# Integration over an Infinite-Dimensional Banach Space and Probabilistic Applications

Authors: Claudio Asci
Journal: International Journal of Analysis
 (2014)

---

## Abstract

We study, for some subsetsI of N*, the Banach space E of bounded real sequences {xn}n∈I. For any integer k, we introduce a measure over (E,B(E)) that generalizes the k-dimensional Lebesgue measure; consequently, also a theory of integration is defined. The main result of our paper is a change of variables' formula for the integration.


---

## Body

## 1. Introduction

In the mathematical literature, some articles introduced infinite-dimensional measures analogous to the Lebesgue one (see, e.g., the paper of Léandre [1], in the context of the noncommutative geometry, that one of Tsilevich et al. [2], which studies a family of σ-finite measures on R+, and that one of Baker [3], which defines a measure on RN* that is not σ-finite).The motivation of this paper follows from the natural extension to the infinite-dimensional case of the results of the article [4], where we estimate the rate of convergence of some Markov chains in [0,p)k to a uniform random vector. In order to consider the analogue random elements in [0,p)N*, it is necessary to overcome some difficulties, for example, the lack of a change of variables’ formula for the integration in the subsets of RN*. A related problem is studied in the paper of Accardi et al. [5], where the authors describe the transformations of generalized measures on locally convex spaces under smooth transformations of these spaces.In our paper, we consider some subsetsI of N*, and we suppose that RI is endowed with the standard infinity-norm generalized to assume the values in [0,+∞]; then, the vector space E of the elements of RI with finite norm is a Banach space with respect to the distance defined by the norm. Observe that although in general it is possible to construct a σ-algebra on RI simply by considering the product indexed by I of the same Borel σ-algebra on R, in this way a product of σ-finite measures μ on R can be defined only if I is finite or μ is a probability measure (by Jessen theorem).To solve this problem and others, in Section2 we use Corollary 4 (that generalizes the Jessen theorem) to define a measure λN,a(k) over (E,B(E)), where k∈N; consequently, we define also a theory of integration. In the case I={1,…,k}, the measure λN,a(k) coincides with the k-dimensional Lebesgue measure on Rk.In Section3, we introduce the determinant of a class of infinite-dimensional matrices, called (m,σ)-standard, and we expose briefly a theory that generalizes the standard theory of the m×m matrices. Moreover, we prove that the determinant of a (m,σ)-standard matrix is equal to the product of its eigenvalues, as in the finite-dimensional case. In Section 4, we present the main result of our paper, that is, a change of variables formula for the integration of the biunique linear functions associated with the (m,σ)-standard matrices (Theorem 29). This result agrees with the analogous finite-dimensional result. In Section 5, we expose an application in the probabilistic framework, that is, the definition of the infinite-dimensional probability density of a random element. Moreover, we prove the formula of the density of such a random element composed with a (m,σ)-standard matrix. In Section 6, we expose some ideas for further study in the mathematical analysis and probability.


## 2. Construction of a Generalized Lebesgue Measure

Suppose thatk∈N, N∈R+, and I={n∈N*:n<M}, where k+1≤M≤+∞ and a=(an:n∈I)∈(R+)I such that there exists ∏n∈Ian∈R+. Moreover, indicate by B, by B(k), by Leb, and by Leb(k), respectively, the Borel σ-algebra on R, the Borel σ-algebra on Rk, the Lebesgue measure on R, and the Lebesgue measure on Rk. Finally, for any topological space E and for any D⊂E, indicate by B(D) the Borel σ-algebra on D.Definition 1.
Define the function∥·∥:RI→[0,+∞] by
(1)∥x∥=sup⁡n∈I|xn|,∀x=(xn:n∈I)∈RI,
and define the following vector space on the field R:
(2)E={x∈RI:∥x∥<+∞}.Remark 2.
E is a Banach space.Proof.
It is easy to prove that∥·∥ is a norm on E; then, E is a metric space with the distance d:E×E→[0,+∞) defined by d(x,y)=∥x-y∥=sup⁡n∈I⁡|xn-yn|, ∀x=(xn:n∈I)∈E, and ∀y=(yn:n∈I)∈E. Moreover, let {xi}i∈N be a Cauchy sequence on E; then, ∀ε>0, ∃i0∈N such that ∀i,j∈N such that i,j≥i0, we have ∥xi-xj∥<ε, and so, ∀n∈I, |(xi)n-(xj)n|<ε. Since R is complete, ∀n∈I, ∃ln∈R such that lim⁡j→+∞⁡(xj)n=ln; then, by setting l=(ln:n∈I)∈RI, we have
(3)|(xi)n-ln|=lim⁡j→+∞|(xi)n-(xj)n|≤ε⟹∥xi-l∥=sup⁡n∈I|(xi)n-ln|≤ε.
This implies that l∈E and lim⁡i→+∞⁡xi=l; then, E is complete, and so it is a Banach space.In order to develop the next arguments, for any setI and for any H⊂I define the projection πH on RH as the function πH:RI→RH given by πH(xn:n∈I)=(xh:h∈H). We will use the following result, whose proof can be found, for example, in Rao [6, page 346].Theorem 3 (Jessen theorem).
LetI be a set and, for any i∈I, let (Ei,Ei,μi) be a probability space. Then, over the measurable space (∏i∈IEi,⨂i∈IEi), there is a unique probability measure μ, indicated by ⨂i∈Iμi, such that, for any H⊂I such that |H|<+∞ and for any A=∏h∈HAh×∏i∈I∖HEi∈⨂i∈IEi, where Ah∈Eh, ∀h∈H, we have μ(A)=∏h∈Hμh(Ah). In particular, if I is countable, then μ(A)=∏i∈Iμi(Ai) for any A=∏i∈IAi∈⨂i∈IEi.Corollary 4.
LetI be a set and, for any i∈I, let (Ei,Ei,μi) be a measure space such that μi is finite. Moreover, suppose that, for some countable set J⊂I, μi is a probability measure for any i∈I∖J and ∏j∈Jμj(Ej)∈R+. Then, over the measurable space (∏i∈IEi,⨂i∈IEi), there is a unique finite measure μ, indicated by ⨂i∈Iμi, such that, for any H⊂I such that |H|<+∞ and for any A=∏h∈HAh×∏i∈I∖HEi∈⨂i∈IEi, where Ah∈Eh, ∀h∈H, one has μ(A)=∏h∈Hμh(Ah)∏j∈J∖Hμj(Ej). In particular, if I is countable, then μ(A)=∏i∈Iμi(Ai) for any A=∏i∈IAi∈⨂i∈IEi.Proof.
For anyi∈I, μi¯=(μi/μi(Ei)) is a probability measure; then, if μ¯=⨂i∈Iμi¯ is the probability measure defined by Theorem 3, the finite measure μ=(∏j∈Jμj(Ej))μ¯ satisfies the statement.Since for anyn∈I∖{1,…,k} the measure (1/2N)Leb(·∩[-Nan,Nan]) is a finite measure over (R,B), from Corollary 4 we can define the σ-finite measure λN,a(k) over (E,B(E)) in the following manner:
(4)λN,a(k)=1(2N)kLeb(k)⊗⨂n∈I∖{1,…,k}12NLeb(·∩[-Nan,Nan]).Remark 5.
For anyN∈R+, we have
(5)λN,a(k)(E)={∏n∈Ianifk=0+∞ifk∈N*.Proof.
IfN∈R+ and k=0, from Corollary 4, we have
(6)λN,a(k)(E)=∏‍n∈I12NLeb([-Nan,Nan])=∏‍n∈Ian.
Analogously, if N∈R+ and k∈N*:
(7)λN,a(k)(E)=1(2N)kLeb(k)(Rk)∏‍n∈I∖{1,…,k}an=+∞.


