# Differentiation Theory over Infinite-Dimensional Banach Spaces

Authors: Claudio Asci
Journal: Journal of Mathematics
 (2016)

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## Abstract

We study, for any positive integerk and for any subset I of N⁎, the Banach space EI of the bounded real sequences xnn∈I and a measure over RI,B(I) that generalizes the k-dimensional Lebesgue one. Moreover, we expose a differentiation theory for the functions defined over this space. The main result of our paper is a change of variables’ formula for the integration of the measurable real functions on RI,B(I). This change of variables is defined by some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms.


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## Body

## 1. Introduction

The aim of this paper is to generalize the results of article [1], where, for any positive integer k and for any subset I of N∗, we study a particular infinite-dimensional measure λN,a,v(k,I) that, in the case I={1,…,k}, coincides with the k-dimensional Lebesgue one on Rk. The measure λN,a,v(k,I) is a product indexed by I of σ-finite measures on the Borel σ-algebra B on R (by using a generalization of the Jessen theorem), and it is defined over the measurable space RI,B(I), and in particular over EI,BI, where B(I) is the product indexed by I of the same σ-algebra B, EI⊂RI is the Banach space of the bounded real sequences xnn∈I, and BI is the restriction to EI of B(I).In the mathematical literature, some articles introduced infinite-dimensional measures analogue of the Lebesgue one (see, e.g., the paper of Léandre [2], in the context of the noncommutative geometry, that one of Tsilevich et al. [3], which studies a family of σ-finite measures on R+, and that one of Baker [4], which defines a measure on RN∗ that is not σ-finite).In paper [1], the main result is a change of variables’ formula for the integration of the measurable real functions on the space EI,BI. This change of variables is defined by a particular class of linear functions over EI, called m,σ-standard. 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 this paper, we prove that the change of variables’ formula given in [1] can be extended by defining some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms.In Section2, we construct the infinite-dimensional Banach space EI, and we define the continuous functions and the homeomorphisms over the open subsets of this space. Moreover, we recall some results about the integration of the measurable real functions defined on a measurable product space. In Section 3, we expose a differentiation theory in the infinite-dimensional context, and in particular we define the functions C1 and the diffeomorphisms. Moreover, we introduce a class of functions, called m,σ-standard, that generalizes the set of the linear m,σ-standard functions given in [1], and we expose some properties of these functions. In Section 4, we present the main result of our paper, that is, a change of variables’ formula for the integration of the measurable real functions on RI,B(I); this change of variables is defined by the biunique, C1 and m,σ-standard functions, with further properties (Theorem 47). This result agrees with the analogous finite-dimensional result. In Section 5, we expose some ideas for further study in the probability theory.


## 2. Construction of an Infinite-Dimensional Banach Space

Henceforth, we will indicate byN∗ and R∗ the sets N∖0 and R∖0, respectively. Let I≠⌀ be a set and let k∈N∗; indicate by τ, by τ(k), by τ(I), by B, by B(k), by B(I), by Leb, and by Leb(k), respectively, the Euclidean topology on R, the Euclidean topology on Rk, the topology ⨂i∈Iτ, the Borel σ-algebra on R, the Borel σ-algebra on Rk, the σ-algebra ⨂i∈IB, the Lebesgue measure on R, and the Lebesgue measure on Rk. Moreover, for any set A⊂R, indicate by B(A) the σ-algebra induced by B on A, and by τ(A) the topology induced by τ on A; analogously, for any set A⊂RI, define the σ-algebra B(I)(A) and the topology τ(I)(A). Finally, if S=∏i∈ISi is a Cartesian product, for any xi:i∈I∈S and for any ⌀≠H⊂I, define xH=(xi:i∈H)∈∏i∈HSi, and define the projection πI,H on ∏i∈HSi as the function πI,H:S→∏i∈HSi given by πI,HxI=xH.Theorem 1.
LetI≠⌀ be a set and, for any i∈I, let Si,Σi,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(Sj)∈R+. Then, over the measurable space ∏i∈ISi,⨂i∈IΣi, 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∖HSi∈⨂i∈IΣi, where Ah∈Σh,∀h∈H, we have μ(A)=∏h∈Hμh(Ah)∏j∈J∖Hμj(Sj). In particular, if I is countable, then μ(A)=∏i∈Iμi(Ai) for any A=∏i∈IAi∈⨂i∈IΣi.Proof.
See the proof of Corollary4 in Asci [1].Henceforth, we will suppose thatI,J are sets such that ⌀≠I,J⊂N∗; moreover, for any k∈N∗, we will indicate by Ik={i1,…,ik} the set of the first k elements of I (with the natural order and with the convention Ik=I if I<k) and analogously Jk; finally, for any j=in∈I, set j=n.Theorem 2.
LetSi,Σi,μi be a probability space, for any i∈I, let S,Σ,μ=∏i∈ISi,⨂i∈IΣi,⨂i∈Iμi, and let f∈L1S,Σ,μ. Moreover, for any H⊂I such that ⌀≠H<+∞, define the measurable function fHc:S,Σ→(R,B) by(1)fHcx=∫SHfxH,xHcdμHxH,where SH,ΣH,μH is the probability space ∏i∈HSi,⨂i∈HΣi,⨂i∈Hμi. Then, μ-a.e. one has limn→+∞⁡fInc=∫Sfdμ.Proof.
See, for example, Theorem3, page 349, in Rao [6].Corollary 3.
