From the hypotheses, linearity of integration and monotone convergence, the collection fL 2 R Σ: P(X 2 L) = L fdµg is a λ-system on D, containing the π-system L. We conclude at once via the π-λ theorem. \A finite (in particular, a probability) measure is determined by its total mass and the values it assumes on a π-system": Proposition 10. Let µ and ν be two finite measures on a measurable space (E; Σ), L ⊂ Σ a π-system with σE(L) = Σ. Suppose µjL[fEg = νjL[fEg. Then µ = ν. Proof. From the hypotheses, continuity from below, and finite additivity of measures, the collection of sets fA 2 Σ: µ(A) = ν(A)g is a λ-system on E containing L. 1But not arbitrary measures; for this reason the result is most often applicable in settings involving fi- nite/probability or at best (via some localization) σ-finite measures. −1 2 d(P◦X ) dµ is the Radon-Nikdoym derivative with respect to µ of the law of X under P on the space (D; Σ) (so P ◦ X−1(L) = P(X 2 L) for L 2 Σ). 2 Remark 11. If in the context of the preceding proposition we drop the assumption that µ and ν are finite, but instead demand that L contains a sequence (Ci)i2N, either monotone or consisting of pairwise disjoint sets, with [i2NCi = E and µ(Ci) = ν(Ci) < 1 for all i 2 N, then still µjL = νjL implies µ = ν. Indeed, since LjCi is a π-system (contained in L; having Ci for an element), σC (LjC ) = σE(L)jC = ΣjC , and µj = νj , we obtain νj = µj for all i 2 , which i i i i LjCi LjCi ΣjCi ΣjCi N implies ν = µ, either by continuity from below or countable additivity, as the case may be. Definition 12. Given a probability space (Ω; F; P), a sub-σ-field A of F, and a collection C = (Cλ)λ2Λ of subsets of F, we say C is an independency (under P) conditionally on A, if for any (non-empty) finite I ⊂ Λ, and then for any choices of Ci 2 Ci, i 2 I, we have, P-a.s. Y PA(\i2I Ci) = PA(Ci): i2I Of course we say subsets B and C of F are independent (under P) conditionally on A, if the family (B; C) consisiting of them alone, is an independency (under P). Further, given a measurable space (E; Σ), Z 2 F=Σ, and B ⊂ F, we say Z is independent of B conditionally on A (relative to Σ under P), if σ(Z) (:= fZ−1(S): S 2 Σg) is independent of B conditionally on A. And so on, and so forth. The qualification \conditionally on A" is dropped when A = f;; Ωg, the trivial σ-field (and then PA(F ) = P(F )). Remark 13. A sub-collection of an independency is an independency. A collection is an indepen- dency if and only if every finite sub-collection thereof is so. \Independence can be raised from π-systems to the σ-fields they generate": Proposition 14. Let (Ω; F; P) be a probability space, A a sub-σ-field of F, C = (Cλ)λ2Λ an independency under P conditionally on A, consisting of π-systems alone. Then (σΩ(Cλ))λ2Λ also is an independency under P conditionally on A. Proof. It will suffice to show that if the finite collection of π-systems (A1;:::; An)(n ≥ 2 an integer) on Ω is an independency, then (σΩ(A1); A2;:::; An) is one also. (Then one can apply mathematical induction.) Consider then L, the collection of all elements L 2 F, such that for all A2 2 A2,..., An 2 An, P-a.s., PA(L \ A2 \···\ An) = PA(L)PA(A2) ··· PA(An).
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