The non-Archimedean analogs of the Bochner-Kolmogorov, Minlos-Sazonov and Kakutani theorems

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The non-Archimedean analogs of the Bochner-Kolmogorov,Minlos-Sazonov and Kakutani theorems.Sergey V.Ludkovsky.17October 2000Abstract Measures on a non-Archimedean Banach space X are considered with values in the real ?eld R and in the non-Archimedean ?elds.The non-Archimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov theorems are given.Moreover,in?nite products of measures are considered and the analog of the Kakutani theorem is given.1Introduction.There are few works about integration in a classical Banach space,that is over the ?eld R of real numbers or the ?eld C of complex numbers [1,2,3,4,25,26].On the other hand,for a non-Archimedean Banach space X (that is over a non-Archimedean ?eld)this theory is less developed.

An integration in X is a very important part of the non-Archimedean anal-ysis.The progress of quantum mechanics and di?erent branches of modern physics related,for example,with theories of elementary particles lead to the necessity of developing integration theory in a non-Archimedean Banach space [9,27].It may also be useful for the development of non-Archimedean analysis.Non-Archimedean functional analysis develops rapidly in recent

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years and has many principal di?erences from the classical functional analy-sis[10,21,22,23,24,27].Topological vector spaces over non-Archimedean ?elds are totally disconnected,classes of smoothness for functions and com-pact operators are de?ned for them quite di?erently from that of the classical case,also the notion of the orthogonality of vectors has obtained quite an-other meaning.In the non-Archimedean case analogs of the Radon-Nikodym theorem and the Lebesgue theorem about convergence are true under more rigorous and another conditions.Especially strong di?erences are for mea-sures with values in non-Archimedean?elds,because classical notions ofσ-additivity and quasi-invariance have lost their meaning.

On the other hand the development of the non-Archimedean functional analysis and its applications in non-Archimedean quantum mechanics[10,27, 28]leads to the necessity of solving such problems.For example,problems related with quantum mechanics on manifolds are related with di?eomor-phism groups,their representations and measures on them[9,12].In articles [11,12,13,14]quasi-invariant measures on di?eomorphism and loop groups and also on non-Archimedean manifolds were constructed.These measures were used for the investigation of irreducible representations of topological groups[12,14,15].The theorems proved in this work enlarge classes of measures on such groups and manifolds,this also enlarges classes of repre-sentations.For example,theorems of the Minlos-Sazonov type characterize measures with the help of characteristic functionals and compact operators. In the non-Archimedean case compact operators are more useful,than nu-clear operators in the classical case.Theorems of the Bochner-Kolmogorov and Kakutani type characterize products of measures and their absolute con-tinuity relative to others measures.

In this article measures are considered on Banach spaces,though the re-sults given below can be developed for more general topological vector spaces, for example,it is possible to follow the ideas of works[16,17,18],in which were considered non-Archimedean analogs of the Minlos-Sazonov theorems for real-valued measures on topological vector spaces over non-Archimedean ?elds of zero characteristic.But it is impossible to make in one article.In this article,apart from articles of M?a drecki,measures are considered also with

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values in non-Archimedean?elds,for the cases of real-valued measures also

Banach spaces over non-Archimedean?elds K of characteristic char(K)>0

are considered.It is well-known,that a real-valued measure m on a locally

compact Hausdor?totally disconnected Abelian topological group G is called

the Haar measure,if

(H)m(x+A)=m(A)for each x∈G and each Borel subset A in G.

For the s-free group G a measure m with values in a non-Archimedean?eld

K s satisfy condition(H)only for an algebra of clopen(closed and open)

subsets A.Indeed,in the last case if a measure is locally?nite andσ-

additive on the Borel algebra of G,then it is purely atomic with atoms being

singletons,so it can not be invariant relative to the entire Borel algebra(see

Chapters7-9[21]).

In§2weak distributions,characteristic functions of measures and their

properties are de?ned and investigated.In§3the non-Archimedean analogs

of the Minlos-Sazonov and Bochner-Kolmogorov theorems are given.Quasi-

measures also are considered.Frequently de?nitions,formulations of state-

ments and their proofs di?er strongly from that of classical.In§4products

of measures are considered together with their density functions.The non-

Archimedean analog of the Kakutani theorem is investigated.

Notations.Henceforth,K denotes a locally compact in?nite?eld with

a non-trivial norm,then the Banach space X is over K.In the present

article measures on X have values in R or in the?eld K s,that is,a?nite

algebraic extension of the s-adic?eld Q s with the certain prime number s.

We assume that K is s-free as the additive group,for example,either K is

a?nite algebraic extension of Q p or char(K)=p and K is isomorphic with

a?eld F p(θ)consisting of elements x= j a jθj,where a j∈F p,|θ|=p?1, F p is a?nite?eld of p elements,p is a prime number and p=s.These

imply that K has the Haar measures with values in R and K s[21].If X is

a Hausdor?topological space with a small inductive dimension ind(X)=0,

then

E denotes an algebra of subsets of X,as a rule E?Bco(X)for K s-valued

measures and E?Bf(X)for real-valued measures,where

Bco(X)denotes an algebra of clopen(closed and open)subsets of X,

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Bf(X)is a Borelσ-?eld of X in§2.1;

Af(X,μ)is the completion of E by a measureμin§2.1;

M(X)is a space of norm-bounded measures on X in§2.1;

M t(X)is a space of Radon norm-bounded measures in§2.1;

c0(α,K)is a Banach space and P L is a projector(?xed relative to a chosen basis)in§2.2;

μL is a projection of a measureμin§2.2;

{μL(n):n}is a sequence of weak distributions in§2.2;

B(X,x,r)is a ball in§2.2;

L(X,μ,K s)in§2.4;

χξis a character with values either in T or T s in§2.6;

θ(z)=?μis a characteristic functional in§2.6;

δ0is the Dirac measure in§2.8;

μ1?μ2is a convolution of measures in§2.11;

ψq,μandτq in§2.14;

C(X,K)is a space of continuous functions from X into K in§2.16;

X?is the topological dual space of X[20];

?C(Y,Γ),τ(Y)in§3.2;

B+,C+in§3.5;

ν?μ,ν～μ,ν⊥μin§4.1.