LetSi,Σi,μi be a measure space such that μi is finite, for any i∈I, and ∏i∈Iμi(Si)∈[0,+∞); moreover, let S,Σ,μ=∏i∈ISi,⨂i∈IΣi,⨂i∈Iμi, let f∈L1S,Σ,μ, and let the measurable function fInc:S,Σ→(R,B) defined by (1), for any n∈N∗. Then, μ-a.e. one has limn→+∞⁡fInc=∫Sfdμ.Proof.
Suppose thatμi(Si)≠0, ∀i∈I; then, μ¯i=μi/μi(Si) is a probability measure, and also μ¯=⨂i∈Iμ¯i, μ¯In=⨂i∈Inμ¯i, and μ¯Inc=⨂i∈Incμ¯i, ∀n∈N∗; then, define the function f¯Inc:S,Σ→(R,B) by(2)f¯Incx=∫SInfxIn,xIncdμ¯InxIn.From Theorem 2, μ--a.e., we have(3)limn→+∞⁡fInc=limn→+∞⁡∏i∈InμiSif¯Inc=∏i∈IμiSi∫Sfdμ¯=∫Sfdμ.
Conversely, ifμi(Si)=0, for some i∈I, then(4)limn→+∞⁡fInc=0=∫Sfdμ.
Thus, sinceμ=∏i∈Iμi(Si)μ-, we obtain the statement.Definition 4.
For any setI≠⌀, define the function ·I:RI→0,+∞ by(5)xI=supi∈I⁡xi,∀x=xi:i∈I∈RI,and define the vector space(6)EI=x∈RI:xI<+∞.Moreover, indicate by BI the σ-algebra B(I)(EI), by τI the topology τ(I)(EI) induced on EI by the product topology τ(I)(RI), and by τ·I the topology induced on EI by the distance d:EI×EI→0,+∞ defined by d(x,y)=x-yI, ∀x,y∈EI; furthermore, for any set A⊂EI, indicate by τ·I(A) the topology induced by τ·I on A. Finally, for any x0∈EI and for any δ>0, indicate by BI(x0,δ) the set x∈EI:x-x0I<δ.Observe thatτIA⊂τ·IA, ∀A⊂EI; moreover, EI is a Banach space, with the norm ·I (see, e.g., the proof of Remark 2 in [1]).Proposition 5.
LetH and I be sets such that ⌀≠H⊂I; then, the function πI,H:RI,τ(I)→RH,τ(H) is continuous and open.Proof.
See, for example, the theory of the product spaces in Weidmann’s book [7].Proposition 6.
LetH and I be sets such that ⌀≠H⊂I, and let π¯I,H:EI→EH be the function given by π¯I,Hx=πI,Hx, for any x∈EI; then,(1)
π¯I,H:EI,τI→EH,τH is continuous and open;(2)
π¯I,H:EI,τ·I→EH,τ·H is continuous and open.Proof of (1).
∀ A ∈ τ H, there exists B∈τ(H) such that A=B∩EH, and so π¯I,H-1(A)=πI,H-1B∩EH∩EI=πI,H-1B∩EI; then, since πI,H-1B∈τ(I) by Proposition 5, we have π¯I,H-1(A)∈τI, and so π¯I,H is a continuous function. Moreover, ∀A∈τI, there exists B∈τ(I) such that A=B∩EI, and so π¯I,H(A)=πI,HB∩EI=πI,HB∩EH; then, since πI,HB∈τ(H) by Proposition 5, we have π¯I,H(A)∈τH, and so π¯I,H is an open function.Proof of (2).
∀ A ∈ τ · H and ∀x=(xi:i∈I)∈π¯I,H-1(A), we have (xi:i∈H)∈A, and so there exists a∈R+ and (yi:i∈H)∈A such that (xi:i∈H)∈∏i∈Hyi-a,yi+a⊂A; let z=(zi:i∈I)∈π¯I,H-1(A) such that zi=xi, ∀i∈I∖H, and zi=yi, ∀i∈H; then, x∈∏i∈Izi-a,zi+a⊂π¯I,H-1(A), and so π¯I,H is a continuous function. Moreover, ∀A∈τ·I and ∀x=xi:i∈H∈π¯I,HA, let y=(yi:i∈I)∈A such that yi=xi, ∀i∈H; since A∈τ·I, there exists a∈R+ and (zi:i∈I)∈A such that y∈∏i∈Izi-a,zi+a⊂A, and so x∈∏i∈Hzi-a,zi+a⊂π¯I,H(A); then, π¯I,H is an open function.Definition 7.
LetU∈τ·J, let x0∈U, let l∈EI, and let φ:U⊂EJ→EI be a function; we say that limx→x0⁡φ(x)=l if, for any neighbourhood M∈τ·I of φ(x0), there exists a neighbourhood N∈τ·J(U) of x0 such that, for any x∈N∖{x0}, we have φ(x)∈M.Definition 8.
LetU∈τ·J and let φ:U⊂EJ→EI be a function; we say that φ is continuous in x0∈U if limx→x0⁡φ(x)=φx0, and we say that φ is continuous in U if φ is continuous in x0 for any x0∈U.Remark 9.
LetU∈τ·J, let V∈τ·I, and let φ:U⊂EJ→V⊂EI be a function; then, φ:U,τ·J(U)→V,τ·I(V) is continuous if and only if φ is continuous in U.Remark 10.
LetH, I, and J be sets such that ⌀≠H⊂I, let U∈τ·J, and let φ:U⊂EJ→EI be a function continuous in U; then, the function πI,H∘φ:U→EH is continuous in U.Proof.
The statement follows from Proposition6.Definition 11.
LetU∈τ·J and let V∈τ·I; a function φ:U⊂EJ→V⊂EI is called homeomorphism if φ is biunique and the functions φ:U,τ·J(U)→V,τ·I(V) and φ-1:V,τ·I(V)→U,τ·J(U) are continuous.