## 3. Infinite-Dimensional Matrices

Definition 6.
LetA=(aij)i,j∈I be a real matrix I×I  (eventually infinite,  if I=N*); then, define the linear function A=(aij)i,j∈I:E→RI, and write x→Ax, in the following manner:
(8)(Ax)i=∑j∈Iaijxj,∀x∈E,∀i∈I,
on condition that, for any i∈I, the sum in (8) converges to a real number.Proposition 7.
LetA=(aij)i,j∈I be a real matrix I×I; then(1)
the linear functionA=(aij)i,j∈I:E→RIgiven by (8) is defined if and only if, for anyi∈I,∑j∈I|aij|<+∞;(2)
A(E)⊂EandAis continuous if and only ifsup⁡i∈I⁡∑j∈I|aij|<+∞; moreover, in this case,∥A∥=sup⁡i∈I⁡∑j∈I|aij|.Proof.
(
1
) Suppose that the function A=(aij)i,j∈I:E→RI is defined; then, ∀i∈I; let x=(xn:n∈I)∈E be such that xn=1 if ain≥0, and xn=-1 if ain<0; since Ax∈RI, we have
(9)∑‍j∈I|aij|=(Ax)i∈R.
Conversely, suppose that ∑j∈I|aij|<+∞, ∀i∈I; then, ∀x∈E and ∀i∈I, ∑j∈I(aijxj)+≤∑j∈I|aij||xj|≤∑j∈I|aij|∥x∥<+∞; analogously, ∑j∈I(aijxj)-<+∞, from which (Ax)i=∑j∈I(aijxj)+-∑j∈I(aijxj)-∈R, and so Ax∈RI.
(
2
) If A(E)⊂E and A is continuous, from the previous arguments, we have that, ∀i∈I, there exists x∈E such that ∥x∥=1 and such that
(10)∑‍j∈I|aij|=(Ax)i≤∥Ax∥≤∥A∥<+∞⟹sup⁡i∈I∑‍j∈I|aij|≤∥A∥<+∞.
Conversely, if sup⁡i∈I⁡∑j∈I|aij|<+∞, ∀x∈E, such that ∥x∥=1, we have
(11)∥Ax∥=sup⁡i∈I|(Ax)i|=sup⁡i∈I|∑‍j∈Iaijxj|≤sup⁡i∈I∑‍j∈I|aij|<+∞⟹∥A∥=sup⁡x∈E:∥x∥=1∥Ax∥≤sup⁡i∈I∑‍j∈I|aij|<+∞.
Finally, if sup⁡i∈I⁡∑j∈I|aij|<+∞, from (10) and (11) we have
(12)∥A∥=sup⁡i∈I∑‍j∈I|aij|.Definition 8.
A linear functionA=(aij)i,j∈I:E→E is called (m,σ)-standard, where m∈I∪{0} and σ:I∖{1,…,m}→I∖{1,…,m} is an increasing function, if(1)
aij=0, ∀(i,j)∉({1,…,m}×I)∪⋃n∈I∖{1,…,m}{(n,σ(n))};(2)
there exists∏n∈I∖{1,…,m}:λn≠0λn∈R*, where λn=an,σ(n), ∀n∈I∖{1,…,m}.Moreover, indicate byAm the matrix (aij)i,j∈{1,…,m}∈Mm(R). Finally, indicate by M(m,σ)(E) the set of the linear (m,σ)-standard functions from E to E.Remark 9.
LetA=(aij)i,j∈I:E→E be a linear (m,σ)-standard function. Then, A is continuous; moreover, σ is biunique if and only if σ(n)=n, ∀n∈I∖{1,…,m}.Proof.
From the point 1 of Definition8,
(13)sup⁡i∈I∑‍j∈I|aij|=sup⁡{sup⁡i∈{1,…,m}∑‍j∈I|aij|,sup⁡n∈I∖{1,…,m}:λn≠0|λn|}.
We havesup⁡i∈{1,…,m}⁡∑j∈I|aij|<+∞ from Proposition 7; moreover, if λn=0 for n sufficiently large, obviously sup⁡n∈I∖{1,…,m}:λn≠0|λn|<+∞; otherwise, consider the subsequence {λnk}k∈N={λn≠0:n∈I∖{1,…,m}}; from the point 2 of Definition 8, we obtain lim⁡k→+∞⁡λnk=1, and so sup⁡n∈I∖{1,…,m}:λn≠0⁡|λn|<+∞ again. Then, sup⁡i∈I⁡∑j∈I|aij|<+∞, from which A is continuous from Proposition 7. Moreover, σ is biunique if and only if σ(n)=n, ∀n∈I∖{1,…,m}, because σ is increasing.Proposition 10.
LetA=(aij)i,j∈I:E→E be a linear (m,σ)-standard function; then, A is biunique if and only if the matrix Am is invertible, an,σ(n)≠0, ∀n∈I∖{1,…,m}, and σ is biunique.Proof.
IfAm is invertible and an,σ(n)≠0, ∀n∈I∖{1,…,m}, let x, y∈E be such that Ax=Ay; from the point 1 of Definition 8, ∀n∈I∖{1,…,m}, we have an,σ(n)xσ(n)=an,σ(n)yσ(n), from which xσ(n)=yσ(n); then, if σ is biunique, we have σ(n)=n, and so (xn:n>m)=(yn:n>m). This implies that Am(tx1,…,xm)=Am(ty1,…,ym), and so (x1,…,xm)=(y1,…,ym); then, x=y; that is, A is injective. Moreover, ∀y∈E, define x∈E in the following manner:
(14)xn=ynann,∀n∈I∖{1,…,m},t(x1,…,xm)=Am-1(t(z1,…,zm)),
where
(15)zi=yi-∑‍n>mainxn,∀i∈{1,…,m}.
It is easy to prove that Ax=y; that is, A is surjective.
Conversely, ifA is biunique, let x, y∈Rm be such that Amx=Amy, and let x¯, y¯∈E be such that x¯n=xn, y¯n=yn, ∀n∈{1,…,m}, and x¯n=y¯n=0, ∀n∈I∖{1,…,m}. We have Amx=π{1,…,m}(Ax¯), Amy=π{1,…,m}(Ay¯), and (Ax¯)n=(Ay¯)n=0, ∀n∈I∖{1,…,m}, from which Ax¯=Ay¯; then, since A is biunique, we have x¯=y¯, and so x=y. Then, the linear function x→Amx is injective; that is, Am is invertible. Moreover, we have an,σ(n)≠0, ∀n∈I∖{1,…,m}; in fact, by supposing by contradiction that an¯,σ(n¯)=0, for some n¯>m, then A(E)⊂{x∈E:xn¯=0}E, and this should contradict the fact that A is surjective. Moreover, σ must be injective; in fact, by supposing that σ(n1)=σ(n2), for some m<n1<n2, then A(E)⊂{x∈E:xn1an2,σ(n2)=xn2an1,σ(n1)}E (a contradiction). Finally, σ must be surjective, because otherwise, ∀y∈E and ∀n¯∈(I∖{1,…,m})∖σ(I∖{1,…,m}), we could choose arbitrarily xn¯∈R in order to determine x=(xn:n∈I)∈E such that Ax=y. Then, A should not be injective (again a contradiction).In order to study the inverse ofA, we must define the following concept, that generalizes the determinant of a m×m matrix (see, e.g., the theory in Lang's book [7]).Definition 11.
