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 ∈ F/B[−∞,∞] against µ (assuming µf ∧ µf < ∞). Finally, for a probability space (Ω, F, P) and a sub-σ-field G of F, PGf := PG(f) := EP[f|G] will denote the conditional + − expectation of an f ∈ F/B[−∞,∞] under P w.r.t. G (assuming Pf ∧ Pf < ∞; in particular, for F ∈ F, PG(F ) := P(F |G) = EP[1F |G] 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 ∈ D, (ii) {A, B} ⊂ D and A ⊂ B, implies B\A ∈ D, and

(iii) whenever (Ai)i∈N is a sequence in D, and Ai ⊂ Ai+1 for all i ∈ N, then ∪i∈NAi ∈ D. Remark 2. Let D ⊂ 2Ω. Then D is a Dynkin system on Ω if and only if (1) Ω ∈ D, (2) A ∈ D implies Ω\A ∈ D, and

(3) whenever (Ai)i∈N is a sequence in D, and Ai ∩ Aj = ∅ for all i 6= j from N, then ∪i∈NAi ∈ D. [If D is a Dynkin system, then for {A, B} ⊂ D with A ∩ B = ∅, A ∪ B = Ω\((Ω\A)\B) ∈ D. Conversely, if (1)-(2)-(3) hold, then for {A, B} ⊂ D with A ⊂ B, B\A = Ω\(A ∪ (Ω\B)) ∈ 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 ∈ L, whenever {A, B} ⊂ 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 := {A ∈ λΩ(L): A ∩ B ∈ λΩ(L) for all B ∈ L}: U is a λ-system, containing L, so λΩ(L) ⊂ U. Now define V := {A ∈ λΩ(L): A ∩ B ∈ λΩ(L) for all B ∈ λΩ(L)}. 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 ∈ 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 ∈ 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 := {A ∈ σΩ(L): P (A)} 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 ∈ F/Σ, f ∈ Σ/B[0,∞]. Suppose L is π-system on D, such that σD(L) = Σ, and suppose that the following holds for all L ∈ L ∪ {D}: Z P(X ∈ 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 {L ∈ R Σ: P(X ∈ L) = L fdµ} 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 µ|L∪{E} = ν|L∪{E}. Then µ = ν. Proof. From the hypotheses, continuity from below, and ﬁnite additivity of measures, the collection of sets {A ∈ Σ: µ(A) = ν(A)} 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 ∈ L) for L ∈ 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)i∈N, either monotone or consisting of pairwise disjoint sets, with ∪i∈NCi = E and µ(Ci) = ν(Ci) < ∞ for all i ∈ N, then still µ|L = ν|L implies µ = ν. Indeed, since L|Ci is a π-system (contained in L; having Ci for an element), σC (L|C ) = σE(L)|C = Σ|C , and µ| = ν| , we obtain ν| = µ| for all i ∈ , which i i i i L|Ci L|Ci Σ|Ci Σ|Ci 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)λ∈Λ 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 ∈ Ci, i ∈ I, we have, P-a.s. Y PA(∩i∈I Ci) = PA(Ci). i∈I

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 ∈ F/Σ, and B ⊂ F, we say Z is independent of B conditionally on A (relative to Σ under P), if σ(Z) (:= {Z−1(S): S ∈ Σ}) is independent of B conditionally on A. And so on, and so forth. The qualification “conditionally on A” is dropped when A = {∅, Ω}, 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 independency 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)λ∈Λ an independency under P conditionally on A, consisting of π-systems alone. Then (σΩ(Cλ))λ∈Λ 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 ∈ F, such that for all A2 ∈ A2,..., An ∈ An, P-a.s., PA(L ∩ A2 ∩ · · · ∩ An) = PA(L)PA(A2) ··· PA(An). It is, from the hypothesis and by properties of conditional expectations, a λ-system, containing the π-system A1. “Algebras are dense in the σ-fields they generate”:

Proposition 15. Let (X, F, µ) be a ﬁnite measure space, A ⊂ F an algebra on X, σX (A) = F. Then A is dense in F for the topology induced by the pseudometric (F × F 3 (F,G) 7→ µ(F 4G)). Proof. The class of sets L := {F ∈ F : ∀ > 0 ∃A ∈ A s.t. µ(F 4A) < } is a λ-system on X, that contains the algebra (and hence π-system) A that generates F: clearly X ∈ L, since X ∈ A and µ(X4X) = µ(∅) = 0; if F ∈ L, > 0 and A ∈ A is such that µ(F 4A) < , then