## 3. Differentiation Theory in the Infinite-Dimensional Context

The following concept generalizes Definition6 in [1] (see also the theory in the Lang’s book [8]).Definition 12.
LetA=aiji∈I,j∈J be a real matrix I×J (eventually infinite); then, define the linear function A=aiji∈I,j∈J:EJ→RI, and write x→Ax, in the following manner:(7)Axi=∑j∈Jaijxj,∀x∈EJ,∀i∈I,on condition that, for any i∈I, the sum in (7) converges to a real number.Definition 13.
LetU∈τ·J; a function φ:U⊂EJ→EI is called differentiable in x0∈U if there exists a linear and continuous function A:EJ→EI defined by a real matrix A=aiji∈I,j∈J, and one has(8)limh→0⁡φx0+h-φx0-AhIhJ=0.Ifφ is differentiable in x0 for any x0∈U, φ is called differentiable in U. The function A is called differential of the function φ in x0, and it is indicated by the symbol dφ(x0).Remark 14.
LetU∈τ·J and let φ,ψ:U⊂EJ→EI be differentiable functions in x0∈U; then, for any α,β∈R, the function αφ+βψ is differentiable in x0, and d(αφ+βψ)(x0)=αdφ(x0)+βdψ(x0).Proof.
Observe that(9)limh→0⁡αφ+βψx0+h-αφ+βψx0-αdφx0+βdψx0hIhJ=limh→0⁡αφx0+h-αφx0-αdφx0h+βψx0+h-βψx0-βdψx0hIhJ≤limh→0⁡αφx0+h-αφx0-αdφx0hIhJ+limh→0⁡βψx0+h-βψx0-βdψx0hIhJ=0⟹dαφ+βψx0=αdφx0+βdψx0.Remark 15.
A linear and continuous functionA=aiji∈I,j∈J:EJ→EI, defined by(10)Axi=∑j∈Jaijxj,∀x∈EJ,∀i∈I,is differentiable and dφ(x0)=A, for any x0∈EJ.Remark 16.
LetU∈τ·J and let φ:U⊂EJ→EI be a function differentiable in x0∈U; then, for any i∈I, the component φi:U→R is differentiable in x0, and dφi(x0) is the matrix Ai given by the ith row of A=dφ(x0). Moreover, if I<+∞ and φi:U⊂EJ→R is differentiable in x0, for any i∈I, then φ:U⊂EJ→EI is differentiable in x0.Remark 17.
LetU∈τ·J and let φ:U⊂EJ→EI be a function differentiable in x0∈U; then, φ is continuous in x0.Proof.
∀ x ∈ U, set(11)σx,x0=fx-fx0-dfx0x-x0.From (8), we have limx→x0⁡σ(x,x0)I=0; moreover,(12)fx-fx0I=dfx0x-x0+σx,x0I≤dfx0x-x0J+σx,x0I,from which limx→x0⁡f(x)=fx0.Definition 18.
LetU∈τ·J and let v∈EJ such that vJ=1; a function φ:U⊂EJ→R is called derivable in x0∈U in the direction v if there exists the limit(13)limt→0⁡φx0+tv-φx0t.
This limit is indicated by(∂φ/∂v)(x0), and it is called derivative of φ in x0 in the direction v. If for some j∈J one has v=ej, where (ej)k=δjk, for any k∈J, indicate (∂φ/∂v)(x0) by (∂φ/∂xj)(x0), and call it partial derivative of φ in x0, with respect to xj. Moreover, if there exists the linear function defined by the matrix Jφ(x0), where Jφ(x0)ij=(∂φi/∂xj)(x0), for any i∈I and j∈J, then Jφ(x0) is called Jacobian matrix of the function φ in x0.Remark 19.
LetU∈τ·J and suppose that a function φ:U⊂EJ→EI is differentiable in x0∈U; then, for any v∈EJ such that vJ=1 and for any i∈I, the function φi:U⊂EJ→R is derivable in x0 in the direction v, and one has(14)∂φi∂vx0=dφix0v.Proof.
∀ h ∈ E J, by setting h=tv, for some matrix A=aiji∈I,j∈J we have(15)limt→0⁡φix0+tv-φix0-Aitvt=limh→0⁡φix0+h-φix0-AihhJ=0.Then, since Aitv=tAiv, we have(16)limt→0⁡φix0+tv-φix0t=Aiv,from which(17)∂φi∂vx0=Aiv=dφix0v.Corollary 20.
LetU∈τ·J and let φ:U⊂EJ→EI be a function differentiable in x0∈U; then, the linear function Jφ(x0):EJ→EI is defined and continuous; moreover, for any h∈EJ, one has dφ(x0)(h)=Jφ(x0)h.Proof.
From Remark19, ∀i∈I and ∀j∈J, we have dφ(x0)ij=dφi(x0)(ej)=(∂φi/∂xj)(x0)=Jφ(x0)ij.Theorem 21.
LetU∈τ·J, let φ:U⊂EJ→EI be a function differentiable in x0∈U, let V∈τ·I such that V⊃φ(U), and let ψ:V⊂EI→EH be a function differentiable in y0=φ(x0). Then, the function ψ∘φ is differentiable in x0, and one has d(ψ∘φ)(x0)=dψ(y0)∘dφ(x0).Proof.
The proof is analogous to that one true in the particular caseH<+∞, I<+∞, J<+∞ (see, e.g., the Lang’s book [9]).Definition 22.