2Weak distributions and families of measures.

2.1.For a Hausdor?topological space X with a small inductive dimension ind(X)=0[5]the Borelσ-?eld is denoted Bf(X).Henceforth,measuresμare given on a measurable space(X,E).The completion of Bf(X)relative toμis denoted by Af(X,μ).The total variation ofμwith values in R on a subset A is denoted by μ|A or|μ|(A)for A∈Af(X,μ).Ifμis non-negative andμ(X)=1,then it is called a probability measure.

We recall that a mappingμ:E→K s for an algebra E of subsets of X is called a measure,if the following conditions are accomplished:

(i)μis additive andμ(?)=0,

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(ii)for each A∈E there exists the following norm

:B?A,B∈E}<∞,

A μ:=sup{|μ(B)|K

s

(iii)if there is a shrinking family F,that is,for each

A,B∈F there exist F?C?(A∩B)and∩{A:A∈F}=?,then lim A∈Fμ(A)=0(see chapter7[21]and also about the completion Af(X,μ) of the algebra E by the measureμ).A measure with values in K s is called a probability measure if X μ=1andμ(X)=1.For functions f:X→K s andφ:X→[0,∞)there are used notations f φ:=sup x∈X(|f(x)|φ(x)), Nμ(x):=inf( U μ:U∈Bco(X),x∈X),where Bco(X)is an algebra of closed and at the same time open(clopen)subsets in X.Tight measures (that is,measures de?ned on E?Bco(X))compose the Banach space M(X) with a norm μ := X μ.Everywhere below there are consideredσ-additive measures with|μ|(X)<∞and X μ<∞forμwith values in R and K s respectively,if it is not speci?ed another.

A measureμon E is called Radon,if for each?>0there exists a compact subset C?X such that μ|(X\C)

2.2.Each Banach space X over K in view of Theorems5.13and5.16[21] is isomorphic with c0(α,K):={x:x=(x j:j∈α),card(j:|x j|K>b)0},whereαis an ordinal,card(A)denotes the cardinality of A, x :=sup(|x j|:j∈α).A dimension of X over K is by the de?nition dim K X:=card(α).For each closed K-linear subspace L in X there exists an operator of a projection P L:X→L.Moreover,an orthonormal in the non-Archimedean sense basis in L has a completion to an orthonormal basis in X such that P L can be de?ned in accordance with a chosen basis.

If A∈Bf(X),then P?1L(A)is called a cylinder subset in X with a base A, B L:=P?1L(Bf(L)),B0:=∪(B L:L?X,L is a Banach subspace,dim K X

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that cl(∪[L(n):n])=X,dim K L(n)=κn for each n,where cl(A)=ˉA denotes a closure of A in X for A?X.We?x a family of projections

P L(m) L(n):L(m)→L(n)such that P L(m)

L(n)

P L(n)

L(k)

=P L(m)

L(k)

for each m≥n≥k.A

projection of the measureμonto L denoted byμL(A):=μ(P?1L(A))for each A∈Bf(L)compose the consistent family:

(1)μL(n)(A)=μL(m)(P?1

L(n)

(A)∩L(m))

for each m≥n,since there are projectors P L(m)

L(n)

,whereκn≤?0and there may be chosenκn

An arbitrary family of measures{μL(n):n∈N}having property(1)is called a sequence of a weak distributions(see also[4,25]).

By B(X,x,r)we denote a ball{y:y∈X, x?y ≤r},which is clopen (closed and open)in X.

2.3.Lemma.A sequence of a weak distributions{μL(n):n}is gen-erated by some measureμon Bf(X)if and only if for each c>0there exists b>0such that||μL(n)|(B(X,0,r)∩L(n))?|μL(n)|(L(n))|≤c and sup n|μL(n)|(L(n))<∞forμwith values in R;

or L(n)\B(X,0,r) μ

L(n)≤c and sup n L(n) μ

L(n)

<∞forμwith

values in K s,where r≥b.

Proof.In the case ofμwith values in R we can use a Hahn decomposition μ=μ+?μ?and substitute everywhere in the proof of Lemma1§2[25]a Hilbert space over R onto X over K,since X is a Radon space in view of Theorem1.2§I.1.3[4],then|μ|(A)=μ+(A)+μ?(A)for A∈Bf(X).

Forμwith values in K s the necessity is evident.To prove the su?ciency it remains only to verify property(2.1.iii),since then X μ=sup n L(n) μ

L(n)

<

∞.Let B(n)∈E(L(n)),A(n)=P?1

L(n)

(B(n)),by Theorem7.6[21]for each c>0there is a compact subset C(n)?B(n)such that B(n)\C(n) μ

L(n)

< c,where B(n)\D(n) μ≤max( B(m)\C(m) μ

L(m)

:m=1,...,n)

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rem(4.8[21])sets A(n)and B(X,0,r)are weakly compact in X,hence, for each r>0there exists n with B(X,0,r)∩A(n)=?.Therefore,

A(n) μ= B(n) μ

L(n)≤ L(n)\B(X,0,r) μ

L(n)

≤c and there exists

lim n→∞μ(A(n))=0,since c is arbitrary.

