Chapter IV Application to Finance 1. The Simple Black Scholes Market
B. The basic extension procedure
We wish to establish some property Q(f) for all nonnegative, F-measurable functions f on Ω and it is easy to establish this property for all indicator functions f = 1A of suitably simple sets A forming a family I of generators for the σ-field F. The extension of the propertyQfrom suchf to all nonnegative,measurablef naturally falls into two steps:
(I) ExtendQfrom generatorsA∈ I to all setsA∈ F.
(II) ExtendQfrom indicator functionsf = 1Ato all nonnegative measurable func- tionsf on Ω.
Usually one also wants to extend Qto suitable measurable functions which are not necessarily nonnegative,but this last extension is often accomplished by merely writing such f as a difference f = f+ −f− of the nonnegative functions f+ = f1[f >0],f−=−f1[f <0]. The purpose of this section is to provide a theorem which makes this extension procedure automatic in all cases to which it can be applied.
Consider step (I) and let us writeQ(A) instead ofQ(1A). SinceQ(A) is known to be true for all setsA in a familyI such thatF =σ(I),it would suffice to show that the family of sets
L0={A∈ F |Q(A) is true} (0) is a σ-field,that is,contains the empty set and is closed under complements and arbitrary countable unions. However one can usually only show that L is closed under disjointcountable unions. This leads to the following definition:
A familyL of subsets of Ω is called aλ-system onΩ,if it contains the empty set and is closed under complements and countabledisjointunions,that is,if
(i) ∅ ∈ L,
(ii) A∈ L ⇒ Ac = Ω\A∈ L,
(iii) If (An)⊆ Lis any disjoint sequence,thenA=
nAn ∈ L.
It is clear from our definition that every σ-field on Ω is a λ-system on Ω. Let us now return to extension step (I). We wish to show that L0 ⊇ F and it is usually easy to prove that L0 is aλ-system containing some familyI of generators forF. ConsequentlyL0contains theλ-systemλ(I) generated byI,that is,the smallestλ- system containingI. Thus the question becomes ifλ(I) =F,that is,λ(I) =σ(I).
It turns out (B.2 below) that this is automatically the case,if the familyI is closed under finite intersections. Let us call π-system on Ω any family of subsets of Ω which is closed under finite intersections.
300 B. The basic extension procedure.
B.1. Let Lbe a λ-system on Ω. Then
(a) E, F ∈ L andE⊆F impliesF\E∈ L. (b) (En)⊆ LandEn↑E implies E∈ L.
(c) IfL is also a π-system, thenLis in fact a σ-field.
Proof. (a) IfE⊆F,thenFc andE are disjoint andF\E= (Fc∪E)c.
(b) Assume that En ∈ L,for each n ≥ 1,and En ↑ E,as n ↑ ∞,that is, E1 ⊆ E2 ⊆ E3 ⊆ . . . and E =
nEn. ThenE is the countable disjoint union E =
n≥1Bn,where B1 =E1 and Bn =En\En−1,for all n >1. According to (a) we haveBn ∈ L,for alln≥1. ThusE∈ L.
(c) Assume now thatL is also aπ-system. Then L is closed under finite unions and also under monotone limits (according to (b)) and consequently under arbitrary countable unions.
B.2. Let I be aπ-system onΩ. Thenλ(I) =σ(I).
Proof. LetL be the λ-systemλ(I) generated by I. ThenLcontainsI. Since the σ-field σ(I) generated by I is aλ-system containing I,we haveL ⊆σ(I). To see the reverse inclusion it will suffice to show thatLis aσ-field. According to B.1.(c), it will suffice to show thatLis a π-system. We do this in three steps:
(i) E∈ I andF ∈ I impliesE∩F ∈ L, (ii) E∈ L andF ∈ I impliesE∩F ∈ L,and (iii) E∈ L andF ∈ LimpliesE∩F ∈ L.
(i) IfE, F ∈ I,thenE∩F ∈ I ⊆ L,sinceI is aπ-system by assumption.
(ii) LetL1={E⊆Ω : E∩F∈ L, ∀F∈ I }. We must show thatL1⊇ L=λ(I).
According to (1) we haveI ⊆ L1. Thus it will suffice to show thatL1is aλ-system.