LetU∈τ·J, let i,j∈J, and let φ:U⊂EJ→R be a function derivable in x0∈U with respect to xi, such that the function ∂φ/∂xi is derivable in x0 with respect to xj. Indicate (∂/∂xj)∂φ/∂xi(x0) by (∂2φ/∂xj∂xi)(x0) and call it second partial derivative of φ in x0 with respect to xi and xj. If i=j, it is indicated by (∂2φ/∂xi2)(x0). Analogously, for any k∈N∗ and for any j1,…,jk∈J, define (∂kφ/(∂xjk⋯∂xj1))(x0) and call it kth partial derivative of φ in x0 with respect to xj1,…,xjk.Definition 23.
LetU∈τ·J and let k∈N∗; a function φ:U⊂EJ→EI is called Ck in x0∈U if, in a neighbourhood V∈τ·J(U) of x0, for any i∈I and for any j1,…,jk∈J, there exists the function defined by x→(∂kφi/∂xjk⋯∂xj1)(x), and this function is continuous in x0; φ is called Ck in U if, for any x0∈U, φ is Ck in x0. Moreover, φ is called strongly C1 in x0∈U if, in a neighbourhood V∈τ·J(U) of x0, there exists the function defined by x→Jφ(x), and this function is continuous in x0, with Jφ(x0)<+∞. Finally, φ is called strongly C1 in U if, for any x0∈U, φ is strongly C1 in x0.Definition 24.
LetU∈τ·J and let V∈τ·I; a function φ:U⊂EJ→V⊂EI is called diffeomorphism if φ is biunique and C1 in U, and the function φ-1:V⊂EI→U⊂EJ is C1 in V.Remark 25.
LetU∈τ·J and let φ:U⊂EJ→EI be a function C1 in x0∈U, where I<+∞, J<+∞; then, φ is strongly C1 in x0.Theorem 26.
LetU∈τ·J, let φ:U⊂EJ→R be a function Ck in x0∈U, let i1,…,ik∈J,  and let j1,…,jk∈J be a permutation of i1,…,ik. Then, one has(18)∂kφ∂xi1⋯∂xikx0=∂kφ∂xj1⋯∂xjkx0.Proof.
The proof is analogous to that one true in the particular caseJ<+∞ (see, e.g., the Lang’s book [9]).Proposition 27.
LetU=∏j∈JUj∩EJ∈τ·J, where Uj∈τ, for any j∈J, and let φ:U⊂EJ→EI be a function C1 in x0∈U, such that(19)φix=∑j∈Jφijxj,∀x=xj:j∈J∈U,∀i∈I,where φij:Uj→R, ∀i∈I and ∀j∈J; moreover, suppose that there exists a neighbourhood V∈τ·J(U) of x0 such that supx∈V⁡Jφ(x)<+∞. Then, φ is continuous in x0; in particular, if φ is strongly C1 in x0 and I<+∞, φ is differentiable in x0.Proof.
Sinceφ is C1 in x0, from (19) there exists a neighbourhood of x0=x0j:j∈J that we can suppose to be V, such that V=∏j∈JVj⊂U, where Vj=x0j-γj,x0j+δj, ∀j∈J, and such that, ∀i∈I and ∀j∈J, φij′ exists in Vj. Let x=xj:j∈J∈V; from the Lagrange theorem, ∀i∈I and ∀j∈J, there exists ξij∈(min⁡x0j,xj,max{x0j,xj})⊂Vj such that φij(xj)-φij(x0j)=φij′ξijxj-x0j. Then,(20)φix-φix0=∑j∈Jφij′ξijxj-x0j⟹φx-φx0=Aφx,x0x-x0,where Aφ(x,x0)=φij′ξiji∈I,j∈J. Thus, if supx∈V⁡Jφ(x)=c<+∞, ∀x∈V, we have(21)Aφx,x0=supi∈I⁡Aφx,x0i≤supi∈I⁡Jφξij:j∈J≤c,from which(22)φx-φx0I≤cx-x0J;then, φ is continuous in x0. Moreover, we have(23)φx-φx0-Jφx0x-x0=Aφx,x0-Jφx0x-x0⟹φx-φx0-Jφx0x-x0Ix-x0J≤Aφx,x0-Jφx0≤supi∈I⁡Jφξij:j∈J-Jφx0.Then, if φ is strongly C1 in x0 and I<+∞, we obtain(24)limx→x0⁡φx-φx0-Jφx0x-x0Ix-x0J=0,and so φ is differentiable in x0, with dφ(x0)=Jφ(x0).Definition 28.
Letm∈N∗, let U=U(m)×∏j∈I∖ImAj∩EI∈τ·I, where U(m)∈τ(m), Aj⊂R is an open interval, for any j∈I∖Im, and let σ:I∖Im→I∖Im be an increasing function. A function φ:U⊂EI→EI is called m,σ-standard if(1)
∀i∈I, there exist some functions φi(m,m):U(m)→R and {φij}j∈I∖Im:Aj→R such that, ∀x∈U, one has φi(x)=φi(m,m)(xi1,…,xim)+∑j∈I∖Imφij(xj);(2)
∀i∈I∖Im and ∀x∈U, one has φi(m,m)(xi1,…,xim)=0 and φij(xj)=0, ∀j≠σ(i);(3)
∀i∈I∖Im, the function gi=φi,σ(i) is constant or injective derivable; moreover, ∀x=(xj:j∈I∖Im)∈∏j∈I∖ImAj, there exists ∏i∈Iφ(-1)i+σigi′(xσ(i))∈R∗, where Iφ={i∈I∖Im:giis injective derivable}.
Ifσ(i)=i, ∀i∈I∖Im, φ is called m-standard; moreover, if the sequence (-1)i+σigi′(xσ(i))i∈Iφ converges uniformly to 1, with respect to x=(xj:j∈I∖Im)∈∏j∈I∖ImAj, then φ is called strongly m,σ-standard.