LetA=(aij)i,j∈I:E→E be a linear (m,σ)-standard function; define the determinant of A, and call it det⁡(m,σ)A, or det⁡A, the real number:
(16)det(m,σ)A={det⁡Am∏‍n∈I∖{1,…,m}λnifσisbiunique0ifσisnotbiunique.Remark 12.
IfA∈M(m1,σ1)(E)∩M(m2,σ2)(E), then det⁡(m1,σ1)A=det⁡(m2,σ2)A.Proof.
Suppose thatm1≤m2; then, we have σ1|I∖{1,…,m2}=σ2. If σ1 is biunique, σ2 is biunique too, and σ1(n)=n, ∀n∈{m1+1,…,m2}; then
(17)det(m1,σ1)A=det⁡Am1∏‍n∈I∖{1,…,m1}λn=det⁡Am1∏‍p∈{m1+1,…,m2}λp∏‍n∈I∖{1,…,m2}λn=det⁡Am2∏‍n∈I∖{1,…,m2}λn=det(m2,σ2)A.
Instead, ifσ1 is not biunique, then either σ2 is not biunique, or σ2 is biunique, but not σ1|{m1+1,...,m2}. In the first case, we have
(18)det(m1,σ1)A=0=det(m2,σ2)A.
In the second case, we have det⁡Am2=0, and so
(19)det(m1,σ1)A=0=det⁡Am2∏‍n∈I∖{1,…,m2}λn=det(m2,σ2)A.Proposition 13.
LetA=(aij)i,j∈I:E→E be a linear (m,σ)-standard function, with σ being biunique, let s,t∈I, s<t, let p=max⁡⁡{t,m}, and let the function τ=σ|I∖{1,…,p}; then(1)
if there existu=(un:n∈I)∈E,v=(vn:n∈I)∈E, andc1,c2∈Rsuch that∑n∈I|un|<+∞,∑n∈I|vn|<+∞,atj=c1uj+c2vj,∀j∈I, by indicating byUandVthe linear functions obtained by substituting thetth row ofAforuandv, respectively, thenUandVare(p,τ)-standard anddet⁡A=c1det⁡U+c2det⁡V;(2)
ifB=(bij)i,j∈I:E→Eis the linear function obtained by exchanging thesth row ofAfor thetth row ofA, thenBis(p,τ)-standard anddet⁡B=-det⁡A;(3)
ifC=(cij)i,j∈I:E→Eis the linear function obtained by substituting thetth row ofAfor thesth row ofA, thenCis(p,τ)-standard anddet⁡C=0.Proof.
(
1
) Since σ is biunique, we have σ(n)=n, ∀n∈I∖{1,…,m}, and so we can prove easily that U and V are (p,τ)-standard; moreover, det⁡A=det⁡Ap∏n∈I∖{1,…,p}λn and det⁡Ap=c1det⁡Up+c2det⁡Vp; then
(20)det⁡A=(c1det⁡Up+c2det⁡Vp)∏‍n∈I∖{1,…,p}λn=c1det⁡Up∏‍n∈I∖{1,…,p}λn+c2det⁡Vp∏‍n∈I∖{1,…,p}λn=c1det⁡U+c2det⁡V.(2) As we observed in the proof of the point 1, B is (p,τ)-standard; moreover, det⁡B=det⁡Bp∏n∈I∖{1,...,p}λn, where Bp is the matrix obtained by exchanging the sth row of Ap for the tth row of Ap; then, det⁡Bp=-det⁡Ap, from which
(21)det⁡B=-det⁡Ap∏‍n∈I∖{1,…,p}λn=-det⁡A.(3) Since the sth row of C and the tth row of C are equal, by exchanging these rows among themselves we obtain again the matrix C; then, from the point 2, we have det⁡C=-det⁡C, from which det⁡C=0.Remark 14.
LetA=(aij)i,j∈I:E→E be a linear (m,σ)-standard function; then, A is biunique if and only if det⁡A≠0.Proof.
IfA is biunique, from Proposition 10  σ is biunique, and so det⁡A=det⁡Am∏n∈I∖{1,…,m}λn; moreover, we have det⁡Am≠0 and λn≠0, ∀n∈I∖{1,…,m}, from which ∏n∈I∖{1,…,m}λn=∏n∈I∖{1,…,m}:λn≠0λn≠0; then, det⁡A≠0.
Conversely, ifdet⁡A≠0, then σ is biunique by definition of det⁡A, and so 0≠det⁡A=det⁡Am∏n∈I∖{1,…,m}λn; this implies that det⁡Am≠0 and λn≠0, ∀n∈I∖{1,…,m}; then, from Proposition 10, A is biunique.Definition 15.
LetA=(aij)i,j∈I:E→E be a linear (m,σ)-standard function; define the I×I matrix cofA=(Aij)i,j∈I by
(22)Aij=(-1)i+jdet⁡(A(1⋯i^⋯∣1⋯j^⋯)),
where A(1⋯i^⋯∣1⋯j^⋯) is the (I∖{i})×(I∖{j}) matrix obtained by deleting the ith row and the jth column of A.Proposition 16.
LetA=(aij)i,j∈I:E→E be a linear (m,σ)-standard function; then, for any i∈I, one has
(23)det⁡A=∑‍j∈IaijAij.Proof.
Suppose thatσ is biunique; then, ∀i∈{1,…,m}, we have
(24)det⁡A=det⁡Am∏‍n∈I∖{1,…,m}λn=∑‍mj=1aij(Am)ij(∏‍n∈I∖{1,…,m}λn)=∑‍mj=1aijAij.
Moreover, ∀i∈{1,…,m} and ∀j>m, the matrix A(1⋯i^⋯∣1⋯j^⋯) is (m-1,σ¯)-standard, where
(25)σ¯:I∖{1,…,m-1}⟶I∖{1,…,m-1}
is not surjective because m∉σ¯(I∖{1,…,m-1}), and so Aij=0; then, det⁡A=∑j∈IaijAij. Finally, ∀i>m, we have aij=0, ∀j≠i; then