Clearly ∅ ∈ L1,since∅ ∈ L. Assume now that E ∈ L1,that is,E∩F ∈ L, for all F ∈ I. We wish to show that Ec ∈ L1. Let F ∈ I be arbitrary. Then Ec∩F =F\E=F\(E∩F). HereF, E∩F ∈ LandE∩F ⊆F. According to B.1.(a) this implies thatEc∩F =F\(E∩F)∈ L. Since this is true for every set F ∈ I,we haveEc∈ L1.
Finally,let (En)⊆ L1be any disjoint sequence andE=
nEn. We must show thatE∈ L1,that is,E∩F ∈ L,for eachF ∈ I. LetF ∈ I. ThenEn∩F ∈ L,since En ∈ L1,for alln≥1. Consequently (En∩F)n is a disjoint sequence contained in theλ-systemL. ThusE∩F =
n(En∩F)∈ L.
(iii) Let L2 ={F ⊆Ω : E∩F ∈ L, ∀E ∈ L }. We must show thatL2 ⊇ L= λ(I). According to (ii) we have I ⊆ L2. Thus it will suffice to show that L2 is a λ-system. This proof is similar to the proof of (ii) and is omitted.
The following is a convenient reformulation of B.2:
B.3π-λTheorem. LetI be aπ-system onΩandL aλ-system onΩ. IfL contains I, thenLcontains theσ-field generated byI.
Theπ-λ-Theorem handles extension step (I) above. All that is necessary is to find a π-systemIof generators forF such that the truth ofQ(A) can be verified for each set A∈ Iand subsequently to show that the familyL0={A∈ F |Q(A) is true} is a λ-system on Ω. Extension step (II) from indicator functions f = 1A to all nonnegative measurable functions f is then usually straightforward. However it is convenient to have a theorem which handles both extension steps (I) and (II) simultaneously. To this end we introduce the following notion:
A family C of nonnegative F-measurable functions on Ω is called aλ-coneon Ω,if it satisfies the following conditions:
(α) C contains the constant function 1.
(β) Iff, g∈ C are bounded andf ≤g theng−f ∈ C. (γ) Iffn∈ C andαn ≥0,for all n≥1,thenf =
nαnfn ∈ C.
B.4 Extension Theorem. LetCbe a λ-cone onΩ. Assume that1A∈ C, for each set Ain someπ-systemI generating theσ-fieldF. ThenC contains every nonnegative measurable functionf onΩ.
Proof. Let L ={A ∈ F |1A ∈ C }. We claim that L is a λ-system on Ω. From (α) and (β) above it follows that 0∈ C and hence ∅ ∈ L. If A∈ L,then 1A ∈ C and so 1Ac = 1−1A ∈ C (according to (β)),that is,Ac ∈ L. Finally,if (An)⊆ L is any disjoint sequence and A =
nAn,then 1An ∈ C,for each n ≥ 1 and so 1A=
n1An ∈ C,that isA∈ L.
ThusL is aλ-system containingI. SinceI is aπ-system by assumption,the π-λTheorem yields that L ⊇σ(I) =F. Thus 1A ∈ C,for every setA∈ F. From (γ) it now follows thatC contains all nonnegative simple functions on Ω.
Let now f be a nonnegative measurable function on Ω and choose a sequence (fn) of simple functions on Ω such that fn ↑ f pointwise,as n ↑ ∞. Using (γ), fk+1−fk∈ C,for allk≥1,and
f = limnfn= limn
f1+n−1
k=1(fk+1−fk)
=f1+∞
k=1(fk+1−fk)∈ C. Let us illustrate this extension procedure in several examples:
302 B. The basic extension procedure.
B.5 Image measure theorem. Let (Ω,F) be a measurable space,that is,Ω a set and F a σ-field on Ω,and X : (Ω,F, P) → (Ω,F) a measurable map.
Then the image PX of the measure P under X is the measure on F defined by PX(A) = P(X−1(A)), A ∈ F. This measure is also called the distribution of X under P.
B.5.0. (a)EPX(f) =EP(f◦X), for each measurable functionf ≥0 onΩ. (b) If f is an arbitrary measurable function on Ω, then f ∈L1(PX) if and only if f ◦X ∈L1(P)and in this case againEPX(f) =EP(f◦X).
