Stochastic processes and their spectral 8 0 representations over non-archimedean fields 0 2 n S.V. Ludkovsky a J 8 25.10.2007 ] R P Abstract . h t The article is devoted to stochastic processes with values in finite- a m and infinite-dimensional vector spaces over infinite fields K of zero [ characteristics with non-trivial non-archimedean norms. For different types of stochastic processes controlled by measures with values in 1 v K and in complete topological vector spaces over K stochastic inte- 9 grals are investigated. Vector valued measures and integrals in spaces 0 over K are studied. Theorems about spectral decompositions of non- 2 1 archimedean stochastic processes are proved. . 1 0 1 8 0 : v 1 Introduction i X r Stochastic integrals and spectral representations of stochastic processes are a widely used over the fields of real and complex numbers [7, 16, 17, 18, 39, 40]. If consider stochastic processes in topological groups or metric spaces it gives some generalization, but many specific features of topological vector spaces and results in them may naturally be missed [35, 34, 18, 40]. At the same time non-archimedean analysis is being fast developed in recent years [22, 37, 38, 41, 11]. It has found applications in non-archimedean 1key words and phrases: stochastic processes, non-archimedean field, zero characteris- tic, random process, linear space, stochastic integral, spectral representation Mathematics Subject Classification 2000: 60G50, 60G51,30G06 1 quantum mechanics and quantum field theory [41, 3, 9, 6, 19]. These parts of mathematical physics heavily depend on probability theory [2]. Then it is very natural in the analysis on totally disconnected topological spaces and totally disconnected topological groups [37, 28]. Stochastic processes on such groups also permit to investigate their isometric representations in non-archimedean spaces. Remind that non-archimedean fields K have non-archimedean norms, for example, for the field of p-adic numbers Q , where p > 1 is a prime number p [22, 37, 42]. Multiplicative norms in such fields K satisfy the strong triangle inequality: |x+y| ≤ max(|x|,|y|) for each x,y ∈ K. Besides locally compact fields we consider also non locally compact fields. For example, the algebraic closure of Q can be supplied with the multiplica- p tive non-archimedean norm and its completion relative to this norm gives the field C of complex p-adic numbers. The field C is algebraically closed and p p complete relative to its norm [22]. Its valuation group Γ := {|z| : z ∈ Cp C ,z 6= 0} is isomorphic with the multiplicative group {px : x ∈ Q}. There p exist larger fields U being extensions of Q such that Γ = {px : x ∈ R}. p p Up There are known extensions with the help of the spherical completions also, if an initial field is not such [37, 38, 8, 11]. Stochastic processes on spaces of functions with domains of definition in a non-archimedean linear space and with ranges in the field of real R or complex numbers C were considered in works [4, 12]-[15, 21, 23]. Another types of non-archimedean stochastic processes are