µ((X\F )4(X\A)) = µ(F 4A) < ; ﬁnally, if (Fi)i∈N is a sequence in L of pairwise disjoint sets, ∞ > 0, then by continuity from above of µ, there is an N ∈ N0 with µ(∪i=N+1Fi) < /2, whilst for i ∞ N each i ∈ {1,...,N}, there is an Ai ∈ A with µ(Fi4Ai) < /2 , whence µ((∪i=1Fi)4(∪i=1Ai)) ≤ ∞ PN µ(∪i=N+1Fi) + i=1 µ(Fi4Ai) < .

2 Monotone classes of sets

We now turn our attention to monotone classes. Deﬁnition 16. Let M ⊂ 2Ω. Then M is called a monotone class on Ω, if

(i) whenever (Ai)i∈N a sequence in M, and Ai ⊂ Ai+1 for all i ∈ N, then ∪i∈NAi ∈ M, and

3 (ii) whenever (Ai)i∈N a sequence in M, and Ai ⊃ Ai+1 for all i ∈ N, then ∩i∈NAi ∈ M. Remark 17. Given M ⊂ 2Ω, there is a smallest (with respect to inclusion) monotone class on Ω containing M. It is the intersection of all the monotone classes on Ω containing M and will be denoted mΩ(M). Remark 18. Let Ω be a set and M be a monotone class on Ω, that contains Ω and is closed under complements and ﬁnite unions. Then M is a σ-algebra on Ω.

Theorem 19. Let A be an algebra on Ω. Then σΩ(A) = mΩ(A).

Proof. Since every σ-algebra on Ω is a monotone class on Ω, the inclusion σΩ(A) ⊃ mΩ(A) is manifest. For the reverse inclusion, it will be sufficient (and, indeed, necessary) to check that mΩ(A) is a σ-algebra on Ω. Then it will be sufficient to show that mΩ(A) is closed under complements and finite unions. Define U := {A ∈ mΩ(M): A ∪ B ∈ mΩ(A) for all B ∈ A}: U is a monotone class containing A, so mΩ(A) ⊂ U. Now define V := {A ∈ mΩ(A): A ∪ B ∈ mΩ(M) for all B ∈ mΩ(M)}. Again V is a monotone class, containing, by what we have just shown, A. But then mΩ(A) ⊂ V. Finally define Z := {A ∈ mΩ(M):Ω\A ∈ mΩ(A)}: Z is a monotone class containing A, so mΩ(A) ⊂ Z and we conclude. Corollary 20 (Halmos’ monotone class theorem). Let A be an algebra on Ω, M a monotone class on Ω, A ⊂ M. Then σΩ(A) ⊂ M. A simple application is as follows. Proposition 21. Let µ and ν be two finite measures on a measurable space (E, Σ), A ⊂ Σ an algebra on Ω with σE(A) = Σ. Suppose µ|A ≤ ν|A. Then µ ≤ ν. Proof. From the hypotheses and the continuity from below and above of finite measures, the collection of sets {A ∈ Σ: µ(A) ≤ ν(A)} is a monotone class on E containing A. Here is another: “(Super-, sub-)martingales remain (super-, sub-)martingales if the underlying filtration is enlarged by an independent σ-field.” Definition 22. Let (T, ≤) be a partially ordered set, (Ω, F, P) a probability space, and let thereon G = (Gt)t∈T be a filtration (i.e. a non-decreasing family of sub-σ-fields of F), and X = (Xt)t∈T a stochastic process with values ([−∞, ∞], B[−∞,∞]) (it means that Xt ∈ F/B[−∞,∞] for all t ∈ T ). Then X is said to be a (G, P)-supermartingale (resp. -submartingale) if it is G-adapted (i.e. Xt ∈ Gt/B[−∞,∞] for each t ∈ T ), the negative (resp. positive) parts of X are P-integrable, and,