Proof. (a) Let C be the family of all nonnegative measurable functions f on Ω which satisfy EPX(f) = EP(f ◦X). For an indicator function f = 1A, A ∈ F, this equality is satisfied by the definition of the image measure PX. Thus 1A ∈ C, for all sets A ∈ F. Moreover it is easily seen that C is a λ-cone on Ω. Property (α) of aλ-cone is trivial and properties (β) and (γ) follow from the linearity and the σ-additivity of the integral (over nonnegative series). Thus C contains every nonnegative measurable function on Ω.
(b) This follows from (a) by writingf =f+−f−.
B.6 Measurability with respect to σ(X). Let (Ω,F) and X : (Ω,F, P)→ (Ω,F) be as above and letσ(X) be theσ-field generated byX on Ω,that is,σ(X) is the smallestσ-field on Ω with respect to whichX is measurable. It is easily seen that σ(X) ={X−1(A)|A∈ F}.
B.6.0. A functionf : Ω→Ris measurable with respect toσ(X)if and only iff has the formf =g◦X, for some measurable functiong: Ω →R.
Proof. Iff =g◦X withgas above,thenf isσ(X)-measurable since a composition of measurable maps is measurable. Conversely letCbe the family of all functionsf on Ω which can be written in the formf =g◦X,withg: Ω→[0,+∞] measurable.
Thus each functionf ∈ C is nonnegative.
We want to show thatCcontains every nonnegative,σ(X)-measurable function f on Ω. Indeed,ifB is any set inσ(X),thenB=X−1(A) and so 1B = 1A◦X,for some set A∈ F. Thus 1B∈ C. MoreoverC is again easily seen to be a λ-cone on Ω.
Only property (β) of aλ-cone is not completely straightforward: Letf, h∈ C be bounded and assume thatf ≤h. Choose a constantM such that 0≤f ≤h≤M and write f = g◦X and h= k◦X,where h, k : Ω → [0,+∞] are measurable.
Then f = f ∧M = (g∧M)◦X and likewise h = (k∧M)◦X. Thus we may assume that g and kare bounded as well,especially finitely valued. In particular then the difference k−g is defined and we have h−f = (k−g)◦X,where the function k−g is measurable on Ω but is not known to be nonnegative. However h−f ≥ 0 implies that h−f = (k−g)+◦X. Thus h−f ∈ C. Applying B.4 to the probability space (Ω, σ(X), P) with I =σ(X) shows thatC contains every nonnegativeσ(X)-measurable function on Ω.
If f is any σ(X)-measurable function on Ω write f =f+−f− and f+ =h1◦X, f− =h2◦X,for some measurable functionsh1, h2 : Ω → [0,+∞]. Note that it does not follow thatf =h◦X with h=h1−h2,since this difference may not be defined on all of Ω.
The sets [f >0],[f <0] are inσ(X) and so there exist setsA1, A2∈ F such that [f >0] =X−1(A1) and [f <0] =X−1(A2). Then [f >0] =X−1(A1\A2) and [f <0] =X−1(A2\A1) and we may therefore assume that the setsA1 andA2
are disjoint.
Note that 1[f >0] = 1A1◦X and sof+ =f+1[f >0] =g1◦X with g1=h11A1. Likewise f− = g2◦X with g2 =h21A2. The measurable functions g1, g2 : Ω → [0,+∞] satisfy g1g2 = 0 and so the difference g = g1−g2 : Ω → R is defined.
Clearlyf =g◦X.
Remark. Let X = (X1, X2, . . . , Xn) : (Ω,F, P) → Rn be a random vector and σ(X1, . . . , Xn) denote the smallestσ-field on Ω making eachXj measurable. Then X is measurable with respect to any σ-field G on Ω if and only if each component Xj is G-measurable. From this it follows that σ(X) = σ(X1, . . . , Xn). Applying B.6.0 we obtain
B.6.1. A functionf : Ω→Ris σ(X1, X2, . . . , Xn)-measurable if and only if f =g(X1, X2, . . . , Xn)for some measurable functiong:Rn→R.