possible depending on a domain of definition, a range of values of functions, values of measures in either the real field or a non-archimedean field [24, 30, 32, 43]. Moreover, a time parameter may be real or non-archimedean and so on, that is a lot of problems for investigations arise. Stochastic processes with values in non-archimedean spaces appear while their studies for non-archimedean Banach spaces, totally disconnected topo- logical groups and manifolds [25]-[29]. Very great importance branching pro- cesses in graphs also have [1, 17, 18]. For finite or infinite graphs with finite degrees of vertices there is possible to consider their embeddings into p-adic graphs, which can be embedded into locally compact fields. Considerations of such processes reduce to processes with values in the field Q of p-adic p numbers. Stochastic processes on p-adic graphs also have applications in analysis of flows of information, mathematical psychology and biology [20]. Morespecificfeaturesarise,whenmeasureswithvaluesinnon-archimedean 2 fields are considered, so this article continuous previous works of the author in this area [25, 30, 31, 32]. In this article representations of stochastic processes with values in finite- and infinite-dimensional vector spaces over infinite fields with non-trivial non-archimedean norms are investigated. Below different types of stochastic processes controlled by measures with values in non-archimedean fields of zero characteristic and stochastic integrals are studied. Theorems about spectral decompositions of non-archimedean stochastic processes are proved (see, for example, §§20-29, 75-82, Lemmas 27 and 29, Theorems 20, 79, 81). Moreover, special features ofthe non-archimedean case areelucidated. These features arise from many differences of the classical over R and C analysis and the non-archimedean analysis. General constructions of the paper are illustrated in Examples 9, 31.1, 40, 74, Theorem 41, etc., where applications to totally disconnected topological groups are discussed as well. Some necessary facts from non-archimedean probability theory and non- archimedean analysis are recalled that to make reading easier (see, for exam- ple, §§1-6 in section 2), as well as developed below, when it is essential. The main results of this paper are obtained for the first time. It is necessary to note that in this article measures and stochastic processes with values not only in non-archimedean fields (see section 2), but also with values in topo- logical linear spaces which may be infinite dimensional over non-archimedean fields are studied (see §§44-74 in section 3). Stochastic processes with values in Qn have natural interesting applica- p tions, for which a time parameter may be either real or p-adic. A random trajectory in Qn may be continuous relative to the non-archimedean norm p in Q , but its trajectory in Qn relative to the usual metric induced by the p real metric may be discontinuous. This gives new approach to spasmodic or jump or discontinuous stochastic processes with values in Qn, when the latter is considered as embedded into Rn. 2 Scalar spectral functions To avoid misunderstandings we first present our notations and definitions and recall the basic facts. 