Furthermore,∀n≥m, define the function φ(n,n):πI,In(U)→Rn by φ(n,n)(x)=φi1(n,n)(x),…,φin(n,n)(x), where(25)φin,nx=φim,mxi1,…,xim+∑j∈In∖Imφijxj,∀x=xj:j∈In∈πI,InU,∀i∈In.Finally, define the m,σ-standard function φ(n):U⊂EI→EI in the following manner:(26)φinx=φin,nxi1,…,xin,∀x=xj:j∈I∈U,∀i∈Im,φinx=φix,∀x=xj:j∈I∈U,∀i∈I∖Im,and indicate φ(m) by φ¯.Remark 29.
Letφ:U⊂EI→EI be a m,σ-standard function. Then,(1)
ifσ is injective, for any n∈N, n≥m, φ is n,σ-standard;(2)
σ is biunique if and only if σ(i)=i, ∀i∈I∖Im;(3)
if∏j∈I∖ImAj⊂EI∖Im, there exist a∈R+ and m0∈N, m0≥m, such that, for any j∈I∖Im0, one has Aj⊂(-a,a).Proof.
Points(1) and (2) follow from the fact that σ is increasing. Moreover, the proof of the point (3) is trivial.Remark 30.
Letφ:U⊂EI→EI be a m,σ-standard function such that ∏j∈I∖ImAj⊂EI∖Im, and σ is injective; then, there exists m1∈N, m1≥m, such that, for any i∈I∖Im1, gi is bounded. In particular, if I=+∞, φ is not surjective.Proof.
Since∏j∈I∖ImAj⊂EI∖Im, from Remark 29, there exists m0∈N, m0≥m, such that, ∀j∈I∖Im0, the set Aj is bounded; then, let H=i∈I∖Im0:giis  not  bounded⊂Iφ; since σ is injective and increasing, we have σ(H)⊂I∖Im0. Moreover, we have H<+∞; indeed, ∀i∈H, the set Aσ(i) is bounded, and so there exists ti∈Aσ(i) such that gi′(ti)>2; by supposing by contradiction H=+∞, ∀x=(xj:j∈I∖Im)∈∏j∈I∖ImAj such that xσ(i)=ti, ∀i∈H, we would obtain ∏i∈Iφgi′(xσ(i))=∏i∈Iφ∖Hgi′(xσ(i))∏i∈Hgi′(ti)=+∞ (a contradiction). Then, there exists m1∈N, m1≥m, such that, ∀i∈I∖Im1, gi is bounded. In particular, ∀i∈I∖Im1, the function gi is not surjective; then, if I=+∞, φ is not surjective.Proposition 31.
Letφ:U⊂EI→EI be a m,σ-standard function; then,(1)
suppose thatφ is injective, πI,H(φ(U))∈τ(H), for any H⊂I∖Im such that 0<H≤2, the function φi:U→R is C1, for any i∈Im, and det⁡Jφ(m,m)(x)≠0, ∀x∈U(m); then, the functions gi, for any i∈I∖Im, and φ(m,m) are injective, and σ is biunique.(2)
ifφ is biunique, then the functions gi, for any i∈I∖Im, φ(m,m), and σ are biunique.Proof of (1).
Suppose thatφ is injective, πI,H(φ(U))∈τ(H), for any H⊂I∖Im such that 0<H≤2, and let i∈I∖Im; we have i∈Iφ, since otherwise we would obtain gi(x)=c, ∀x∈Aσ(i), for some c∈R, from which π{i}(φ(U))=c∉τ=τ({i}), and this should contradict the assumption; then, gi is injective. Moreover, σ must be injective; in fact, by supposing by contradiction that σ(i1)=σ(i2), for some m<i1<i2, then(27)πI,i1,i2φU=y1,y2∈R2:y1=gi1x,y2=gi2x,for  somex∈Aσi1=y1,y2∈R2:y1∈gi1Aσi1,y2=gi2gi1-1y1∉τ2=τi1,i2(a contradiction). Moreover, σ is surjective; in fact, suppose by contradiction that there exists n∈I∖Im∖σI∖Im; since φ is injective, ∀y∈φ(U), there is a unique x=(xj:j∈I)∈U such that φ(x)=y, and so(28)yi=φix=φim,mxi1,…,xim+φinxn+∑j∈I∖Im∪nφijxj,∀i∈Im⟹φim,mxi1,…,xim+φinxn-yi+∑j∈I∖Im∪nφijxj=0,∀i∈Im.Then, consider the function F:U(m)×An→Rm defined by(29)Fiz,t=φim,mz+φint-yi+∑j∈I∖Im∪nφijxj,∀z∈Um,∀t∈An,∀i∈Im.From (28) and by assumption, we have(30)Fxi1,…,xim,xn=0,det⁡∂F∂zxi1,…,xim,xn≠0;then, there exist a neighbourhood V1⊂An of xn and a neighbourhood V2⊂U(m) of (xi1,…,xim) such that, ∀t∈V1, there exists a unique z=f(t)∈V2 such that Fz,t=0 (a contradiction with the uniqueness of x=(xj:j∈I)∈U such that φ(x)=y). Finally, let x,y∈U(m) be such that φ(m,m)(x)=φ(m,m)(y), and let x¯,y¯∈U be such that x¯i=xi and y¯i=yi, ∀i∈Im, x¯i=y¯i, ∀i∈I∖Im. We have(31)φix¯=φim,mx+∑j∈I∖Imφijx¯j=φim,my+∑j∈I∖Imφijy¯j=φiy¯,∀i∈Im;φix¯=gix¯σi=giy¯σi=φiy¯,∀i∈I∖Im,from which φ(x¯)=φ(y¯); then, since φ is injective, we have x¯=y¯, and so x=y. Then, the function φ(m,m) is injective.Proof of (2).