2.4.De?nition and notations.A functionφ:X→R(or K s)of the formφ(x)=φS(P S x)is called a cylinder function ifφS is a Bf(S)-measurable (or E(S)-measurable respectively)function on a?nite-dimensional over K space S in X.ForφS∈L1(S,μ,R)forμwith values in R orφS∈L(S,μS,K s):=L(μS)forμwith values in K s we may de?ne an integral by a sequence of weak distributions{μS(n)}:

X

φ(x)μ?(dx):= φS(n)(x)μS(n)(dx),

where L(μ)is the Banach space of classes ofμ-integrable functions(f=gμ-almost everywhere,that is, A μ=0,A:={x:f(x)=g(x)}isμ-negligible)

with the following norm f := g N

μ

[1,21,25].

2.5.Lemma.A subset A?X=c0(ω0,K)is relatively compact if and only if A is bounded and for each c>0there exists a?nite-dimensional over K subspace L?X such thatˉA?L c:={y∈X:d(y,L):=inf{ x?y : x∈L}≤c}.

Proof.If A is bounded and for each c>0there exists L c withˉA?L c, then there is a sequence{k(j):j∈N}?Z such that lim j→∞k(j)=∞,ˉA?{x∈X:|x j|≤p?k(j),j=1,2,...}=:S,but X is Lindel¨o f, S is sequentially compact,henceˉA is compact(see§3.10.31[5]).IfˉA is compact,then for each c>0there exists a?nite number m such that ˉA? m j=1B(X,x j,c),where x j∈X.Therefore,ˉA?L c for L=sp K(x j: j=1,...,m):=(x= m j=1b j x j:b j∈K).

2.6.Remarks and de?nitions.As an additive group K is isomorphic with Q n p with n∈N:={1,2,...}.The topologically adjoint space over Q p (that is,of continuous linear functionals f:K→Q p)is isomorphic with Q n p [8].For x and z∈Q n p we denote by z(x)the following sum n j=1x j z j,where x=(x j:j=1,...,n),x j∈Q p.Each number y∈Q p has a decomposition y= l a l p l,where min(l:a l=0)=:ord p(y)>?∞(ord(0):=∞)[20], a l∈(0,1,...,p?1),we de?ne a symbol{y}p:= l<0a l p l for|y|p>1and

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{y}p=0for|y|p≤1.

For a locally compact?eld K with a characteristic char(K)=p>0 letπj(x):=a j for each x= j a jθj∈K(see Notation).All continuous charactersχ:K→C(orχ:K→C s)have the formχ=χξ(x)= exp{2πiη(ξ(x))},whereπj:K→R,η(x):={x}p andξ∈Q n p?=Q n p for char(K)=0,η(x):=π0(x)/p andξ∈K?=K for char(K)=p>0, x∈K,i=(?1)(1/2)(see§25[8]),exp:C→C.Eachχis locally constant, henceχ:K→T(orχ:K→T s)is also continuous,where T denotes the discrete group of all roots of1(by multiplication),T s denotes its subgroup of elements with orders that are not degrees s m of s,m∈N.

For a measureμwith values in R or K s there exists a characteristic functional(that is,called the Fourier-Stieltjes transformation)θ=θμ: C(X,K)→C or C s:

(2)θ(f):= Xχe(f(x))μ(dx),

where e=(1,...,1),x∈X,f is in the space C(X,K)of continuous functions from X into K,in particular for z=f in the topologically conjugated space X?over K,z:X→K,z∈X?,θ(z)=:?μ(z).It has the folowing properties:

(3a)θ(0)=1forμ(X)=1

andθ(f)is bounded on C(X,K);

|θ(f)|=1for probability measures;

(3b)sup

f

(4)θ(z)is weakly continuous,that is,(X?,σ(X?,X))-continuous,

σ(X?,X)denotes a weak topology on X?,induced by the Banach space X over K.To each x∈X there corresponds a continuous linear functional x?:X?→K,x?(z):=z(x),moreover,θ(f)is uniformly continuous relative to the norm on

|f(x)|K<∞};

C b(X,K):={f∈C(X,K): f :=sup

x∈X

(5)θ(z)is positive de?nite on X?and on C(X,K)

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forμwith values in[0,∞).

Property(4)follows from Lemma2.3,boundedness and continuity ofχe and the fact that due to the Hahn-Banach theorem there is x z∈X with z(x z)=1for z=0such that z|(X?L)=0and

θ(z)= Xχe(P L(x))μ(dx)= Lχe(y)μL(dy),

where L=Kx z,also due to the Lebesgue theorem2.4.9[6]for real measures (or from Exer.7.F[21]forμwith values in K s,see also§4.2[26]).Indeed, for each c>0there exists a compact subset S?X such that|μ|(X\S)

Property(5)is accomplished,since

N

θ(f l?f j)αlˉαj= X|N j=1αjχe(f j(x))|2μ(dx)≥0,

l,j=1

particularly,for f j=z j∈X,whereˉαj is a complex conjugated number to αj.

We call a functionalθ?nite-dimensionally concentrated,if there exists L?X,dim K L0and δ>0in view of Theorem I.1.2[4](or Theorem7.6[21])and Lemma2.5there exists a?nite-dimensional over K subspace L and compact S?Lδsuch that X\S μ

This de?nition is correct,since L?X,X has the isometrical em-bedding into X?as the normed space associated with the?xed basis of X,such that functionals z∈X separate points in X.If z∈L,then |θ(z)?θL(z)|≤c×b×q,where b= X μ,q is independent of c and b.Each characteristic functionalθL(z)is uniformly continuous by z∈L relative to the norm ? on L,since|θL(z)?θL(y)|≤| S′∩L[χe(z(x))?χe(y(x))]μL(dx)| +| L\S′[χe(z(x))?χe(y(x))]μL(dx)|,where the second term does not exceed

L

an uniformly equicontinuous by x∈S′family relative to z∈B(L,0,1).