1. Definitions. Let G be a completely regular totally disconnected topological space, let also R be its covering ring of subsets in G, S{A : A ∈ 3 R} = G. We call the ring separating, if for each two distinct points x,y ∈ G there exists A ∈ R such that x ∈ A, y ∈/ A. A subfamily S ⊂ R is called shrinking, if an intersection of each two elements from A contains an element from A. If A is a shrinking family, f : R → K, where K = R or K is the field with the non-archimedean norm, then it is written lim f(A) = 0, if A∈A for each ǫ > 0 there exists A ∈ A such that |f(A)| < ǫ for each A ∈ A with 0 A ⊂ A . 0 A measure µ : R → K is a mapping with values in the field K of zero characteristic with the non-archimedean norm satisfying the following prop- erties: (i) µ is additive; (ii) for each A ∈ R the set {µ(B) : B ∈ R,A ⊂ B} is bounded; (iii)ifAistheshrinkingfamilyinRandTA∈AA = ∅,thenlimA∈Aµ(A) = 0. Measures on Bco(G) are called tight measure, where Bco(G) is the ring of clopen (simultaneously open and closed) subsets in G. For each A ∈ R there is defined the norm: kAk := sup{|µ(B)| : B ⊂ µ A,B ∈ R}. For functions f : G → X, where X is a Banach space over K and φ : G → [0,+∞) define the norm kfk := sup{|f(x)|φ(x) : x ∈ G}. φ More generally for a complete locally K-convex space X with a family of non-archimedean semi-norms S = {u} [33] define the family of semi-norms kfk := sup{u(f(x))φ(x) : x ∈ G}. Recall that a subset V in X is called φ,u absolutely K-convex or a K-disc, if VB + VB ⊆ V, where B := {x ∈ K : |x| ≤ 1}. Translates x+V of absolutely K-convex sets are called K-convex, where x ∈ X. A topological vector space over K is called K-convex, if it has a base of K-convex neighborhoods of zero (see 5.202 and 5.203 [33]). A semi- norm u in X is called non-archimedean, if u(x+y) ≤ max[u(x),u(y)]for each x,y ∈ X. A topological vector space X over K is locally K-convex if and only if its topology is generated by a family of non-archimedean semi-norms. Therefore, acompleteK-convexspaceistheprojectivelimitofBanachspaces over K (see 6.204, 6.205 and 12.202 [33]). Put also N (x) := inf{kUk : x ∈ U ∈ R} for each x ∈ G. If a function f µ µ is a finite linear combination over the field K of characteristic functions Ch A of subsets A ⊂ G from R, then it is called simple. A function f : G → X is called µ-integrable, if there exists a sequence f ,f ,... of simple functions 1 2 such that there exists lim kf −f k = 0 for each u ∈ S. n→∞ n Nµ,u The space L(µ,X) = L(G,R,µ,X) of all µ-integrable functions with 4 values in X is K-linear. At the same time RGPnj=1ajChAj(x)µ(dx) := Pnj=1ajµ(Aj) for simple functions extends onto L(µ,X), where aj ∈ X, A ∈ R for each j. j Put R := {A : A ⊂ G,Ch ∈ L(µ,K)}. For A ∈ R let µ¯(A) := µ A µ R χ (x)µ(dx). G A For 1 ≤ q < ∞ denote by kfk := [sup |f(x)|qN (x)]1/q for a simple function f : G → X, when q x∈G µ X is the Banach space, or kfk := [sup u(f(x))qN (x)]1/q for each u ∈ S, when X is the com- q,u x∈G