P-a.s., PGs Xt ≤ Xs (resp. PGs Xt ≥ Xs) whenever s ≤ t are from T . X is said to be a (G, P)- martingale if it is both a supermartingale and a submartingale. Proposition 23. In the context of the previous definition if further H is a sub-σ-field of F independent of G∞ ∨ σ(X), then X is a (G ∨ H, P)-supermartingale (resp. submartingale, martingale), provided it is a (G, P)-supermartingale (resp. submartingale, martingale). Here G ∨H = (Gt ∨H)t∈T and G∞ = ∨t∈T Gt. Proof. The submartingale case follows upon replacing X by −X; the martingale case upon con- sidering both, so we may restrict to supermartingales. Asssuming X is one; let s ≤ t be from T . We may assume without loss of generality X is integrable (X ∧ N is also a supermartingale for each N ∈ N0 and one may pass to the limit via monotone convergence). Then the class of sets C ∈ Gs ∨ H for which P[Xt; C] ≤ P[Xs; C] is a monotone class by dominated convergence (here one uses the integrability of Xt and Xs). Moreover, it contains the algebra n L := {∪i=1Ai ∩ Bi : {A1,...,An} ⊂ Gs,A1 ...,An pairwise disjoint, {B1,...,Bn} ⊂ H, n ∈ N0} n Pn that generates Gs ∨ H on Ω: (under the obvious provisos) P[Xt; ∪i=1AiBi] = i=1 P[Xt; AiBi] = Pn Pn n i=1 P[Xt; Ai]P[Bi] ≤ i=1 P[Xs; Ai]P[Bi] = ··· = P[Xs; ∪i=1AiBi]. (To see that L is an algebra consider the partitions induced on Ω by finite subsets of Gs ∪ H (and Gs in particular). Use the fact that the algebra generated by Gs ∪ H is equal to the union of the algebras generated by finite subsets of Gs ∪ H.)

4 The next slight extension (to include conditional independence) of Kolmogorov’s zero-one law is also easily proven using the above techniques.

Proposition 24. Let (Ω, F, P) be a probability space, let C be a sub-σ-ﬁeld of F and let ξ = (ξi)i∈I be a family of π-systems in F, conditionally independent given C. Then

\ −1 lim sup ξ := σΩ(∪i∈I\F ξi) ⊂ C ∨ P ({0, 1}). F ⊂I,F ﬁnite

Remark 25. Often one takes C = {∅, Ω}, the trivial σ-field, in which case the conclusion says that the tail σ-field lim sup ξ is P-trivial. Proof. We omit making explicit some details for brevity; the reader is asked to fill them in as necessary. By an application of Dynkin’s lemma, for each finite F ⊂ I, σΩ(∪i∈I\F ξi), and hence lim sup ξ, is seen to be conditionally independent of σΩ(∪i∈F ξi) given C. Then lim sup ξ is independent of the algebra ∪F ⊂I,F finiteσΩ(∪i∈F ξi) given C. By an application of Dynkin’s or Halmos’ result, lim sup ξ is independent of σΩ(∪i∈I ξi) given C. In particular, lim sup ξ is independent of itself 2 given C. Then, if D ∈ lim sup ξ, P(D|C) = P(D|C) a.s., i.e. P(D|C) = 1C a.s. for some C ∈ C. In particular, P(D\C) = P(P(D|C); Ω\C) = 0 and P(D ∩ C) = P(P(D|C); C) = P(C), which implies that P(D4C) = 0.

3 Monotone classes of functions

The following few results deal with monotone classes of functions. For a class H of [−∞, ∞]-valued functions we will use the notations Hb := {f ∈ H : f is bounded} and H+ := {f ∈ H : f ≥ 0}; for a σ-ﬁeld G in particular, we write G+ := G/B[0,∞] and Gb := (G/BR)b. Here is ﬁrst what is typically behind “the usual arguments of linearity, monotone convergence and π/λ-considerations”. Proposition 26. Let H be a class of [−∞, ∞]-valued functions on Ω and C ⊂ 2Ω. Denote B := σΩ(C). Suppose as follows. (i) C is a π-system on Ω.

(ii) 1C ∈ H for all C ∈ C ∪ {Ω}.

(iii) Hb is a vector space over R.

(iv) Hb+ is closed under nondecreasing bounded limits.