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Therefore,

θn(z)

(6)θ(z)=lim

n→∞

for each?nite-dimensional over K subspace L,whereθn(z)is uniformly equicontinuous and?nite-dimensionally concentrated on L(n)?X,z∈X, cl( n L(n))=X,L(n)?L(n+1)for every n,for each c>0there are n and q>0such that|θ(z)?θj(z)|≤cbq for z∈L(j)and j>n,q=const>0 is independent of j,c and b.Let{e j:j∈N}be the standard orthonormal basis in X,e j=(0,...,0,1,0,...)with1in j-th 788c851cb7360b4c2e3f6479ing countable addi-tivity ofμ,local constantness ofχe,considering all z=be j and b∈K,we get thatθ(z)on X is non-trivial,whilstμis a non-zero measure,since due to Lemma2.3μis characterized uniquely by{μL(n)}.Indeed,forμwith values in R a measureμV on V,dim K V

F(g)(z):=lim

r→∞ B(V,0,r)χe(z(x))g(x)m(dx),

z∈V,g∈L(V,μV,C s),m is the Haar measure on V either with values in R or K s respectively.Therefore,the mappingμ→θμis injective.

2.7.Proposition.Let X=K j,j∈N,

(a)μandνbe real probability measures on X,supposeνis symmetric. Then X?μ(x)ν(dx)= X?ν(x)μ(dx)∈R and for each0

μ([x∈X:?ν(x)≤l])≤ X(1??μ(x))ν(dx)/(1?l).

(b).For each real probability measureμon X there exists r>p3such that for each R>r and t>0the following inequality is accomplished:μ([x∈X: x ≥tR])≤c X[1??μ(yξ)]ν(dy),

whereν(dx)=C×exp(?|x|2)m(dx),m is the Haar measure on X with values in[0,∞),m(B(X,0,1))=1,ν(X)=1,2>c=const≥1is independent on t,c=c(r)is non-increasing whilst r is increasing,C>0.

Proof.(a).Recall thatνis symmetric,ifν(B)=ν(?B)for each B∈Bf(X).Therefore, Xχe(z(x))ν(dx)= Xχe(?z(x))ν(dx),that is equivalent

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to X sin(2π{z(x)}p)ν(dx)=0or?ν(z)∈R.If0

(b).Letν(dx)=γ(x)m(dx),whereγ(x)=C×exp(?|x|2),C>0,ν(X)=1.Then F(γ)(z)=:?γ(z)≥0,and?γ(0)=1andγis the continuous positive de?nite function withγ(z)→0whilst|z|→∞.In view of(a):μ([x: x ≥tR])≤ X[1??μ(yξ)]ν(dy)/(1?l),where|ξ|=1/t,t>0, l=l(R).Estimating integrals,we get(b).

2.8.Lemma.Let in the notation of Proposition2.7νξ(dx)=γξ(x)m(dx),γξ(x)=C(ξ)exp(?|xξ|2),νξ(X)=1,ξ=0,then a measureνξis weakly con-verging to the Dirac measureδ0with the support in0∈X for|ξ|→∞. Proof.We have:C(ξ)?1=C q(ξ)?1= l∈Z[p lq?p(l?1)q]exp(?p2l|ξ|2)<∞,where the sum by l<0does not exceed1,q=jn,j=dim K X, K.Here K is considered as the Banach space Q n p with the n=dim Q

p

following norm|?|p equivalent to|?|K,for x=(x1,...,x j)∈X with x l∈K as usually|x|p=max1≤l≤j|x l|p,for y=(y1,...,y n)∈K with

.Further,p l+s x l=0exp(2πi ?s?1i=l x i p i+s) y l∈Q p:|y|p:=max1≤l≤n|y l|Q

p

= 1p l+s exp(2πiφ)dφ+β(s),where s+l<0,lim s→?∞(β(s)p?s?l)=0,there-fore,sup[|?γ1(z)|R|z|X:z∈X,|z|≥p3]≤2.Then taking0=ξ∈K and carrying out the substitution of variable for continuous and bounded func-tions f:X→R we get lim|ξ|→∞ X f(x)νξ(dx)=f(0).This means thatνξis weakly converging toδ0for|ξ|→∞.

2.9.Theorem.Letμ1andμ2be measures in M(X)such that?μ1(f)=?μ2(f)for each f∈Γ.Thenμ1=μ2,where X=c0(α,K),α≤ω0,Γis a vector subspace in a space of continuous functions f:X→K separating points in X.

Proof.Let at?rstα<ω0,then due to continuity of the convolution γξ?μj byξ,and Proposition4.5§I.4[26]and Lemma2.8we getμ1=μ2, since the familyΓgenerates Bf(X).Now letα=ω0,A={x∈X: (f1(x),...,f n(x))∈S},νj be an image of a measureμj for a mapping x→(f1(x),...,f n(x)),where either S∈Bf(K n)or S∈E(K n),f j∈X?→X?. Then?ν1(y)=?μ1(y1f1+...+y n f n)=?μ2(y1f1+...+y n f n)=?ν2(y)for each y=(y1,...,y n)∈K n,consequently,ν1=ν2on E.Further we can use the

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Prohorov theorem3.4§1.3[26],since compositions of f∈Γwith continuous functions g:K→R or g:K→K s respectively generate a family of real-valued or K s-valued functions correspondingly separating points of X.

2.10.Proposition.Letμl andμbe measures in M(X l)and M(X) respectively,where X l=c0(αl,K),αl≤ω0,X= n1X l,n∈N.Then the condition?μ(z1,...,z n)= n l=1?μl(z l)for each(z1,...,z n)∈X?→X?is equivalent toμ= n l=1μl.

Proof.Letμ= n l=1μl,then?μ(z1,...,z n)= Xχe( z l(x l)) n l=1μl(dx l) = n l=1 X lχe(z l(x l))μl(dx l).The reverse statement follows from Theorem2.9.