(26)∑‍j∈IaijAij=aiiAii=aii(-1)2idet⁡Am∏‍n∈I∖{1,…,m,i}λn=det⁡Am∏‍n∈I∖{1,…,m}λn=det⁡A.
Instead, ifσ is not biunique, ∀i,j∈{1,…,m}, the matrix A(1⋯i^⋯∣1⋯j^⋯) is (m-1,σ^)-standard, where σ^(n)=σ(n+1), ∀n>m-1; then, σ^ is not biunique, from which Aij=0. Moreover, ∀i∈{1,…,m} and ∀j>m, as in the case σ being biunique, we have Aij=0. Finally, ∀i>m, we have aij=0, ∀j≠σ(i); then
(27)∑‍j∈IaijAij=ai,σ(i)Ai,σ(i).
Moreover, the matrix A(1…i^…∣1…σ(i)^…) is (m,σ~)-standard, where the function σ~:I∖{1,…,m,i}→I∖{1,…,m,σ(i)} is not biunique; in fact, in this case necessarily σ(i)=i, and so σ should be biunique (a contradiction); then, we have Ai,σ(i)=0, from which
(28)det⁡A=0=∑‍j∈IaijAij.Corollary 17.
LetA=(aij)i,j∈I:E→E be a biunique and linear (m,σ)-standard function; then, A-1:E→E is a linear (m,σ)-standard function A-1=(bij)i,j∈I; moreover
(29)A-1=1det⁡A(cofA)t.Proof.
From Proposition16, we have
(30)∑‍n∈IainAin=det⁡A.
Moreover, we have
(31)∑‍n∈IainAjn=0,∀i,j∈I,i≠j;
in fact, from Proposition 16, the left side of (31) is equal to det⁡C, where C is the (p,τ)-standard matrix obtained by substituting the ith row of A for the jth row of A; then, from Proposition 13, we have det⁡C=0. This implies that
(32)∑‍n∈IainAjn=(det⁡A)δij,∀i,j∈I,
where δij is the Kronecker symbol, and so
(33)(A(cofA)t)ij=(det⁡A)δij,∀i,j∈I,
from which the formula (29) follows. Moreover, as we observed in the proof of Proposition 16, ∀i∈{1,…,m} and ∀j>m, we have Aij=0; finally, ∀i,j>m such that i≠j, the matrix A(1⋯i^⋯∣1⋯j^⋯) is (m,σ¯)-standard, where σ¯:I∖{1,…,m,i}→I∖{1,…,m,j} is not surjective because i∉σ¯(I∖{1,…,m,i}), and so Aij=0 again; from formula (29), this implies that A-1 is (m,σ)-standard.Definition 18.
Define the function∥·∥:CI→[0,+∞] by
(34)∥x∥=sup⁡n∈I|xn|,∀x=(xn:n∈I)∈CI,
and define the following vector space on the field C, with the norm ∥·∥:
(35)F={x∈CI:∥x∥<+∞}⊃E.Definition 19.
LetA=(aij)i,j∈I be a real matrix I×I; then, define the linear function A=(aij)i,j∈I:F→CI and write x→Ax, in the following manner:
(36)(Ax)i=∑‍j∈Iaijxj,∀x∈F,∀i∈I,
on condition that, for any i∈I, the sum in (36) converges to a complex number.Proposition 20.
LetA=(aij)i,j∈I be a real matrix I×I; then(1)
the linear functionA=(aij)i,j∈I:F→CIgiven by (36) is defined if and only if, for anyi∈I,∑j∈I|aij|<+∞.(2)
A(F)⊂FandAis continuous if and only ifsup⁡i∈I⁡∑j∈I|aij|<+∞; moreover, in this case∥A∥=sup⁡i∈I⁡∑j∈I|aij|.Proof.
The proof is analogous to that one of Proposition7.Definition 21.
LetV be a vector space on C, and let T:V→V be a linear function; indicate by VP(T) the set of the eigenvalues of T.Proposition 22.
LetA=(aij)i,j∈I:E→E be a linear (m,σ)-standard function, with σ biunique; then, by considering A as a function from F to F, one has
(37)VP(A)=VP(Am)∪{λn:n∈I∖{1,…,m}}.
Moreover
(38)det⁡A=∏‍λ∈VP(A)λ.Proof.
Letλ∈C be an eigenvalue of Am, let x∈Cm∖{0} be the corresponding eigenvector, and let y∈CI∖{0} be such that yn=xn, ∀n∈{1,…,m}, and yn=0, ∀n∈I∖{1,…,m}. We have (Ay)n=(Ax)n=(λx)n=(λy)n, ∀n∈{1,…,m}, and (Ay)n=0=(λy)n, ∀n∈I∖{1,…,m}, from which Ay=λy, and so λ∈VP(A). Moreover, ∀n∈I∖{1,…,m}, since σ is biunique, from the Remark 9, we have σ(n)=n. If ain=0, ∀i∈{1,…,m}, let x∈RI∖{0} be such that xi=δin, ∀i∈I; we have Ax=λnx, and so λn∈VP(A). Otherwise, suppose that ain≠0 for some i∈{1,…,m}; if λn∈VP(Am), then λn∈VP(A) by the previous arguments; conversely, if (Am-λnIm)x≠0, ∀x∈Cm∖{0}, the matrix (Am-λnIm) is invertible and so there exists x∈Rm∖{0} such that Amx-λnx=(t-a1n,…,-ain,…,-amn); then, by considering y∈RI∖{0} such that yi=xi, ∀i∈{1,…,m}, yi=δin, ∀i∈I∖{1,…,m}, we have Ay=λny, and so λn∈VP(A). Then
(39)VP(Am)∪{λn:n∈I∖{1,…,m}}⊂VP(A).
Conversely, ifλ∈VP(A), we have Ax=λx, for some x∈CI∖{0}, and so, ∀n∈I∖{1,…,m}, λnxn=(Ax)n=λxn; then, by supposing λ∉{λn:n∈I∖{1,…,m}}, we have xn=0, from which xn≠0 for some n∈{1,…,m}. Moreover, we have
(40)Am(x1,…,xm)t=((Ax)1,…,(Ax)m)t=λ(x1,…,xm)t,
and so λ∈VP(Am). Then, we have
(41)VP(A)⊂VP(Am)∪{λn:n∈I∖{1,…,m}},
from which (37) follows. Moreover, since σ is biunique, from (37), we have
(42)det⁡A=det⁡Am∏‍n∈I∖{1,…,m}λn=∏‍λ∈VP(A)λ.