µ plete K-convex space. The completion of the space of all simple functions by k∗k or by {k∗k : u ∈ S} denote by Lq(µ,X), where L(µ,X) = L1(µ,X). q q,u LetGbeatotallydisconnectedcompletelyregularspace,letalsoB (G)be c a covering ring of clopen compact subsets in G, suppose that µ : B (G) → K c is a finitely-additive function such that its restriction µ| for each A ∈ B (G) A c is a measure on a separating covering ring R(G)| , where B (G)| = R| , A c A A R| := {E ∈ R : E ⊆ A}. A A measure η : R → K is called absolutely continuous relative to a measure µ : R → K, if there exists a function f ∈ L(µ,K) such that η(A) = R Ch (x)f(x)µ(dx) for each A ∈ R, denote it by η (cid:22) µ. If η (cid:22) µ G A and µ (cid:22) η, then we say that η and µ are equivalent η ∼ µ. AK-valuedmeasure P onR(X)we callaprobability measure ifkXk =: P kPk = 1 and P(X) = 1 (see [32]). The following statements from the non-archimedean functional analysis proved in [37] are useful. 2. Lemma Let µ be a measure on R. There exists a unique function N : G → [0,∞) such that µ (1) kCh k = kAk ; A Nµ µ (2) if φ : G → [0,∞) and kCh k ≤ kAk for each A ∈ R, then φ ≤ N ; A φ µ µ N (x) = inf kAk for each x ∈ X. µ x∈A,A∈R µ 3. Theorem. Let µ be a measure on R. Then R is a covering ring of µ G and µ¯ is a measure on R that extends µ. µ 4. Lemma. If µ is a measure on R, then N = N and R = R . µ µ¯ µ µ¯ 5. Theorem. Let µ be a measure on R, then N is upper semi- µ continuous and for every A ∈ R and ǫ > 0 the set {x ∈ A : N (x) ≥ ǫ} is µ µ R -compact. µ 6. Theorem. Let µ be a measure on R, let also S be a separating covering ring of G which is a sub-ring of R and let ν be a restriction of µ µ 5 onto S. Then S = R and ν¯ = µ¯. ν µ 7. Notations and definitions. Let (Ω,A,P) - be a probability space, where Ω is a space of elementary events, A is a separating covering ring of events in Ω, R(Ω) ⊆ A ⊆ R (Ω), P : A → K is a probability, K is a P non-archimedean field of zero characteristic, char(K) = 0, complete relative to its multiplicative norm, K ⊃ Q , 1 < p is a prime number, Q is the field p p of p-adic numbers. Denote by ξ a random vector (a random variable for n = 1) with values in Kn or in a linear topological space X over K such that it has the probability distribution P (A) = P({ω ∈ Ω : ξ(ω) ∈ A}) for each A ∈ R(X), where ξ ξ : Ω → X, ξ is (A,R(X))-measurable, where R(X) is a separating covering ringofX suchthatR(X) ⊂ Bco(X),Bco(X)denotestheseparatingcovering ring of all clopen (simultaneously closed and open) subsets in X. That is, ξ−1(R(X)) ⊂ A. If T is a set and ξ(t) is a random vector for each t ∈ T, then ξ(t) is called a random function (or stochastic function). Particularly, if T is a subset in a field, then ξ(t) is called a stochastic process, while t ∈ T is interpreted as the time parameter. As usually put M(ξk) := R ξk(ω)P(dω) for a random variable ξ and Ω k ∈ N whenever it exists. Random vectors ξ and η with values in X are called independent, if P({ξ ∈ A,η ∈ B}) = P({ξ ∈ A})P({η ∈ B}) for each A,B ∈ R(X). 