2.11.Proposition.Let X be a Banach space over K;supposeμ,μ1 andμ2are probability measures on X.Then the following conditions are equivalent:μis the convolution of two measuresμj,μ=μ1?μ2,and?μ(z)=?μ1(z)?μ2(z)for each z∈X.

Proof.Letμ=μ1?μ2.This means by the de?nition thatμis the image of the measureμ1?μ2for the mapping(x1,x2)→x1+x2,x j∈X,consequently,?μ(z)= X×Xχe(z(x1+x2))(μ1?μ2)(d(x1,x2))= 2l=1 Xχe(z(x l))μl(dx l) =?μ1(z)?μ2(z).On the other hand,if?μ1?μ2=μ,then?μ=(μ1?μ2)∧and due to Theorem2.9above for real measures,or Theorem9.20[21]for measures with values in K s,we haveμ=μ1?μ2.

2.12.Corollary.Letνbe a probability measure on Bf(X)andμ?ν=μfor eachμwith values in the same?eld,thenν=δ0.

Proof.If z0∈X?→X?and?μ(z0)=0,then from?μ(z0)?ν(z0)=?μ(z0)it follows that?ν0(z0)=1.From the property2.6(6)we get that there exists m∈N with?μ(z)=0for each z with z =p?m,since?μ(0)=1.Then

?ν(z+z0)=1,that is,?ν|(B(X,z

0,p?m))=1.Sinceμare arbitrary we get

?ν|X=1,that is,ν=δ0due to§2.6and§2.9for K s-valued measures and real-valued measures.

2.1

3.Corollary.Let X and Y be a Banach space over K,(a)μand νbe probability measures on X and Y respectively,suppose T:X→Y is a continuous linear operator.A measureνis an image ofμfor T if and only if ?ν=?μ?T?,where T?:Y?→X?is an adjoint operator.(b).A characteristic functional of a real measureμon Bf(X)is real if and only ifμis symmetric.

Proof follows from§2.6and§2.9.

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2.14.De?nition.We say that a real probability measureμon Bf(X)

for a Banach space X over K and0

ψq,μ(z)= X|z(x)|qμ(dx)<∞for each z∈X?.The weakest vector topology in X?relative to which all(ψq,μ:μ)are continuous is denoted byτq.

2.15.Theorem.A characteristic functional?μof a real probability

Radon measureμon Bf(X)is continuous in the topologyτq for each q>0.

Proof.For each c>0there exists a compact S?X such thatμ(S)>

1?c/4and

|1??μ(z)|≤| S(1?χe(z(x)))μ(dx)|+| X\S(1?χe(z(x)))μ(dx)|≤|1??μc(z)|+c/2, whereμc(A)=(μ(A∩S)/μ(S)and A∈Bf(X);further analogously to the proof of IV.2.3[26].

2.16.Proposition.For a completely regular space X with ind(X)=0

the following statements are accomplished:

(a)if(μβ)is a bounded net of measures in M(X)that weakly converges

to a measureμin M(X),then(?μβ(f))converges to?μ(f)for each continuous

f:X→K;if X is separable and metrizable then(?μβ)converges to?μ

uniformly on subsets that are uniformly equicontinuous in C(X,K);

(b)if M is a bounded dense family in a ball of the space M(X)for mea-

sures in M(X),then a family(?μ:μ∈M)is equicontinuous on a locally

K-convex space C(X,K)in a topology of uniform convergence on compact

subsets S?X.

Proof.(a).Functions exp(2πiη({f(x)}))are continuous and bounded on

X,where?μ(f)= Xχe(f(x))μ(dx).Then(a)follows from the de?nition of the weak convergence and Proposition1.3.9[26],since sp C{exp(2πi{f(x)}p):

{exp(2πiη(f(x)):f∈C(X,K}

f∈C(X,K)}is dense in C(X,C)and sp C

s

is dense in C(X,C s).

(b).For each c>0there exists a compact subset S?X such that

|μ|(S)>|μ(X)|?c/4for real-valued measures or μ|(X\S)

valued measures.Therefore,forμ∈M and f∈C(X,K)with|f(x)|K<

c<1for x∈S we get|μ(X)?Re(?μ(f)|=2| X sin2(πη(f(x)))μ(dx)|

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analogously to the proof of Proposition IV.3.1[26],since X is the T1-space and for each point x and each closed subset S in X with x/∈S there is a continuous function h:X→B(K,0,1)such that h(x)=0and h(S)={1}.

2.17.Theorem.Let X be a Banach space over K,η:Γ→C be

a continuous positive de?nite function,(μβ)be a bounded weakly relatively compact net in the space M t(X)of Radon norm-bounded measures and there exists limβ?μβ(f)=γ(f)for each f∈Γand uniformly on compact subsets of the completion?Γ,whereΓ?C(X,K)is a vector subspace separating points in X.Then(μβ)weakly converges toμ∈M t(X)with?μ|Γ=γ.

Proof is analogous to the proof of Theorem IV.3.1[26]and follows from Theorem2.9above and for K s-valued measures using the non-Archimedean Lebesgue convergence theorem(see Ch.7[21]).

2.18.Theorem.(a).A bounded family of measures in M(K n)is weakly relatively compact if and only if a family(?μ:μ∈M)is equicontinuous on K n.

(b).If(μj:j∈N)is a bounded sequence of measures in M t(K n),γ: K n→C is a continuous(and in addition positive de?nite for real-valuedμj) function,?μj(y)→γ(y)for each y∈K n(and uniformly on compact subsets in K n for K s-valued measures),then(μj)weakly converges to a measureμwith?μ=γ.