## 4. Change of Variables’ Formula

Definition 23.
Letk∈N, let M,N∈R+, and let a=(an:n∈I)∈(R+)I such that ∏n∈Ian∈R+; define the following sets in B(E):
(43)EN,a(k)=Rk×∏‍n∈I∖{1,…,k}[-Nan,Nan];EM,N,a(k)=[-M,M]k×∏‍n∈I∖{1,…,k}[-Nan,Nan].Definition 24.
Leta=(an:n∈I)∈(R+)I and b=(bn:n∈I)∈(R+)I be such that ∏n∈Ian∈R+, ∏n∈Ibn∈R+; define ab∈(R+)I in the following manner:
(44)ab=(anbn:n∈I).Proposition 25.
LetA=(aij)i,j∈I:E→E be a biunique and linear (m,σ)-standard function; then, for any a=(an:n∈I)∈(R+)I such that ∏n∈Ian∈R+, there exists b=(bn:n∈I)∈(R+)I such that ∏n∈Ibn∈R+ and such that, for any k∈N, k≥m, and for any N∈R+, one has
(45)A-1(EN,a(k))=EN,b(k).Proof.
From Corollary17, A-1=(bij)i,j∈I:E→E is a linear (m,σ)-standard function. By setting ρn=bnn, ∀n>m, from (29), we have
(46)ρn=1ann=1λn⟹∏‍n∈I∖{1,…,m}ρn=∏‍n∈I∖{1,…,m}1λn∈R*.
Set b=(bn:n∈I)∈(R+)I such that
(47)bn=1,∀n∈{1,…,m},(bn:n>m)=(an:n>m)(|ρn|:n>m).
By definition of b, we have
(48)∏‍n∈Ibn=(∏‍n∈I∖{1,…,m}an)(∏‍n∈I∖{1,…,m}1|λn|)∈R+;
moreover, for any k∈N, k≥m, and for any N∈R+, we have A-1(EN,a(k))⊂EN,b(k). Analogously, it is possible to prove that A(EN,b(k))⊂EN,c(k), where
(49)(cn:n>m)=(bn:n>m)(|λn|:n>m)=(an:n>m).
Moreover, since k≥m, we have EN,c(k)=EN,a(k), and so EN,b(k)⊂A-1(EN,a(k)), from which (45) follows.Lemma 26.
LetA=(aij)i,j∈I:E→E be a biunique and linear (m,σ)-standard function; then, for any M1∈R+ and for any a=(an:n∈I)∈(R+)I such that ∏n∈Ian∈R+, there exist M2,M3∈R+ and b=(bn:n∈I)∈(R+)I, c=(cn:n∈I)∈(R+)I such that ∏n∈Ibn∈R+, ∏n∈Icn∈R+, and such that, for any k∈N and for any N∈R+, one has
(50)A-1(EM1,N,a(k))⊂EM2,N,b(k);(51)A(EM2,N,b(k))⊂EM3,N,c(k).
Moreover, (cn:n>m)=(an:n>m).Proof.
From the Banach theorem of the open function (see also the exercise 5.14 in [8]), A-1 is continuous; then, ∀N∈R+ and ∀x∈EM1,N,a(k), we have
(52)∥A-1(x)∥≤∥A-1∥∥x∥≤∥A-1∥max⁡⁡{M1,N,∥a∥}.
Set M2=∥A-1∥max⁡⁡{M1,N,∥a∥} and b=(bn:n∈I)∈(R+)I such that
(53)bn=M2N,∀n∈{1,…,m},(bn:n>m)=(an:n>m)(|ρn|:n>m),
where ρn, ∀n∈I, is defined as in the proof of Proposition 25. By definition of b, we have
(54)∏‍n∈Ibn=(M2N)m(∏‍n∈I∖{1,…,m}an)(∏‍n∈I∖{1,…,m}1|λn|)∈R+,
and (50) holds. Analogously, it is possible to prove (51); moreover
(55)(cn:n>m)=(bn:n>m)(|λn|:n>m)=(an:n>m).Remark 27.
LetA=(aij)i,j∈I:E→E be a linear (m,σ)-standard function; then, A is B(E)/B(E)-measurable.Proof.
Letτ be the topology induced by the norm ∥·∥ on E; then, since A is continuous by Remark 9, ∀B∈τ we have A-1(B)∈τ⊂B(E). Moreover, since σ(τ)=B(E), we have A-1(B)∈B(E), ∀B∈B(E).Proposition 28.
Letμ1 and μ2 be two measures on a measurable space (S,Σ) that coincide on a π-system I on S; then, if σ(I)=Σ and μ1(S)=μ2(S)<+∞, then μ1 and μ2 coincide on Σ.Proof.