Then Bb ⊂ H. If furthermore H+ is closed under nondecreasing limits, then B+ ⊂ H. Proof. Any nonnegative function is the nondecreasing limit of nonnegative simple functions (uniformly, in particular boundedly, on any set on which the function is bounded) and any bounded function is the diﬀerence of its bounded positive and its bounded negative part. So it is enough to show that every indicator of an element of E is in H. But the set of all E for which this is true is a Dynkin system containing the π-system C. In the context of the simple Markov property: Proposition 27. Let ≤ be a linear order on F , (Ω, F, P) a probability space, and let thereon G = (Gt)t∈F be a ﬁltration, and X = (Xt)t∈F a stochastic process with values in a measurable space (E, E), adapted to G. Let C be a π-system on E generating E. Then the statements

(i) PGs (Xt ∈ C) = PXs (Xt ∈ C), P-a.s., whenever s ≤ t are from F and C ∈ C; and ⊗F≥s (ii) PGs [Θ ◦ (Xu)u∈F≥s ] = PXs [Θ ◦ (Xu)u∈F≥s ], P-a.s., whenever s ∈ F and Θ ∈ (E )+, are equivalent.3

3 For s ∈ F , F≥s := {t ∈ T : t ≥ s}.

5 Remark 28. Using properties of conditional expectations, (i)-(ii) are seen to be equivalent to (iii): for all s ∈ F , σ(Xt : t ∈ F≥s) [or just each Xt, as t runs over F≥s] is independent of Gs conditionally on Xs. Assume now G is the natural filtration of X: Gt = σ(Xs : s ≤ t) for t ∈ T . Then, since, as we have already observed, conditional independence can be raised from π-systems to the σ-fields they generate, we may also (iv): in (iii) replace Gs by ∪Aσ(Xt : t ∈ A) where A runs over the finite subsets of F≤s, and still an equivalent condition obtains. Finally, if in addition E is denumerable and E = 2E discrete, then the above are in fact equivalent to

(v) P(Xtn+1 = in+1|Xtn = in,...,Xt0 = i0) = P(Xtn+1 = in+1|Xtn = in), whenever P(Xtn = in,...,Xt0 = i0) > 0, t0 ≤ · · · ≤ tn+1 are from F , and {i0, . . . , in+1} ⊂ E, n ∈ N0. Proof. That (ii) implies (i) is immediate. Conversely, assuming the ﬁrst condition, we prove by induction that for all n ∈ N0, all g1, . . . , gn nonnegative measurable maps, and all s ≤ t1 ≤ t2 ≤ · · · ≤ tn from F , the equality under the second condition holds true with Θ = g1 ◦ prt1 ··· gn ◦ prtn . It is clear for n = 0 (when Θ ≡ 1). Let n ∈ N, suppose the claim valid for n − 1. Note that by Proposition 26, (i) implies that

PGs [f ◦ Xt] = PXs [f ◦ Xt], P-a.s., whenever s ≤ t are from F and f is a measurable nonnegative map. Then P [g ◦ X ··· g ◦ X ] = P [g ◦ X ··· g ◦ X P [g ◦ X ]] = P [g ◦ Gs 1 t1 n tn Gs 1 t1 n−1 tn−1 Gtn−1 n tn Gs 1 X ··· g ◦ X P [g ◦ X ]] = P [g ◦ X ··· g ◦ X P [g ◦ X ]] = P [g ◦ t1 n−1 tn−1 Xtn−1 n tn Xs 1 t1 n−1 tn−1 Xtn−1 n tn Xs 1 X ··· g ◦ X P [g ◦ X ]] = P [g ◦ X ··· g ◦ X · g ◦ X ]. t1 n−1 tn−1 Gtn−1 n tn Xs 1 t1 n−1 tn−1 n tn It remains to apply yet again the previous proposition. “σ-ﬁelds are the embodiement of information”: Proposition 29 (The Doob-Dynkin lemma). Let Ω be a set, (E, E) a measurable space, X :Ω → E, 4 f :Ω → [−∞, ∞]. Then f ∈ σ(X)/B[−∞,∞] if and only if there is a ξ ∈ E/B[−∞,∞] with f = ξ ◦X.

Remark 30. As a further consequence, an analogous claim is true if Rn (or any Borel measurable subset of Rn, or of [−∞, ∞]) replaces [−∞, ∞]. Proof. Suﬃciency is immediate. Necessity. The class of nonnegative functions f for which the property holds is seen to be closed under nonnegative linear combinations and nondecreasing limits: in not so many words; if fn = ξn ◦X, with fn ↑ f, then f = (lim supn→∞ ξn)◦X. By Proposition 26, the property holds for all nonnegative f, since it clearly holds for indicators of the elements of −1 + − σ(X) = {X (A): A ∈ E}: 1X−1 (A) = 1A ◦X. One concludes via the decomposition f = f −f : + − + − − again in not so many words; if f = ξ+◦X and f = ξ−◦X, then f = f −f = (ξ+∨0−ξ ∨0)◦X, where one sets, e.g. ∞ − ∞ = 0.