8. Definition. Let {Ω,R,P} be a probability space with a probability measure with values in a non-archimedean field K complete relative to its multiplicative norm, K ⊃ Q . Consider a set G and a ring J of its subsets. p Let ξ(A) = ξ(ω,A), ω ∈ Ω, be a K - valued random variable for each A ∈ J such that (M1) ξ(A) ∈ Y, ξ(∅) = 0, where Y = L2(Ω,R,P,K); (M2) ξ(A ∪ A ) = ξ(A ) + ξ(A ) mod(P) for each A ,A ∈ J with 1 2 1 2 1 2 A ∩A = ∅; 1 2 (M3) M(ξ(A )ξ(A )) = µ(A ∩A ); 1 2 1 2 (M4) M(ξ(A )ξ(A )) = 0 for each A ∩ A = ∅, A ,A ∈ J, that is 1 2 1 2 1 2 ξ(A ) and ξ(A ) are orthogonal random variables, where µ(A) ∈ K for each 1 2 A,A ,A ∈ J. 1 2 The family of random variables {ξ(A) : A ∈ J} satisfying Conditions (M1 − M4) we shall call the elementary orthogonal K-valued stochastic measure. 9. Example. If ξ(A) has a zero mean value Mξ(A) = 0 for each 6 A ∈ J, while ξ(A ) and ξ(A ) are independent random variables for A ,A ∈ 1 2 1 2 J with A ∩ A = ∅, then they are orthogonal, since M(ξ(A )ξ(A )) = 1 2 1 2 (Mξ(A ))(Mξ(A )). 1 2 10. Lemma. The function µ from Definition 8 is additive. Proof. Since ξ(A) ∈ Y = L2(Ω,R,P,K) for each A ∈ J, then there ex- istsMξ(A) = R ξ(ω,A)P(dω),sincesup |ξ(ω,A)|2N2(ω) ≤ sup |ξ(ω,A)|2N (ω) Ω x∈G P x∈G P for the probability measure P having N (x) ≤ 1 for each x ∈ G, that is, p L1(Ω,R,P,K) ⊂ L2(Ω,R,P,K). Therefore, from Conditions (M2,M4) for each A ,A ∈ J with the void 1 2 intersection A ∩A = ∅ the equalities follow: 1 2 M(ξ2(A ∪A )) = M[(ξ(A )+ξ(A ))2] 1 2 1 2 = M[ξ2(A )+2ξ(A )ξ(A )+ξ2(A )] = Mξ2(A )+Mξ2(A ). In view of (M3) 1 1 2 2 1 2 this gives µ(A ∪A ) = µ(A )+µ(A ). 1 2 1 2 11. Note. Suppose that µ has an extension to a measure on the sep- arating covering ring R(G), G is a totally disconnected completely regular space, where J ⊂ R (G). µ 12. Definitions. Letarandomfunctionξ(t)bewithvaluesinacomplete linear locally K-convex space X over K, t ∈ T, where (T,ρ) is a metric space with a metric ρ. Then ξ(t) is called stochastically continuous at a point t , 0 if for each ǫ > 0 there exists lim P({u(ξ(t)−ξ(t )) > ǫ}) = 0 for each ρ(t,t0)→0 0 u ∈ S. If ξ(t) is stochastically continuous at each point of a subset E in T, then it is called stochastically continuous on E. If lim sup P({u(ξ(t)) > R}) = 0 for each u ∈ S, then a random R→∞ t∈E function ξ(t) is called stochastically bounded on E. Let L0(R(G),X) denotes the class of all step (simple) functions f(x) = Pmk=1ckChAk(x), where ck ∈ X, Ak ∈ R(G) for each k = 1,...,m ∈ N, A ∩A = ∅ for each k 6= j. Then the non-archimedean stochastic integral k j by the elementary orthogonal stochastic measure ξ(A) of f ∈ L0(R(G),X) is defined by the formula: (SI) η(ω) := RGf(x)ξ(ω,dx) := Pmk=1ckξ(ω,Ak). 13. Lemma. Let f,g ∈ L0(R(G),K), where f(x) = Pmk=1ckChAk(x) andg(x) = Pmk=1dkChAk(x), then M(RGf(x)ξ(dx)RGg(y)ξ(dy))= Pmk=1ckdkµ(Ak) and there exists a K-linear embedding of L0(R(G),K) into L2(µ,K). Proof. In view of Conditions (M1,M2) there exists R f(x)ξ(ω,dx) ∈ G Y = L(P). Since RGf(x)ξ(dx)RGg(y)ξ(dy) = Pmk,j=1ckdjξAk)ξ(Aj, then 7 M(RGf(x)ξ(dx)RGg(y)ξ(dy))= Pnk,j=1ckdjM(ξ(Ak)ξ(Aj)) = Pmk=1ckdkµ(Ak) due to Conditions (M3,M4), since A ∩ A = ∅ for each j 6= k. This k j gives the K-linear embedding θ of L0(R(G),K) into L2(µ,K) such that θ(f) = Pnk=1ckChAk(x) and (i) kθ(f)k = [maxm |c |2sup N (x)]1/2 = [maxm |c |2kA k ]1/2 < 2 k=1 k x∈Ak µ k=1 k k µ ∞ due to Lemma 2. 