(c).A bounded sequence of measures(μj)in M t(K n)weakly convereges to a measureμin M t(K n)if and only if for each y∈K n there exists lim j→∞?μj(y)=?μ(y).

(d).If a bounded net(μβ)in M t(K n)converges uniformly on each bounded subset in K n,then(μβ)converges weakly to a measureμin M t(K n),where n∈N.

Proof.(a).This follows from the Prohorov theorem1.3.6[26]and Propo-sitions2.7,2.16.

(b).We have the following inequality:lim m sup j>mμj([x∈K n:|x|≥tR])≤2 K n(1?Re(η(ξy)))ν(dy)with|ξ|=1/t due to§2.7and§2.8for real-valued measures.Due to the non-Archimedean Fourier transform and the Lebesgue convergence theorem[21]for K s-valued measures and from the condition lim R→∞sup|y|>R|γ(y)|R n=0it follows,that for each?>0there

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exists R0>0such that lim m sup j>m μj|{x∈K n:|x|>R} ≤2sup|y|>R|γ(y)|RR0.In view of Theorem2.17(μj)converges weakly toμwith ?μ=γ.(c,d).These may be proved analogously to IV.3.2[26].

2.19.Corollary.If(?μβ)→1uniformly on some neighbourhood of0in K n for a bounded net of measuresμβin M t(K n),then(μβ)converges weakly toδ0.

2.20.De?nition.A family of probability measures M?M t(X)for

a Banach space X over K is called planely concentrated if for each c>0 there exists a K-linear subspace S?X with dim K S=n1?c.The Banach space M t(X)is supplied with the following norm μ :=|μ|(X).

2.21.Lemma.Let S and X be the same as in§2.20;z1,...,z m∈X?be a separating family of points in S.Then a set E:=S c∩(x∈X:|z j(x)|≤r j; j=1,...,m)is bounded for each c>0and r1,...,r m∈(0,∞).

Proof.A space S is isomorphic with K n,consequently,p(x)=max(|z j|: j=1,...,m)is a norm in S equivalent to the initial norm.

2.22.Theorem.Let X be a Banach space over K with a familyΓ?X separating points in M?M t(X).Then M is weakly relatively compact if and only if a family{μz:μ∈M}is weakly relatively compact for each z∈Γand M is planely concentrated,whereμz is an image measure on K of a measure μinduced by z.

Proof follows from Lemmas2.5,2.21and the Prohorov theorem(see also Theorem1.3.7[26]with a substitution[?r j,r j]onto B(K,0,r j)).

2.2

3.Theorem.For X andΓthe same as in Theorem2.22a sequence {μj:j∈N}?M t(X)is weakly convergent toμ∈M t(X)if and only if for each z∈Γthere exists lim j→∞?μj(z)=?μ(z)and a family{μj}is planely concentrated.

Proof follows from Theorems2.17,18,22(see also Theorem IV.3.3[26]).

2.24.Proposition.Let X be a weakly regular space with ind(X)=0,Γ?C(X,K)be a vector subspace separating points in X,(μn:n∈N)?M t(X),μ∈M t(X),lim n→∞?μn(f)=?μ(f)for each f∈Γ.Then(μn)is weakly convergent toμrelative to the weakest topologyσ(X,Γ)in X relative to which all f∈Γare continuous.

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Proof follows from Theorem2.18and is analogous to the proof of Propo-sition IV.3.3[26].

3The non-Archimedean analogs of the Minlos-Sazonov and Bochner-Kolmogorov theorems.

3.1.Let(X,U)= λ(Xλ,Uλ)be a product of measurable completely regular Radon spaces(Xλ,Uλ)=(Xλ,Uλ,Kλ),where Kλare compact classes approx-imating from below each measureμλon(Xλ,Uλ),that is,for each c>0and elements A of an algebra Uλthere is S∈Kλ,S?A with A\S μ

λ

Theorem.Each bounded quasi-measureμwith values in K s on(X,U) (that is,μ|U

λ

is a bounded measure for eachλ)is extendible to a measure on an algebra Af(X,μ)?U,where an algebra U is generated by a family(Uλ:

λ∈Λ).

Proof.We have2.1(i)by the condition and X μ<∞,if2.1(iii)is satis?ed.It remains to prove2.1(iii).For each sequence(A n)?U with n A n=?and each c>0for each j∈N we choose K j∈K,where the compact class K is generated by(Kλ)(see Proposition1.1.8[4]),such that K j?A j and A j\K j μ

3.2.De?nition.Let X be a Banach space over K,then a mapping f:X→C is called pseudocontinuous,if its restriction f|L is uniformly continuous for each subspace L?X with dim K L

of mappings f:Y→K of a set Y into a?eld K.We denote by?C(Y,Γ)the minimalσ-algebra(that is called cylinder)generated by an algebra C(Y,Γ)

of subsets of the form C f

1,...,f n;E :={x∈X:(f1(x),...,f n(x))∈S},where

S∈Bf(K n),f j∈Γ.We supply Y with a topologyτ(Y)which is generated

by a base(C f

1,...,f n;E :f j∈Γ,E is open in K n).

3.3.Theorem.Non-Archimedean analog of the Bochner-Kolmogorov theorem.Let X be a Banach space over K,X a be its algebraically adjoint

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K-linear space(that is,of all linear mappings f:X→K not necessarily continuous).A mappingθ:X a→C is a characteristic functional of a probability measureμwith values in R[or K s]and is de?ned on?C(X a,X)

[or C(X a,X)]if and only ifθsatis?es conditions2.6(3,5)for(X a,τ(X a))

and is pseudocontinuous on X a[orθsatis?es2.6(3,6)for(X a,τ(X a)and

is pseudocontinuous on X a respectively].

Proof.(I).For dim K X=card(α)

Kα,hence the statement of theorem for a measureμwith values in K s follows

from Theorem9.20[21]and Theorems2.9and2.18above,sinceθ(0)=1and

|θ(z)|≤1for each z.