B.7 Uniqueness of finite measures. LetIbe aπ-system on Ω which generates theσ-field F and contains the set Ω. Then
B.7.0. If the finite measures P, P on F satisfy P(A) =P(A), for all setsA ∈ I, then P=P.
Proof. The family L={A∈ F |P(A) =P(A)} is a λ-system containing I. To see that Lis closed under complements we use Ω∈ I ⊆ L and the finiteness ofP andP. By theπ-λTheoremL ⊇ F. ThusP(A) =P(A),for all setsA∈ F,that is, P=P.
B.8 Fubini’s theorem. Let (Ωj,Fj, Pj), j = 1,2,be probability spaces and (Ω,F, P) the product space (Ω1,F1, P1)×(Ω2,F2, P2). In other words Ω = Ω1×Ω2, F =F1× F2 the productσ-field,that is,the σ-field generated by the measurable rectanglesA1×A2 withAj∈ Fj,j= 1,2,andP =P1×P2 the product measure, that is,the unique probability measure on F satisfyingP(A) =P1(A1)P2(A2),for each measurable rectangleA=A1×A2∈ F.
For a functionf : Ω→Randx∈Ω1,y∈Ω2we define the sectionsfx: Ω2→R andfy: Ω1→R byfx(y) =fy(x) =f(x, y).
304 B. The basic extension procedure.
B.8.0 Fubini’s theorem. Let f : Ω → [0,+∞] be measurable with respect to the productσ-fieldF. Then the sections fx and fy areF2-measurable respectivelyF1- measurable, for each x∈Ω1 respectivelyy∈Ω2 and we have
Ω1
EP2(fx)P1(dx) =EP(f) =
Ω2
EP1(fy)P2(dy). (1) Proof. LetCbe the family of all nonnegative measurable functionsf on the product Ω such that the sectionsfxand fy areF2-measurable respectivelyF1-measurable, for each x∈ Ω1 respectively y ∈ Ω2 and such that equation (1) holds. From the linearity and σ-additivity of the integral,it follows that C is a λ-cone on Ω. We wish to show thatCcontains every nonnegative measurable functionf on Ω. Since the measurable rectangles A = A1×A2, Aj ∈ Fj, j = 1,2,form a π-system of generators for the product σ-field F,it will now suffice to show that C contains every such measurable rectangle A(or rather its indicator function 1A).
Since 1A(x, y) = 1A1(x)1A2(y),the sections (1A)x and (1A)y are given by (1A)x= 1A1(x)1A2 and (1A)y = 1A2(y)1A1,for eachx∈Ω1, y∈Ω2. The measur- ability claim follows immediately and equation (1) reduces to the definition of the product measure P.
B.9 Approximation of sets by generators. Let us now show that the sets in the σ-field F = σ(A) generated by some field of sets A can be approximated by sets inAin the following sense:
B.9.0. Let (Ω,F, P)be a probability space andA ⊆ F a field of sets generating the σ-field F. Then, for each setE ∈ F andC >0, there exists a setA∈ A such that P(A∆E)< C.
Proof. LetL be the family of all sets E ⊆Ω which can be approximated by sets in A as in B.9.0. We wish to show thatL ⊇ F. SinceL contains theπ-systemA generating F,it will suffice to show that Lis a λ-system of subsets of Ω. Indeed, we have∅ ∈ A ⊆ Land the equality Ac∆Ec =A∆E shows that L is closed under complements. It remains to be shown only thatLis closed under countable disjoint unions.
Let (En)n≥1⊆ Lbe a disjoint sequence,E=
nEnandC >0 be arbitrary. As
nP(En) = P(E)≤1 we can chooseN such that
n>NP(En)< C/2. For 1≤ n≤N chooseAn∈ Asuch thatP
An∆En
< C/2n+1and set A=
n≤NAn∈ A. Then,from the inclusion
A∆E⊆
n≤N
An∆En
∪
n>NEn we obtainP(A∆E)≤
n≤NP An∆En
+
n>NP(En)< C. ThusE∈ L. B.10 Independence. LetGbe a sub-σ-field ofF. Recall that an eventB ∈ Fis called independent ofG,ifP(A∩B) =P(A)P(B),for all eventsA∈ G. Likewise a sub-σ-fieldS of F is called independent ofG if each eventB ∈ S is independent of G. Finally a random vectorX is called independent of G if theσ-field σ(X) is independent ofG.