See, for example, Theorem 3.3 in Billingsley [9].Now, we can prove the main result of our paper, that generalizes the change of variables formula for the integration of a biunique linear function onRm with values in Rm (see, e.g., Lang's book [10]).Theorem 29 (change of variables’ formula).
LetA=(aij)i,j∈I:E→E be a biunique and linear (m,σ)-standard function, let a=(an:n∈I)∈(R+)I be such that ∏n∈Ian∈R+, and let b∈(R+)I be the sequence defined by Proposition 25. Then, for any k∈N, k≥m, for any N∈R+, for any B∈B(E), and for any measurable function f:(E,B(E))→(R,B) such that f+ (or f-) is λN,a(k)-integrable, one has
(56)∫BfdλN,a(k)=∫A-1(B)f(A)|det⁡A|dλN,b(k).Proof.
∀
n
∈
N, let hn:E→E be the biunique and linear (m,σ)-standard function given by
(57)(hn(x))i=(An(π{1,…,n}(x)))i,∀x∈E,∀i∈{1,…,n};(hn(x))i=λixi,∀x∈E,∀i∈I∖{1,…,n}.
Moreover, ∀M1∈R+ and ∀a=(an:n∈I)∈(R+)I such that ∏n∈Ian∈R+, let M2(n), M3(n) be the constants, and let b(n), c(n) be the sequences defined by Lemma 26 and the function hn; finally, consider the analogous constants M2, M3, and the sequences b, c defined by A. Observe that M2(n)≤M2, (b(n))i≤bi, ∀i∈I, ∀n∈N. Suppose that n≥k≥m and N∈R+; then, ∀B=∏p∈IBp, where Bp∈B([-M1,M1]), ∀p∈{1,…,k}, Bp∈B([-Nap,Nap]), ∀p>k, we have
(58)∫BdλN,a(k)=∫(B1×⋯×Bk)×∏q>kBqd((⨂kp=112NLeb)⊗(⨂q>k12NLeb|B([-Naq,Naq])))=∫(B1×⋯×Bn)×∏q>nBqd((⨂np=112NLeb)⊗(⨂q>n12NLeb|B([-Naq,Naq])))=∫B1×⋯×Bnd(⨂np=112NLeb)×∫∏q>nBqd(⨂q>n12NLeb|B([-Naq,Naq]))=∫An-1(B1×⋯×Bn)|det⁡An|d(⨂np=112NLeb)×∫∏q>n1/λqBq∏‍q>n|λq|d(⨂q>n12NLeb|B([-Nbq,Nbq]))=∫hn-1(B)|det⁡hn|d((12NLeb|B([-Nbq,Nbq])⨂np=112NLeb)⊗(⨂q>n12NLeb|B([-Nbq,Nbq])))=∫hn-1(B)|det⁡hn|d((⨂kp=112NLeb)⊗(⨂q>k12NLeb|B([-Nbq,Nbq])))=∫hn-1(B)|det⁡hn|dλN,b(k).
Consider the measures μ1 and μ2 on B(EM1,N,a(k)) defined by
(59)μ1(B)=∫BdλN,a(k);μ2(B)=∫hn-1(B)|det⁡hn|dλN,b(k).
From (58), μ1 and μ2 coincide on the set I={B∈B(EM1,N,a(k)):B=∏p∈IBp}; since I is a π-system on EM1,N,a(k) such that σ(I)=B(EM1,N,a(k)) and since μ1(EM1,N,a(k))=μ2(EM1,N,a(k))=(M1/N)k∏p>kap<+∞, from Proposition 28, we have that ∀B∈B(EM1,N,a(k)):
(60)∫EM1,N,a(k)1BdλN,a(k)=∫EM2,N,b(k)1B(hn)|det⁡hn|dλN,b(k).
This implies that if φ:(EM3,N,a(k),B(EM3,N,a(k)))→([0,+∞),B([0,+∞))) is a simple function such that φ(x)=0, ∀x∉EM1,N,a(k), we have
(61)∫EM1,N,a(k)φdλN,a(k)=∫EM2,N,b(k)φ(hn)|det⁡hn|dλN,b(k).
Then, if l:(EM3,N,a(k),B(EM3,N,a(k)))→([0,+∞),B([0,+∞))) is a measurable function such that φ(x)=0, ∀x∉EM1,N,a(k), and {φi}i∈N is a sequence of increasing positive simple functions over EM3,N,a(k) such that lim⁡i→+∞⁡φi=l, φi(x)=0, ∀x∉EM1,N,a(k), ∀i∈N, from Beppo Levi theorem we have
(62)∫EM1,N,a(k)ldλN,a(k)=lim⁡i→+∞∫EM1,N,a(k)φidλN,a(k)=lim⁡i→+∞∫EM2,N,b(k)φi(hn)|det⁡hn|dλN,b(k)=∫EM2,N,b(k)l(hn)|det⁡hn|dλN,b(k)=lim⁡n→+∞∫EM2,N,b(k)l(hn)|det⁡hn|dλN,b(k).