14. Note. Denote by L2(R(G),K) the completion of L0(R(G),K) by the norm k∗k induced from L2(µ,K). 2 15. Definition. Let L0(ξ,X) denotes the class of all step (simple) functions f(x) = Pmk=1ckξ(Ak), where ck ∈ X, Ak ∈ R(G) for each k = 1,...,m ∈ N, A ∩ A = ∅ for each k 6= j. Then the non-archimedean k j stochastic integral by the elementary orthogonal stochastic measure ξ(A) of f ∈ L0(ξ,X) is defined by the formula: (SI) η(ω) := RGf(x)ξ(ω,dx) := Pmk=1ckξ(ω,Ak). 16. Lemma. Let f,g ∈ L0(ξ,K), where f(x) = Pmk=1ckξ(Ak) and g(x) = Pmk=1dkξ(Ak), then M(RGf(x)ξ(dx)RGg(y)ξ(dy)) = Pmk=1ckdkµ(Ak) and there exists a K-linear embedding of L0(R(G),K) into L2(P,K). Proof. In view of Conditions (M1,M2) there exists R f(x)ξ(ω,dx) ∈ G Y = L(P,K). But f is the step function, hence (i) kfkL2(P,K) = [maxmk=1|ck|2supω∈Ω|ξ2(ω,Ak)|NP(x)]1/2 = [maxm |c |2kξ(∗,A )k2 ]1/2 < ∞ k=1 k k L2(P) and inevitably f ∈ L2(P). Thus the mapping ψ(f) := Pmk=1ckChAk(x) gives the K-linear embedding of L0(ξ,K) into L2(P,K). The second statement is verified as in Lemma 13 due to Formulas 12,15(SI). 17. Note. Denote by L2(ξ,K) the completion of L0(ξ,K) by the norm k∗k induced from L2(P,K). 2 18. Corollary. The mappings 12(SI) and 15(SI) and Conditions (M1− M4) induce an isometry between L2(R(G),K) and L2(ξ). Proof. The valuation group Γ := {|z| : z ∈ K,z 6= 0} is contained K in (0,∞). In view of Theorem 5, Lemma 10 and Note 11 without loss of generality for a step function f we take a representation with A ∈ R(G) k such that kA k = |µ(A )| for each k = 1,...,m. The family of all such step k µ k functions is everywhere dense in L2(R(G),K). SinceM(ξ2(A)) = µ(A)foreachA ∈ R(G),thenN (x) = inf kAk , µ A∈R(G),x∈A µ where kAk = sup{|µ(B)| : B ∈ R(G),B ⊂ A} = sup{|M(ξ2(B))| : µ B ∈ R(G),B ⊂ A}. On the other hand, M(ξ2(B)) = R ξ2(ω,B)P(dω), Ω 8 |M(ξ2(B))| ≤ sup |ξ2(ω,B)|N (ω). By our supposition µ is the measure, ω∈Ω P hence taking a shrinking family S in R(G) such that T A = {x} we get A∈S N (x) = inf [sup sup |ξ(ω,B)|2N (ω)]. µ A∈R(G),x∈A B∈R(G),B⊂A ω∈Ω P Thus Nµ(x) = infA∈R(G),x∈A[supB∈R(G),B⊂Akξ2(∗,B)kL2(P)] and kAkkµ = kξ(∗,A )k2 for each k = 1,...,m due to Lemma 2 and due to the choice k L2(P) kA k = |µ(A )| above. k µ k The mapping ψ from §16 also is K-linear from L0(ξ) into L0(R(G),K) such that ψ is the isometry relative to k ∗ kL2(P) and k ∗ kL2(µ) due to For- mulas 13(i) and 16(i) and Lemma 2. Two spaces L2(P) and L2(µ) are com- plete by their definitions, consequently, ψ has the K-linear extension from L2(R(G),K) onto L2(ξ) which is the isometry between L2(R(G),K) and L2(ξ). 19. Definition. If f ∈ L2(R(G),K), then put by the definition: η = ψ(f) = R f(x)ξ(dx). G The random variable η we call the non-archimedean stochastic integral of the function f by measure ξ. Taking a limit in L2(P,X) we denote also by l.i.m.. 20. Theorems. 