B.10.0. If the event B satisfies P(A∩B) = P(A)P(B), for all events A in some π-system generating the σ-fieldG, thenB is independent ofG.
Proof. Fix B and let I ⊆ G be a π-system with G =σ(I) such that P(A∩B) = P(A)P(B),for all A ∈ I. We have to show that this equality holds for all sets A ∈ G. Let L be the family of all setsA ∈ F such that P(A∩B) =P(A)P(B).
We haveL ⊇ I and want to show thatL ⊇ G. By the π-λTheorem it will suffice to show thatLis a λ-system of subsets of Ω.
Clearly∅ ∈ L. IfA∈ L,thenP(Ac∩B) =P
B\(A∩B)
=P(B)−P(A∩B) = P(B)−P(A)P(B) =P(Ac)P(B). ThusAc∈ L. Finally,let (An)⊆ Lbe a disjoint sequence and A=
nAn. ThenP(A∩B) =P
n(An∩B)
=
nP(An∩B) =
nP(An)P(B) =P(A)P(B). ThusA∈ L.
B.10.1. Let A,B,S be sub-σ-fields ofF and assume thatS is independent ofA. (a) IfB is independent ofσ(A ∪ S)thenS is independent ofσ(A ∪ B).
(b) S is independent ofσ(A ∪ N), whereN is the family ofP-null sets.
Proof. (a) Assume that B is independent ofσ(A ∪ S). The family I ={A∩B | A∈ A, B∈ B }is aπ-system generating the σ-fieldσ(A ∪ B). According to B.10.0 it will now suffice to show thatP(S∩A∩B) =P(S)P(A∩B),for all setsA∈ A andB ∈ B. Indeed,for suchAandB we have
P(S∩A∩B) =P(S∩A)P(B) =P(S)P(A)P(B) =P(S)P(A∩B), where the first equality uses the independence ofB fromσ(A ∪ S),the second the independence ofS fromAand the third the independence ofBfromA.
(b) Let B =σ(N). Then the σ-field B consists of the P-null sets and their com- plements and is therefore independent of every otherσ-field. According to (a),Sis independent ofσ(A ∪ B) =σ(A ∪ N).
C. Positive semideịnite matrices. Let{e1, e2, . . . , en} denote the standard basis of Rn. Elements ofRn are viewed as column vectors and t, C denote the trans- pose of a vector t respectively matrix C. Recall that a real n×n matrix C is calledsymmetricif it satisfiesC=C in which case it admits an orthonormal basis {f1, f2, . . . , fn} ⊆Rn consisting of eigenvectors of C. Let λ1, . . . , λn be the asso- ciated eigenvalues and let the eigenvectorsfj be numbered such thatλ1, . . . λk&= 0 andλk+1=λk+2=. . .=λn= 0.
Given that this is the case,let U be the n×n matrix whose columns are the eigenvectors fj: cj(U) = U ej = fj, j = 1,2, . . . , n. Then U is an orthog- onal matrix,that is,U is invertible and U−1 = U. We claim that U diago- nalizes the matrix C in the sense that U−1CU = diag(λj),where diag(λj) de- notes the diagonal matrix with entriesλ1,. . .,λn down the main diagonal. Indeed, U−1CU ei=U−1Cfi=λiU−1fi =λiei=diag(λj)ei,for alli= 1,2, . . . , n.
Recall that an n×nmatrixC is calledpositive semidef initeif it satisfies (Ct, t) =tCt=n
i,j=1Cijtitj≥0, ∀t= (t1, t2, . . . , tn)∈Rn.
306 D. Kolmogoroff Existence Theorem.
By contrast to the case of complex scalars this does not imply that the matrixCis symmetric,as the example of the matrixC=1
0 2 1
shows.
IfC is symmetric and positive semidefinite,then λj=λjfj2= (λjfj, fj) = (Cfj, fj)≥0 and it follows thatλ1, λ2, . . . λk>0 andλk+1=λk+2=. . .=λn= 0.