Suppose thatgi is injective, ∀i∈I∖Im, and let x,y∈U be such that φ(x)=φ(y); then, ∀i∈I∖Im we have gi(xσ(i))=φi(x)=φi(y)=gi(yσ(i)), from which xσ(i)=yσ(i); then, if σ is biunique, from Remark 29, we have σ(i)=i, and so (xi:i∈I∖Im)=(yi:i∈I∖Im). This implies that(32)φim,mxi1,…,xim=φix-∑j∈I∖Imφijxj=φiy-∑j∈I∖Imφijyj=φim,myi1,…,yim,∀i∈Im⟹φm,mxi1,…,xim=φm,myi1,…,yim,and so, if φ(m,m) is injective, we have (xi1,…,xim)=(yi1,…,yim); then, x=y; that is, φ is injective.Proposition 32.
Letφ:U⊂EI→EI be a m,σ-standard function; then:(1)
Suppose thatφ is biunique, the function φi:U→R is C1, for any i∈Im, and det⁡Jφm,mx≠0, for any x∈U(m). Then, the functions gi, for any i∈I∖Im, σ and φ(m,m) are biunique.(2)
If the functionsgi, for any i∈I∖Im, σ and φ(m,m) are biunique, then φ is biunique.Proof of (1).
Ifφ is biunique, ∀H⊂I∖Im such that 0<H≤2; we have πI,HφU=RH∈τ(H); then, from Proposition 31, the functions gi, ∀i∈I∖Im, and φ(m,m) are injective, and σ is biunique; thus, ∀i∈I∖Im, we have σ(i)=i. Moreover, ∀i∈I∖Im, gi is surjective, since φ is surjective. Furthermore, ∀y=(yi1,…,yim)∈Rm, let (xj:j∈I∖Im)∈∏j∈I∖ImAj∩EI∖Im and let y¯∈EI be such that(33)y¯i=yi+∑j∈I∖Imφijxj,∀i∈Im,y¯i=gixi,∀i∈I∖Im;moreover, let x¯=φ-1(y¯)∈U, from which y¯i=gi(x¯i), ∀i∈I∖Im. Since gi is injective, we have x¯i=xi, ∀i∈I∖Im; then, ∀i∈Im, we have(34)φim,mx¯i1,…,x¯im+∑j∈I∖Imφijxj=y¯i=yi+∑j∈I∖Imφijxj⟹yi=φim,mx¯i1,…,x¯im⟹y=φm,mx,where x=(x¯i1,…,x¯im)∈U(m); then, φ(m,m) is surjective.Proof of (2).
If the functionsgi, ∀i∈I∖Im, σ and φ(m,m) are biunique, from Proposition 31, we obtain that φ is injective. Moreover, ∀y∈EI, define x∈U(m)×∏j∈I∖ImAj in the following manner:(35)xi=gi-1yi∈Ai,∀i∈I∖Im,xi1,…,xim=φm,m-1zi1,…,zim∈Um,where(36)zi=yi-∑j∈I∖Imφijxj,∀i∈Im.Let x0=(x0,i:i∈I)∈U; ∀i∈I∖Im, we have(37)xi=gi-1yi-x0,i+x0,i≤gi-1yi-gi-1gix0,i+x0,i;moreover, the function gi-1:R→Ai is derivable, and(38)gi-1′t=1gi′gi-1t∈R∗,∀i∈I∖Im,∀t∈R;then, the Lagrange theorem implies that, for some ξi∈min{yi,gi(x0,i)},max{yi,gi(x0,i)}, we have(39)gi-1yi-gi-1gix0,i=gi-1′ξiyi-gix0,i;thus, from (37) and (38), we obtain(40)xi≤yi-gix0,igi′gi-1ξi+x0,i.
Moreover, we have∏i∈I∖Imgi′(gi-1(ξi))=∏i∈Iφgi′gi-1ξi∈R+, from which there exists i0∈I∖Im such that ∀i∈I∖Im, i>i0, we have gi′gi-1ξi>1/2; then, there exists c∈R+ such that supi∈I∖Im⁡gi′gi-1ξi-1≤c, and so formula (40) implies(41)supi∈I∖Im⁡xi≤cyI+φx0I+x0I<+∞;then, we have x∈EI, from which x∈U. Finally, it is easy to prove that φ(x)=y, and so φ is surjective.Remark 33.
Letφ:U⊂EI→EI be a m,σ-standard function such that φij(xj)=0, for any xj∈Aj, for any i∈Im, and for any j∈I∖Im; then,(1)
ifφ is injective, and πI,HφU∈τ(H), for any H⊂I∖Im such that 0<H≤2, then the functions gi, for any i∈I∖Im, and φ(m,m) are injective, and σ is biunique.(2)
ifφ is biunique, then the functions gi, for any i∈I∖Im, φ(m,m) and σ are biunique.Proof of (1).
Suppose thatφ is injective; by proceeding as in the proof of Proposition 31, we have that the functions gi, ∀i∈I∖Im, φ(m,m), and σ are injective. Moreover, σ is surjective; in fact, suppose by contradiction that there exists n∈I∖Im∖σI∖Im, and let x=(xj:j∈I); moreover, ∀t∈An, t≠xn, let x¯=(x¯j:j∈I)∈U be the sequence defined by x¯j=xj, ∀j≠n, and x¯n=t; we have(42)φix=φim,mxi1,…,xim=φim,mx¯i1,…,x¯im=φix¯,∀i∈Im,φix=gixσi=gix¯σi=φix¯,∀i∈I∖Im;thus, we have φ(x)=φ(x¯), and so φ is not injective (a contradiction).Proof of (2).