(II).We consider now the case ofμwith values in R andα<ω0.

In§2.6(see also§2.16-18,24)it was proved thatθ=?μhas the desired properties for real probability measuresμ.On the other hand,there is

θwhich satis?es the conditions of the theorem.Letθξ(y)=θ(y)hξ(y),

where hξ(y)=F[C(ξ)exp(? xξ 2)](y)(that is,the Fourier transform by

x),νξ(Kα)=1,νξ(dx)=C(ξ)exp(? xξ 2)m(dx)(see Lemma2.8),ξ=

0.Thenθξ(y)is positive de?nite and is uniformly continuous as a prod-

uct of two such functions.Moreover,θξ(y)∈L1(Kα,m,C).Forξ=0

a function fξ(x)= Kαθξ(y)χe(x(y))m(dy)is bounded and continuous,a

function exp(? xξ 2)=:s(x)is positive de?nite.Sinceνξis symmet-

ric and weakly converges toδ0,hence there exists r>0such that for each|ξ|>r we have?γξ(y)= KαC(ξ)exp(? xξ 2p)exp(2πiηy(x)))m(dx)

= cos(2πη(y(x)))exp(? xη 2p)C(ξ)m(dx)/2>1?1/R for|y|≤R,con-sequently,?γξ(y)=?ζ2ξ(y)for|y|≤R,where?ζξis positive de?nite uni-

formly continuous and has a uniformly continuous extension on Kα.There-

fore,for each c>0there exists r>0such that νξ?κξ?κξ

for each|ξ|>r,whereκξ(dx)=ζξ(x)m(dx)is aσ-additive non-negative measure.Hence due to corollary from Proposition IV.1.3[26]there exists

r>0such that Kαθξ(y)χe(?x(y))νj(dy)≥0for each|j|>r,conse-quently,fξ(x)=lim|j|→∞ Kαθξ(y)χe(?x(y))νj(dy)≥0.From the equality

F[F(γξ)(?y)](x)=γξ(x)and the Fubini theorem it follows that fξχe(y(x))h j(x)m(dx) = θξ(u+y)νj(du).For y=0we get lim|ξ|→∞ fξ(x)m(dx)= f(x)m(dx)

=lim|ξ|→∞lim|j|→∞ fξ(x)h j(x)m(dx)and lim|ξ|lim|j|| Kαθξ(u)νj(du)|≤1.

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From Lemma2.8it follows that?f(y)=θ(y),since by Theorem2.18θ= lim|ξ|→∞θξis a characteristic function of a probability measure on Bf(Kα), where f(x)= Kαθ(y)χe(?x(y))m(dy).

(III).Now letα=ω0.It remains to show that the conditions imposed on θare su?cient,because their necessity follows from the modi?cation of2.6 (since X has an algebraic embedding into X a).The space X a is isomorphic with KΛwhich is the space of all K-valued functions de?ned on the Hamel basisΛin X.The Hamel basis exists due to the Kuratowski-Zorn lemma (that is,each?nite system of vectors inΛis linearly independent over K, each vector in X is a?nite linear combination over K of elements fromΛ). Let J be a family of all non-void subsets inΛ.For each A∈J there exists a functionalθA:K A→C such thatθA(t)=θ( y∈A t(y)y)for t∈K A. From the conditions imposed onθit follows thatθA(0)=1,θA is uniformly continuous and bounded on K A,moreover,it is positive de?nite(or due to 2.6(6)for each c>0there are n and q>0such that for each j>n and z∈K A the following inequality is satis?ed:

(7)|θA(z)?θj(z)|≤cbq,

moreover,L(j)?K A,q is independent on j,c and b.From(I,II)it follows that on Bf(K A)there exists a probability measureμA such that?μA=θA. The family of measures{μA:A∈J}is consistent and bounded,since μA=μE?(P A E)?1,if A?E,where P A E:K E→K A are the natural projectors.Indeed,in the case of measures with values in R eachμA is the probability measure.For measures with values in K s this is accomplished due to conditions(7),2.6(6)for X a and due to Theorem9.20[21].

In view of Theorem1.1.4[4](or Theorem3.1above)on a cylinderσ-algebra of the space KΛthere exists the unique measureμsuch thatμA=μ?(P A)?1for each A∈J,where P A:KΛ→K A are the natural projectors. From X a=KΛit follows thatμis de?ned on?C(X a,X)(or on C(X a,X) for K s-valued measures).Forμon?C(X a,X)or C(X a,X)there exists its extension on Af(X,μ)such that Af(X,μ)?Bco(X)(see§2.1).

3.4.De?nition.[23]A continuous linear operator T:X→Y for Banach spaces X and Y over K is called compact,if T(B(X,0,1))=:S

18

is a compactoid,that is,for each neighbourhood U ?0in Y there exists a ?nite subset A ?Y such that S ?U +co (A ),where co (A )is the least K -absolutely convex subset in V containing A (that is,for each a and b ∈K with |a |≤1,|b |≤1and for each x,y ∈V the following inclusion ax +by ∈V is accomplished).

3.5.Let B +be a subset of non-negative functions which are Bf (X )-measurable and let C +be its subset of non-negative cylinder functions.By ?B +we denote a family of functions f ∈B +such that f (x )=lim n g n (x ),g n ∈C +,g n ≥f .For f ∈?B +let X f (x )μ?(dx )=inf g ≥f,g ∈C + X g (x )μ?(dx ).For f ∈L (X,μ,K s )and K s -valued measure μlet X f (x )μ?(dx )=lim n →∞ X g n (x )μ?(dx )for norm-bounded sequence of cylinder functions g n from L (X,μ,K s )converging to f uniformly on compact subsets of X .Due to the Lebesgue converging theorem this limit exists and does not depend on a choice of {g n :n }.