1. For a step function f(x) = Pnk=1akChAk(x), where a ∈ K, A ∈ R(G), n = n(f) ∈ N, the stochastic integral is given by the k k formula: η = R f(x)ξ(dx) = Pnk=1akξ(Ak). 2. For each f,g ∈ L2(R(G),K) there is the identity: M(R f(x)ξ(dx)R g(y)ξ(dy))= R f(x)g(x)µ(dx). G G G 3. For each f,g ∈ L2(R(G),K) and α,β ∈ K the stochastic integral is K-linear: R [αf(x)+βg(x)]ξ(dx) = αR f(x)ξ(dx)+βR g(x)ξ(dx). G G G 4. For each sequence of functions f ∈ L2(G,R(G),µ,K) such that n limn→∞kf −fnkL2(µ,K) = 0 there is exists the limit: R f(x)ξ(dx) = l.i.m. R f (x)ξ(dx). G n→∞ G n 5. There exists an extension of ξ from R onto R (G). µ Proof. Statements of (1) and (3) follow from the consideration above. To finish the proof of (2) it is sufficient to show that fg ∈ L1(µ,K), if f and g ∈ L2(µ,K), where µ is the measure on G. Since 2|f(x)g(x)| ≤ |f(x)|2 + |g(x)|2 for each x ∈ G, then 2sup |f(x)g(x)|N (x) ≤ sup (|f(x)|2 + x∈G µ x∈G |g(x)|2)N (x) ≤ kfk2 +kgk2 , consequently, f(x)g(x) is µ-integrable. µ L2(µ) L2(µ) 4. Fromlim [sup |f(x)−f (x)|2N (x)] = 0andM[R (f−f )(x)ξ(dx)R (f− n→∞ x∈G n µ G n G 9 f )(y)ξ(dy)]= R (f−f )2(x)µ(dx)itfollows,thatlim M[(R (f−f )(x)ξ(dx))2] = n G n n→∞ G n 0, that is l.i.m. R f (x)ξ(dx) = R f(x)ξ(dx) due to Corollary 18. n→∞ G n G ˜ 5. Extend now the stochastic measure ξ from R(G) to ξ on R (G). If µ A ∈ R (G), then Ch ∈ L(G,R(G),µ,K). Since Ch ∈ L(G,R(G),µ,K), µ A A ˜ then sup N (x) < ∞. Put ξ(A) := R Ch (x)ξ(dx) = R ξ(dx) for each x∈A µ G A A A ∈ R (G), consequently, µ ˜ (1) ξ is defined on R (G). µ ˜ Therefore, ξ(A) = ξ(A) for each A ∈ R(G). Foreach A,B ∈ R (G) there µ exist sequences of simple functions fn = Pkak,nChAk,n, gm = Plbl,mChBl,m with a ,b ∈ K, A ,B ∈ R(G) such that lim kCh − f k = k,n l,m k,n l,m n→∞ A n L(µ) ˜ ˜ 0 and lim kCh − g k = 0. Since M(a ξ(A )b ξ(B )) = m→∞ B m L(µ) k,n k,n l,m l,m a b µ(A ∩B ) for each k,n,l,m, then k,n l,m k,n l,m ˜ ˜ (2) M(ξ(A)ξ(B)) = µ¯(A ∩ B) for each A,B ∈ R (G), where µ¯ is the µ extensionofthemeasureµfromR(G)onR (G). IfS ⊂ R (G)isashrinking µ µ family such that T A = ∅, then A∈S (3) l.i.m. ξ˜(A) = 0 due to Corollary 18, since M[ξ˜(A)]2 = µ¯(A) and A∈S lim µ¯(A) = 0 due to Theorem 3. A∈S 21. Definition. A randomfunction of sets satisfying conditions 20.5(1− 3) is called the orthogonal stochastic measure. 22. Corollary. Let ξ and ξ˜ be as in Theorem 20.5, then L2(ξ,K) = L2(ξ˜,K). 23. Note. If ξ is an orthogonal stochastic measure with a structure measureµonR (G)andg ∈ L2(µ,K),thenputρ(A) := R Ch (x)g(x)ξ(dx) µ G A for each A ∈ R (G) and ν(A) := R g2(x)µ(dx). µ A 24. Lemma. Iff ∈ L2(ν,K), thenf(x)g(x) ∈ L2(µ,K)and R f(x)ρ(dx) = G R f(x)g(x)ξ(dx). G Proof. In view of Theorems 20 for each A,B ∈ R (G) there is the µ equality M[ρ(A)ρ(B)] = M[R Ch (x)g(x)ξ(dx)R Ch (y)g(y)ξ(dy) G A G B = R g2(x)µ(dx) = ν(A∩B). A∩B Since g ∈ L2(µ,K), then ν is the measure absolutely continuous relative to µ on Rµ(G). If f(x) = PkakChAk(x) is a simple function with ak ∈ K and Ak ∈ Rµ(G), then RGf(x)ρ(dx) = PkakRGChAk(x)g(x)ξ(dx) = Pkakρ(Ak) = RGf(x)g(x)ξ(dx), since sup |f(x)g(x)|2N (x) ≤ [max |a|2]sup |g(x)|2N (x) < ∞. x∈G µ k k x∈G µ Iff isafundamentalsequenceofsimplefunctionsinL2(ν,K),thenM[(R (f − n G n 10