Sinceφ is surjective, the functions gi, ∀i∈I∖Im, and φ(m,m) are surjective; moreover, ∀H⊂I∖Im such that 0<|H|≤2, we have πI,H(φ(U))=RH∈τ(H); then, since φ is injective, from point 1, the functions gi, ∀i∈I∖Im, and φ(m,m) are injective, and σ is biunique.Corollary 34.
Letφ:U⊂EI→EI be a m,σ-standard function such that πI,HφU∈τ(H), for any H⊂I∖Im such that 0<H≤2, and φ¯ is injective; then, the functions φ, φ(n), and φ(n,n), for any n∈N, n≥m, are injective.Proof.
Observe thatφ¯ is m,σ-standard, and πI,Hφ¯U=πI,HφU∈τ(H), ∀H⊂I∖Im such that 0<H≤2; then, from Remark 33, we have that the functions gi, ∀i∈I∖Im, and φ(m,m) are injective, and σ is biunique; then, from Proposition 31, φ is injective; analogously, since ∀n∈N, n≥m, the function φ(n) is n,σ-standard, from Proposition 31  φ(n) is injective, Moreover, we have πI,H(φ(n)(U))=πI,H(φ(U))∈τ(H), ∀H⊂I∖In such that 0<H≤2; then, from Remark 33, φ(n,n)=φ(n)(n,n) is injective.Proposition 35.
Letφ:U⊂EI→EI be a function C1 in x0=(x0,j:j∈I)∈U and m,σ-standard. Then, for any n≥m, the function φ(n,n):πI,In(U)→Rn is C1 in x0,j:j∈In, the function φ(n):U⊂EI→EI is C1 in x0, there exists the function Jφ(n)(x0):EI→EI, and it is continuous. Moreover, if φ is C1 in x0 and strongly m,σ-standard, then φ(n) is differentiable in x0. Finally, if φ is strongly C1 in x0 and strongly m,σ-standard, then φ is differentiable in x0.Proof.
By assumption, there exists a neighbourhoodV=∏j∈IVj∈τ·I(U) of x0 such that, ∀i,j∈I, there exists the function x→∂φi(x)/∂xj on V, and this function is continuous in x0; then, ∀x=xj:j∈In∈∏j∈InVj, let x∈V such that (xj:j∈In)=x; since φ is a m,σ-standard function, ∀i,j∈In, we have(43)∂φin,nx∂xj=∂φix∂xj,from which φ(n,n) is C1 in x0,j:j∈In. Moreover, ∀x∈V, we have(44)∂φinx∂xj=∂φix∂xjifi,j∈I×Im∪I∖Im×I∖Im0ifi,j∈Im×I∖Im,and so φ(n) is C1 in x0. Furthermore, ∀i∈I, the function φi(n):U⊂EI→R is differentiable in x0, since φi(n) depends only on a finite number of variables, and so there exists the function Jφi(n)(x0):EI→R; moreover, since ∀i∈I∖Im we have Jφi(n)(x0)=gi′(x0,σ(i)) and since the sequence gi′(x0,σ(i))i∈Iφ is convergent, this implies that supi∈I∖Im⁡gi′(x0,σ(i))<+∞, and so supi∈I⁡Jφi(n)(x0)<+∞; then, there exists the function Jφ(n)(x0):EI→EI, and it is continuous.
Moreover, suppose thatφ is C1 in x0 and strongly m,σ-standard; then, the sequence (-1)i+σigi′(xσ(i))i∈Iφ converges uniformly to 1, with respect to x=(xj:j∈I∖Im)∈∏j∈I∖ImAj; thus, ∀ε>0, there exists i¯∈I∖Im such that, ∀i∈Iφ, i>i¯, and ∀x∈U, we have(45)-1i+σigi′xσi-1<ε2.Observe that, since ∀i≤i¯ the function φi(n) is differentiable in x0, there exists a neighbourhood N∈τ·I(U) of x0 such that, ∀x∈N∖{x0}, we have(46)supi∈I:i≤i¯⁡φinx-φinx0-Jφinx0x-x0x-x0J<ε.Moreover, ∀i∈I such that i>i¯, gi is derivable in Aσ(i) and so, from the Lagrange theorem, there exists ξi∈(min{x0,σ(i),xσ(i)},max{x0,σ(i),xσ(i)}) such that(47)gixσi-gix0,σi=gi′ξixσi-x0,σi,from which(48)φinx-φinx0-Jφinx0x-x0x-x0J=gixσi-gix0,σi-gi′x0,σixσi-x0,σix-x0J=-1i+σigi′ξi--1i+σigi′x0,σixσi-x0,σix-x0J≤-1i+σigi′ξi-1+-1i+σigi′x0,σi-11Iφi<ε.Then, from (46) and (48), we have(49)φnx-φnx0-Jφnx0x-x0Ix-x0J<ε,and so φ(n) is differentiable in x0.
Finally, ifφ is strongly C1 in x0 and strongly m,σ-standard, the function ψ=φ-φ¯:U⊂EI→EI given by(50)ψix=∑j∈I∖Imφijxj∀i∈Im,∀x∈U0∀i∈I∖Im,∀x∈Uis strongly C1 in x0, and so it is differentiable in x0 from Proposition 27; then, since φ¯ is differentiable in x0, this is true for φ=ψ+φ¯ too, from Remark 14.Proposition 36.
Letφ:U⊂EI→EI be a function C1 and m,σ-standard; then, φ:U→RI is B(I)(U),B(I)-measurable.Proof.