In particular, the formula (62) is true for any continuous and bounded function l:EM3,N,a(k)→[0,1]. In this case, let {fn}n∈N be the sequence of the measurable functions fn:(EM2,N,b(k),B(EM2,N,a(k)))→(R,B) given by
(63)fn(x)=l(hn(x))|det⁡hn|,∀x∈EM2,N,b(k),∀n∈N.
Since det⁡hn=det⁡A, ∀n≥m, we have |fn|≤g, where g:(EM2,N,b(k),B(EM2,N,a(k)))→([0,+∞),B([0,+∞))) is the measurable function defined by
(64)g(x)=|det⁡A|,∀x∈EM2,N,b(k).
Moreover
(65)∫EM2,N,b(k)gdλN,b(k)=|det⁡A|λN,b(k)(EM2,N,b(k))=|det⁡A|(2M2)k(2N)k∏‍p>k(12NLeb([-Nbp,Nbp]))=|det⁡A|M2kNk∏‍p>kbp<+∞.
Moreover, we have lim⁡n→+∞⁡hn=A, and so lim⁡n→+∞⁡fn=l(A)|det⁡A|; then, from the dominated convergence theorem,
(66)lim⁡n→+∞∫EM2,N,b(k)l(hn)|det⁡hn|dλN,b(k)=∫EM2,N,b(k)l(A)|det⁡A|dλN,b(k).
Then, from (62) we have
(67)∫EM1,N,a(k)ldλN,a(k)=∫EM2,N,b(k)l(A)|det⁡A|dλN,b(k).
LetB=∏p∈IBp∈B(EM1,N,a(k)), where Bp=(ap,bp), ∀p∈I; moreover, ∀n∈N*, consider the continuous function ln:EM3,N,a(k)→[0,1] defined by
(68)ln(x)={1ifx∈∏‍p∈I(ap+δpn,bp-δpn)∥x-x2∥∥x1-x2∥ifx∈B∖∏‍p∈I(ap+δpn,bp-δpn)0ifx∉B,
where δp=(bp-ap)/2, ∀p∈I, x1=r∩∂(∏p∈I(ap+(δp/n),bp-(δp/n))), x2=r∩∂B, where r is the half-line with initial point ∏p∈I((ap+bp)/2) and containing x. Since {ln}n∈N is an increasing positive sequence such that lim⁡n→+∞⁡ln=1B, from Beppo Levi theorem and (67), we have
(69)∫BdλN,a(k)=lim⁡n→+∞∫EM1,N,a(k)lndλN,a(k)=lim⁡n→+∞∫EM2,N,b(k)ln(A)|det⁡A|dλN,b(k)=∫A-1(B)|det⁡A|dλN,b(k).
Moreover, Proposition 28 again implies that the formula (69) is true ∀B∈B(EM1,N,a(k)). Consider the measures μ and υ on B(EN,a(k)) defined by
(70)μ(B)=∫BdλN,a(k),υ(B)=∫A-1(B)|det⁡A|dλN,b(k),
and set Bn=B∩En,N,a(k), ∀n∈N*, ∀B∈B(EN,a(k)). Since Bn⊂Bn+1, A-1(Bn)⊂A-1(Bn+1), ⋃n∈N*Bn=B, and ⋃n∈N*A-1(Bn)=A-1(B), from the continuity property of μ and υ and (69), we have
(71)∫BdλN,a(k)=lim⁡n→+∞∫BndλN,a(k)=lim⁡n→+∞∫A-1(Bn)|det⁡A|dλN,b(k)=∫A-1(B)|det⁡A|dλN,b(k).
Then,∀D∈B(EN,a(k)),
(72)∫B1DdλN,a(k)=∫B∩DdλN,a(k)=∫A-1(B∩D)|det⁡A|dλN,b(k)=∫A-1(B)1A-1(D)|det⁡A|dλN,b(k)=∫A-1(B)1D(A)|det⁡A|dλN,b(k).
Thus, by proceeding as in the proof of the formula (62), for any measurable function f:(EN,a(k),B(EN,a(k)))→([0,+∞),B([0,+∞))), we obtain
(73)∫BfdλN,a(k)=∫A-1(B)f(A)|det⁡A|dλN,b(k).
Then, if f:(EN,a(k),B(EN,a(k)))→(R,B) is a measurable function such that f+ (or f-) is λN,a(k)-integrable:
(74)∫BfdλN,a(k)=∫Bf+dλN,a(k)-∫Bf-dλN,a(k)=∫A-1(B)f+(A)|det⁡A|dλN,b(k)-∫A-1(B)f-(A)|det⁡A|dλN,b(k)=∫A-1(B)f(A)|det⁡A|dλN,b(k).
Finally, suppose thatB∈B(E) and f:(E,B(E))→(R,B) is a measurable function such that f+ (or f-) is λN,a(k)-integrable; from formula (74), Proposition 25 and definitions of λN,a(k) and λN,b(k) given by (4), we have
(75)∫BfdλN,a(k)=∫B∩EN,a(k)fdλN,a(k)=∫A-1(B∩EN,a(k))f(A)|det⁡A|dλN,b(k)=∫A-1(B)f(A)|det⁡A|dλN,b(k).