3.6.Lemma.A sequence of a weak distributions (μL (n ))of probability Radon measures is generated by a real probability neasure μon Bf (X )of a Banach space X over K if and only if there exists

(8)lim

|ξ|→∞ X G ξ(x )μ?(dx )=1,

where X G ξ(x )μ?(dx ):=S ξ({μL (n ):n })and S ξ({μL (n )}):=lim n →∞ L (n )F n (γξ,n )(x )μL (n )(dx ),γξ,n (y ):= m (n )l =1γξ(y l ),F n is a Fourier transformation by (y 1,...,y n ),y =(y j :j ∈N ),y j ∈K ,γξ(y l )are the same as in Lemma 2.8for K 1;here m (n )=dim K L (n )

Proof.If a sequence of weak ditributions is generated by a measure μ,then in view of 2.6(3-6),Lemmas 2.3,2.5,2.8,Propositions 2.10and 2.16,Corollary 2.13,the Lebesgue convergence theorem and the Fubini theorem,also from the proof of Theorem 3.3and the Radon property of μit follows that there exists r >0such that X G ξ(x )μ?(dx )= X G ξ(x )μ(dx )=lim n →∞ L (n )γξ,n (y )?μL (n )(y )m L (n )(dy ),since lim j →∞x j =0for each x =(x j :j )∈X .In addition,lim |ξ|→∞S ξ({μL (n )})= X μ(dx )=1.Indeed,for each c >0and d >0there exists a compact

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V c?X with μ|(X\V

c)

n>n0.Therefore,choosing suitable sequences of c(n),d(n),V c(n)and L(j n) we get that[ L(n)γξ,n(y)?μL(n)(y)m L(n)(dy):n∈N]is a Cauchy sequence, where m L(n)is the real Haar measure on L(n),the latter is considered as

Q m(n)b

p ,b=dim Q

p

K,m(B(L(n),0,1)=1.Here we use Gξ(x)for a formal

expression of the limit Sξas the integral.Then Gξ(x)(modμ)is de?ned evidently as a function forμor{μL(n):n}with a compact support,also forμwith a support in a?nite-dimensional subspace L over K in X.By the de?nition supp(μL(n):n)is compact,if there is a compact V?X with supp(μL(n))?P L(n)V for each n.That is,condition(8)is necessary.

On the other hand,if(8)is satis?ed,then for each c>0there exists r>0such that| X Gξ(x)μ?(dx)?1|

?

?1|r,consequently, there exists n0such that for each n>n0the following inequality is satis?ed: |1? X F n(γξ,n)(x)μ?(dx)|≤| μ|(L(n)∩B(X,0,R)) ?1|+

sup

|x|>R

|F n(γξ,n)(x)| μL(n)|(L(n)\B(X,0,R)) .

Therefore,from lim R→∞sup|x|>R|F n(γξ,n)(x)|=0and from Lemma2.3the statement of Lemma3.6follows.

3.7.Notes and de?nitions.Suppose X is a locally convex space over

a locally compact?eld K with non-trivial non-Archimedean valuation and X?is a topologically adjoint space.The minimumσ-algebra with respect to which the following family{v?:v?∈X?}is measurable is called aσ-algebra of cylinder sets.For a K s-valued measureμon X a completion of a linear space of characteristic functions{ch U:U∈Bco(X)}in L(X,μ,K s) is denoted by Bμ(X).Then X is called a RS-space(or KS-space)if on X?there exists a topologyτsuch that the continuity of each positive de?nite function f:X?→C(or f:X?→C s with f C0<∞respectively) is necessary and su?cient for f to be a characteristic functional of a non-negative measure(a tight measure of?nite norm correspondingly).Such topology is called the R-Sazonov(or K-Sazonov)type topology.The class of RS-spaces(and KS-spaces)contains all separable locally convex spaces

20

over K.For example,l∞(α,K)=c0(α,K)?,whereαis an ordinal[21].In particular we also write c0(K):=c0(ω0,K)and l∞(K):=l∞(ω0,K),where

ω0is the?rst countable ordinal.

Let n K(l∞,c0)denotes the weakest topology on l∞for which all func-tionals p x(y):=sup n|x n y n|are continuous,where x= n x n e n∈c0and

y= n y n e?n∈l∞,e n is the standard base in c0.Such topology n K(l∞,c0)

is called the normal topology.The induced topology on c0is denoted by

n K(c0,c0).

3.8.Theorem.Let f:l∞(K)→C(or f:l∞(K)→C s)be a functional

such that

(i)f is positive de?nite(or f(0)=1and f C0≤1),

(ii)f is continuous in the normal topology n K(l∞,c0),then f is the char-acteristic functional of a probability measure on c0(K).

Proof.Ifνis the Haar measure on K n,then on Bco(K n)it takes values

in Q.Therefore,Lemma4.1[18]is transferable onto the case of K s-valued measures,since Q?K s.Therefore,analogously to Equation(4.1)of Lemma

4.2[18]we have

(i)P{|V1|K

B of subsets of a set?,f W is a characteristic function of W=(V1,...,V n).

To continue the proof we need the following statements.

3.9.Lemma Let f:c0(K)→C s be a function satisfying the following

two conditions:

(i)|f(x)|≤1for each x∈c0(K),

(ii)f is continuous at zero in the topology n K(c0,c0),

then for each?>0there existsλ(?)∈c0(K)such that|1?f(x)|

for each x∈c0(K).

Proof.In view of continuity for each?>0there exists y(?)∈c0such

that|1?f(x)|

that|πK|=p?1.If x∈c0is such that pλ(?)(x)

?+pλ(?)(x).If pλ(?)(x)≥p,then|1?f(x)|≤2≤p

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