We come now to the main result of this section. Theorem 31 (Monotone class theorem for functions – multiplicative case). Let K and H be classes of bounded R-valued functions on Ω, denote by B := σ(K) the smallest σ-algebra w.r.t. which all elements of K are measurable. Suppose as follows. (i) K is a multiplicative class (i.e. {f, g} ⊂ K ⇒ fg ∈ K).

(ii) K ∪ {1Ω} ⊂ H.

(iii) H is a vector space over R.

(iv) H+ is closed under nondecreasing bounded limits.

Then Bb ⊂ H. 4Here σ(X) = X−1E = {X−1(A): A ∈ E} is the σ-ﬁeld generated by X (relative to E).

6 Remark 32. It is worth pointing out that the smallest σ-field w.r.t. which all the (bounded) continuous functions are measurable (i.e. the Baire σ-field) coincides with the Borel σ-field on metrizable topological spaces: for a metric d on X and a closed set F ⊂ X, the map (X 3 x 7→ d(x, F )) is continuous and its zero set is F .

Proof. We may assume 1Ω ∈ K (if K is a multiplicative class then so is K∪{1Ω}). By Proposition 26 it is enough to show that 1C ∈ H for all C belonging to some π-system C on Ω generating B. Let A0 denote the algebra generated by K. Since K is already closed under multiplication, A0 is simply the linear span of K. Consequently A0 ⊂ H.

Lemma 33. If H is as stated, then H is closed for k · ksup, i.e. it is closed under uniform convergence.

Proof. Suppose (fn)n∈N is a sequence in H and fn → f pointwise uniformly on Ω as n → ∞. By −n passing to a subsequence if necessary, one can arrange that kfn+1 − fnksup ≤ 2 , n ∈ N. Deﬁne 1−n gn := fn − 2 + kf1ksup + 1, n ∈ N. Then gn ∈ H and kgnksup ≤ kfnksup + kf1ksup + 1, n ∈ N −n so the sequence (gn)n∈N is uniformly bounded. Also gn+1 − gn = fn+1 − fn + 2 ≥ 0, n ∈ N and g1 = f1 + kf1k ≥ 0. It follows that limn→∞ gn = limn→∞ fn + kf1ksup + 1 ∈ H and thus limn→∞ fn ∈ H.

This lemma thus tells us that the uniform closure A of A0 is again a subset of H and it is trivial to check that it is itself in turn an algebra. Referring to Weierstrass’ Theorem, let pn, n ∈ N be a sequence of real polynomials deﬁned on [−1, 1] converging uniformly to ([−1, 1] 3 s 7→ |s|). Then if 0 6≡ f ∈ A, pn ◦ (f/kfksup) → |f|/kfksup uniformly as n → ∞. Hence, since A is an algebra closed for uniform convergence, one has |f| ∈ A. It follows, moreover, that for {f, g} ⊂ A, f ∨ g = [|f − g| + f + g]/2 and f ∧ g = [f + g − |f − g|]/2 are elements of A in turn. Take now {f1, . . . , fm} ⊂ A and (b1, . . . , bm) a sequence of real numbers. Then the function

m Y + gn := [n(fk − bk) ] ∧ 1 k=1 Qm 1 is in A for each natural n and gn ↑ k=1 {fk>bk} pointwise (and boundedly), as n → ∞, and hence the latter function is in H. As an application of the latter consider:

Proposition 34. Two laws on (R, BR) with the same characteristic function, coincide. Proof. If µ and ν are two such laws, then clearly the class H of bounded R-valued Borel measurable functions f, for which R fdµ = R fdν, is a vector space, containing the constants, and closed under nondecreasing limits of nonnegative functions converging to a bounded function (monotone convergence). Moreover it contains ﬁnite linear combinations of the functions sin(a·), cos(a·), a ∈ R, whose totality is closed under multiplication. Finally, the latter generate the Borel σ-ﬁeld. There are many variations on Theorem 31. We content ourselves by making explicit two of them. Theorem 35 (Monotone class theorem for functions – algebra case). Let A and H be classes of bounded R-valued functions on Ω; denote by B := σ(A) the smallest σ-algebra w.r.t. which all elements of A are measurable. Suppose as follows.