It is now easily seen that C can be written as C = QQ,for some n×n matrix Q. IndeedQ=U diag(
λj) yields such a matrixQ: FromU−1CU =diag(λj) it follows that
C=U diag(λj)U−1=U diag(λj)U =
U diag(
λj)
U diag(
λj)
=QQ. Indeed it is even true thatChas a positive squareroot (i.e.,Qabove can be chosen to be symmetric and positive semidefinite). We do not need this. The relation C=QQwill be the key in the proof of the existence of Gaussian random variables with arbitrary parametersm∈Rn and Ca symmetric,positive semidefiniten×n matrix.
Let us note that the matrixQ=U diag(
λj) satisfiesrange(Q) =range(C).
Indeed,using the equality CU = U diag(λj),we have range(C) = range(CU) = range(U diag(λk)) =span{U e1, U e2, . . . , U ek}=range(Q).
D. Kolmogoroff Existence Theorem.
Compact classes and countable additivity. Let E be a set. A family K0 of subsets of E has thef inite intersection property,ifK0∩K1∩. . .∩Kn &=∅,for each finite subfamily{K0, K1, . . . , Kn} ⊆ K0.
Acompact classonEis now a familyKof subsets ofEsuch that
K∈K0K&=∅, for each subfamilyK0⊆ K,which has the finite intersection property,that is,
K0⊆ KandK0∩. . .∩Kn&=∅,
for each finite subfamily{K0, . . . , Kn} ⊆ K0, ⇒
K∈K0K&=∅.
In more familiar terms: If we setS ={Kc =E\K : K∈ K },thenKis a compact class onEif and only if every cover ofEby sets inS has a finite subcover. Thus the family of closed subsets of a compact topological spaceE is always a compact class onE. Similarly the family of all compact subsets of a Hausdorff spaceE is also a compact class on E. Here the Hausdorff property is needed to make all compact sets closed. These compact classes are closed under finite unions.
D.1. Let K be a compact class on the setE. Then (a) Every subfamily ofK is again a compact class on E.
(b) There is a topology onE in which E is compact and such that K is contained in the family of all closed subsets ofE.
(c) There is a compact class K1 on E such that K ⊆ K1 and K1 is closed under finite unions.
(d) The family of all finite unions of sets inK is again a compact class onE.
Proof. (a) is clear. (b) Set S ={Kc |K ∈ K },then every cover of E by sets in K has a finite subcover. By the Alexander Subbasis Theorem E is compact in the topology generated by S as a subbasis onE. Clearly every setK ∈ Kis closed in this topology. (c) LetK1 be the family of all closed subsets ofE in the topology of (c). (d) follows from (c) and (a).
D.2. Let Abe a field of subsets of E,à:A →[0,+∞)a finite, finitely additive set function on AandK ⊆ Aa compact class on E. Ifà is inner regular with respect toK in the sense that
à(A) = sup{à(K)|K∈ K, K⊆A}, for all sets A∈ A, then àis countably additive on A.
Proof. Sinceàis already finitely additive andà(E)<+∞,the countable additivity ofàis implied by the following condition:
(Dn)∞n=1⊆ A, Dn↓ ∅ ⇒ à(Dn)↓0. (0) To verify (0) consider such a sequence (Dn)∞n=1⊆ Aand letC >0 be arbitrary. For eachn≥1 choose a setKn ∈ Ksuch that
Kn⊆Dn and à(Dn\Kn)< C/2n.
As
n≥1Dn=∅it follows that
n≥1Kn=∅. SinceKis a compact class,it follows that
n≤NKn =∅,for some finite numberN. Thenn≥N implies Dn ⊆DN =DN \N
j=1Kj=N
j=1(DN \Kj)⊆N
j=1(Dj\Kj) and so à(Dn)≤N
j=1à(Dj\Kj)<N
j=1C/2j< C. Thusà(Dn)→0,asn↑ ∞. Products. LetE be a compact Hausdorff space,E the Borel σ-field onE and I any index set. The product Ω =EI is then the family of all functions ω:I →E.
We write ω(t) = ωt, t∈I,andω = (ωt)t∈I. Equipped with the product topology Ω is again a compact Hausdorff space by Tychonoff’s Theorem.
For t ∈ I we have the projection (coordinate map) πt : ω ∈ Ω → ω(t)∈ E.