## 5. Probabilistic Applications

Definition 30.
Let(Ω,F,P) be a probability space; a random element X:(Ω,F,P)→(E,B(E)) is called λN,a(k)-continuous if there exists a measurable function fX:(E,B(E))→([0,+∞),B([0,+∞))) such that, for any A∈B(E),
(76)P(X∈A)=∫AfXdλN,a(k).
The function fX is called infinite-dimensional probability density of X.Theorem 31.
LetA=(aij)i,j∈I:E→E be a biunique and linear (m,σ)-standard function, let a=(an:n∈I)∈(R+)I be such that ∏n∈Ian∈R+, and let b∈(R+)I be the sequence defined by Proposition 25. Then, for any k∈N, k≥m, for any N∈R+, and for any λN,b(k)-continuous random element X:(Ω,F,P)→(E,B(E)), the random element T=A∘X:(Ω,F,P)→(E,B(E)) is λN,a(k)-continuous and one has
(77)fT(t)=fX(A-1(t))|det⁡A-1|,∀t∈E.Proof.
∀
B
∈
B
(
E
), we have
(78)P(T∈B)=E[1B(T)]=E[1B(A(X))]=∫E1B(A(x))fX(x)dλN,b(k)(x)=∫A-1(B)fX(A-1(A(x)))|det⁡A-1||det⁡A|dλN,b(k)(x)=∫BfX(A-1(t))|det⁡A-1|dλN,a(k)(t)(fromTheorem29).


## 6. Problems for Further Study

A natural extension of this paper is the generalization of Theorem29 by considering the measurable and C1-invertible functions A:E→E. As in the finite case, we can define the infinite-dimensional Jacobian matrix of these functions and the determinant of this Jacobian, if it is a (m,σ)-standard matrix.Moreover, from Definition30 and Theorem 31, in the probabilistic context it is possible to introduce many random elements that generalize the well-known continuous random vectors in Rm (e.g., the Gaussian random elements in E defined by the (m,σ)-standard matrices) and to develop a theory and some applications in the statistical inference.In particular, as we point out in the introduction, we can generalize the paper [4] by considering the recursion {Xn}n∈N on ∏i∈N*[0,p) defined by
(79)Xn+1=AXn+Bn(mod⁡p),
where X0=x0∈E, A is a (m,σ)-standard matrix, p∈R+, and {Bn}n∈N is a sequence of independent and identically distributed random elements on E. Our target is to prove that, with some assumptions on the law of Bn, the sequence {Xn}n∈N converges with geometric rate to a random element with law ⨂i∈N*(1/p)Leb|B([0,p)). Moreover, we wish to quantify the rate of convergence in terms of A, p, m and the law of Bn and to prove that if A has an eigenvalue that is a root of 1, then O(p2) steps are necessary to achieve randomness. We hope to develop these ideas in a further paper.



---

*Source: 404186-2014-06-15.xml*