(i) A is an algebra (i.e. it is closed under multiplication and linear combinations) and 1Ω ∈ A. (ii) A ⊂ H. (iii) H is closed under uniform convergence.

(iv) H+ is closed under nondecreasing bounded limits.

7 Then Bb ⊂ H. Proof. Replacing A by its uniform closure, the proof proceeds exactly as in the last paragraph of Theorem 31.

Theorem 36 (Monotone class theorem for functions – lattice case). Let L and H be classes of bounded R-valued functions on Ω; denote by B := σ(L) the smallest σ-algebra w.r.t. which all elements of L are measurable. Suppose as follows.

(i) L ⊂ Bb+ is a cone closed under ∧ (i.e. {f, g} ⊂ L ⇒ f ∧ g ∈ L) and 1Ω ∈ L. (ii) L ⊂ H.

(iii) H is a vector space over R.

(iv) H+ is closed under nondecreasing bounded limits.

Then Bb ⊂ H.

Proof. A := L − L is a vector lattice over R (vector space over R closed under ∧ and ∨) containing the constants ((f − g) ∧ (h − j) = [(f + j) ∧ (h + g)] − (g + j) whenever {f, g, h, j} ⊂ L); so is its uniform closure K. By Lemma 33, H ⊃ K. The map ([−1, 1] 3 x 7→ x2) can be approximated by a sequence of linear combinations of maps of the form ([−1, 1] 3 x 7→ (x − c)+ = (x − c) ∨ 0) and ([−1, 1] 3 x 7→ (x − c)− = (c − x) ∨ 0), c ∈ R (consider the diﬀerences ([−1, 1] 3 x 7→ (x − c)+ − (x − d)+ = (x − c)+ ∧ (d − c)) for real 0 ≤ c < d . . . ). Similarly as in the proof of Theorem 31 (there for |f|), it now follows that, together with any f ∈ K, also f 2 ∈ K. By polarization (fg = ((f + g)2 − (f − g)2)/4) we ﬁnd that K is a multiplicative class. It remains to apply Theorem 31. For instance, this form is ideally suited to proving uniqueness in the Danielli theorem:

Proposition 37. Let A be a linear space of bounded real-valued functions on Ω, which is closed under ∧ and contains the constant functions. Let λ be a linear real-valued function on A, that is nondecreasing (i.e. nonnegative on the cone A+ of nonnegative elements of A) and satisﬁes for every sequence (hn)n∈ in A+ with limn→∞ hn = 0, limn→∞ λ(hn) = 0. N R Then there exists at most one measure µ on σ(A) for which λ(h) = hdµ for all h ∈ A+. Remark 38. Danielli’s theorem also gives the existence of such a µ.

Proof. Let µ1 and µ2 be two such measures. Necessarily µ1 and µ2 are ﬁnite. Furthermore, A+ is a cone, stable under ∧, of bounded [0, ∞)-valued functions on Ω, containing 1Ω. Clearly σ(A) = R R σ(A+). The class H of bounded real-valued σ(A)-measurable functions h for which hdµ1 = hdµ2 is a vector space over R closed under limits of nonnegative functions nondecreasing to a bounded function. It contains A+. Here is another application:

Proposition 39. Let (Ω, F, P) be a probability space, {X,Y } ⊂ F/B[0,∞]. Suppose E[X ∧ a] = E[Y ∧ a] for all a ∈ [0, ∞). Then X and Y have the same law.

Proof. The class of f ∈ (B[0,∞])b for which E[f ◦ X] = E[f ◦ Y ] is a vector space closed under nondecreasing nonnegative limits. By assumption and linearity it contains the cone of piecewise aﬃne continuous ultimately constant nondecreasing functions mapping [0, ∞] → [0, ∞) (consider suitable sums of a constant and of the functions ([0, ∞) 3 x 7→ α((x ∧ b) − (x ∧ a)) = α((x − a)+ ∧ (b − a))) for real 0 ≤ a < b, 0 < α . . . ). This cone contains 1[0,∞] and generates the Borel σ-ﬁeld on [0, ∞].