The product σ-field EI on Ω = EI is then defined to be the σ-field σ(πt, t ∈ I) generated by the coordinate mapsπt. It is characterized by the following universal property: a map X from any measurable space into
EI,EI
is measurable if and only ifπt◦X is measurable for eacht∈I.
More generally,for all subsets H ⊆J ⊆I,we have the natural projections πH: Ω =EI →ΩH =EH and πJ H : ΩJ =EJ→ΩH=EH which are measurable with respect to the product σ-fields and satisfy
πH =πJ H ◦πJ, H ⊆J ⊆I.
In this notationπt=πH,whereH ={t}. LetH(I) denote the family of allf inite subsets ofI. For each setH ={t1, t2, . . . , tn} ∈ H(I) we have
πH(ω) = (ωt1, ωt2, . . . , ωtn)∈ΩH =EH.
308 D. Kolmogoroff Existence Theorem.
IfH ∈ H(I) andBH ∈ EH,then the subset Z=π−H1(BH)⊆Ω is called thef inite dimensional cylinder with base BH. This cylinder is said to be represented on the set H ∈ H(I). The cylinderZ =π−H1(BH) also satisfies
Z=πH−1(BH) =π−J1
π−J H1(BH)
=πJ−1(BJ),
where BJ =π−J H1(BH) ∈ EJ. In other words,the cylinder Z = πH−1(BH) can be represented on every setJ ∈ H(I) withJ ⊇H. Thus any two cylindersZ1,Z2can be represented on the same setH ∈ H(I). Since
πH−1(BH)c
=πH−1(BHc ) and π−H1(BH)∩π−H1(CH) =πH−1(BH∩CH) it follows that the family Z = {πH−1(BH) ⊆ Ω | H ∈ H(I) andBH ∈ EH} of finite dimensional cylinders is a field of subsets of Ω. Clearly the finite dimensional cylinders generate the product σ-fieldEI.
If we merely need a π-system of generators for the product σ-field EI we can manage with a far smaller family of sets. A f inite dimensional rectangleis a set Z of the form
Z =
t∈Hπ−t1(Et) =
t∈H
πt∈Et ,
where H ∈ H(I) andEt ∈ E,for all t∈ H. Thus Z is the cylinder based on the rectangle BH =
t∈HEt∈ EH. The finite dimensional rectangles inEI no longer form a field but they are still aπ-system generating the product σ-fieldEI.
Indeed,the setH in the definition ofZ can be enlarged,by settingEt=E for the new elementst,without alteringZ. Thus any two finite dimensional rectangles can be represented on the same set H ∈ H(I) and from this it follows easily that the intersection of any two finite dimensional rectangles is another such rectangle.
Each finite dimensional rectangle is in EI and thus the σ-field G generated by the finite dimensional rectangles satisfies G ⊆ EI. On the other hand each coordinate mapπtisG measurable and this impliesEI ⊆ G. ThusEI =G.
Finite dimensional rectangles are extremely basic events and a σ-field on the product space EI will not be useful unless it contains them all. In this sense the product σ-field EI is the smallest useful σ-field on EI. It has the following desirable property: a probability measure Q onEI is uniquely determined by its values on finite dimensional rectangles in EI. Usually,when such a measure Qis to be constructed to reflect some probabilistic intuition,it is clear what Q has to be on finite dimensional rectangles and this then determinesQon all ofEI. If such uniqueness does not hold,the problem arises which among all possible candidates best reflects the underlying probabilistic intuition.
The product topology on Ω =EI provides us with two moreσ-fields on Ω,the Baireσ-field (theσ-field generated by the continuous (real valued) functions on Ω) and the Borel σ-fieldB(Ω) (theσ-field generated by the open subsets of Ω).
Let us say that a function f = f(ω) on Ω depends only on countably many coordinates of the pointω = (ωt)t∈I ∈Ω if there exists a countable subsetI0⊆I such thatf(ω) =f(˜ω),for allω,ω˜∈Ω withω|I0= ˜ω|I0.
Likewise a subsetA⊆Ω is said to depend only on countably many coordinates if this is true of its indicator function 1A,equivalently,if there exists a countable subsetI0⊆I such thatω∈A⇐⇒ω˜ ∈A,for allω,ω˜ ∈Ω withω|I0 = ˜ω|I0.