Leti∈I, and let n∈N, n≥max{i,m}; from Proposition 35, the function φi(n,n):πI,In(U)→R is C1. Then, ∀B∈τ, we have φi(n,n)-1(B)∈τ(n)⊂B(n); moreover, if we consider φi(n,n) as a function from U to R, we have πI,Inφin,n-1B=RI∖In, and so φi(n,n)-1(B)∈B(I); then, since σ(τ)=B, ∀B∈B, we have φi(n,n)-1(B)∈B(I), and so φi(n,n)-1(B)∈B(I)(U). Moreover, since limn→+∞⁡φi(n,n)=φi, the function φi is B(I)(U),B-measurable. Let(51)ΣI=B=∏i∈IBi:Bi∈B,∀i∈I;∀B∈Σ(I), we have(52)φ-1B=⋂i∈Iφi-1Bi∈BIU.Finally, since σΣ(I)=B(I), ∀B∈B(I), we have φ-1(B)∈B(I)(U).Proposition 37.
Letφ:U⊂EI→EI be a m,σ-standard function, such that φ¯:U→φ¯(U) is a homeomorphism. Then, the functions φ(m,m):U(m)→φ(m,m)U(m) and gi:Ai→gi(Ai), ∀i∈I∖Im, are homeomorphisms, and σ is biunique.Proof.
Sinceφ¯ is a homeomorphism, we have φ¯(U)∈τ·I; then, from Proposition 6, ∀H⊂I∖Im such that 0<H≤2, we have πI,Hφ¯U∈τ·H=τ(H); thus, since φ¯ is injective, from Remark 33, the functions gi, ∀i∈I∖Im, and φ(m,m) are injective, and σ is biunique; then, the functions gi and gi-1, ∀i∈I∖Im, are derivable, and so they are continuous. Moreover, we have φ(m,m)U(m)=πI,Imφ¯(U)∈τ(m), gi(Ai)=πI,{i}φ¯(U)∈τ, ∀i∈I∖Im. Finally, from Remark 10, the function πI,Im∘φ:U,τ·I(U)→Rm,τ(m) is continuous; then, ∀B∈τ(m), we have φ(m,m)-1(B)×∏j∈I∖ImAj=πI,Im∘φ-1(B)∈τ·I(U), from which φ(m,m)-1(B)∈τ(m)(U(m)); then, φ(m,m) is continuous; analogously, we can prove that the function φ(m,m)-1 is continuous.Proposition 38.
Letφ:U⊂EI→EI be a m,σ-standard function such that πI,H(φ(U))∈τ(H), for any H⊂I∖Im such that 0<H≤2. Then, φ¯:U→φ¯(U) is a diffeomorphism if and only if the functions φ(m,m):U(m)→φ(m,m)U(m) and gi:Ai→gi(Ai), ∀i∈I∖Im, are diffeomorphisms, and σ is biunique.Proof.
We haveπI,H(φ¯(U))=πI,H(φ(U))∈τ(H), for any H⊂I∖Im such that 0<H≤2. If φ¯ is a diffeomorphism, then φ¯ is injective, and so, from Remark 33, the functions gi, ∀i∈I∖Im, and φ(m,m) are injective, and σ is biunique. Moreover, φ¯ is C1 in U, and so, from Proposition 35, φ(m,m)=φ¯(m,m) is C1 in U(m)=πI,ImU; analogously, since φ¯-1:φ¯(U)→U is C1 in φ¯(U), then φ(m,m)-1=φ¯-1(m,m) is C1 in φ(m,m)U(m)=πI,Imφ¯(U)∈τ(m); then, φ(m,m) is a diffeomorphism. Moreover, ∀i∈I∖Im, ∀x∈Ai, let x¯=(x¯j:j∈I)∈U such that x¯i=x; we have gi′(x)=∂φ¯i/∂xi(x¯), and so gi is C1 in Ai; analogously, ∀i∈I∖Im, ∀y∈gi(Ai), let y¯=(y¯j:j∈I)∈φ¯(U) such that y¯i=y; we have gi-1′(y)=∂φ¯-1i/∂yi(y¯), and so gi-1 is C1 in gi(Ai)=πI,{i}φ¯(U)∈τ; then, gi is a diffeomorphism.
Conversely, if the functionsφ(m,m) and gi, ∀i∈I∖Im, are diffeomorphisms and σ is biunique, then φ¯ is injective from Remark 33; moreover, ∀x=(xj:j∈I)∈U, set x=(xj:j∈Im)∈U(m); we have(53)∂φ¯ix∂xj=∂φim,mx∂xjifi,j∈Im×Imgi′xiifi∈I∖Im,j=i0otherwise,and so φ¯ is C1 in U; analogously, ∀y=(yj:j∈I)∈φ¯(U), set y=(yj:j∈Im)∈πI,Imφ¯(U); we have(54)∂φ¯-1iy∂yj=∂φ-1im,my∂yjifi,j∈Im×Imgi-1′yiifi∈I∖Im,j=i0otherwise,and so φ¯-1 is C1 in φ¯(U); then, φ¯ is a diffeomorphism.Proposition 39.
Letφ:U⊂EI→EI be a m,σ-standard function such that φi:U→R is C1, for any i∈Im; moreover, suppose that φ¯ is injective and φ¯(U)∈τ·I; then,(1)
ifφ(m,m):U(m),τ(m)(U(m)))→(φ(m,m)U(m),τ(m)φ(m,m)U(m) is an open function, then, for any n∈N, n≥m, the function φ(n,n):πI,InU→φ(n,n)πI,InU is a homeomorphism, and one has φ(n)(U)∈τ·I.(2)
ifφ¯:U→φ¯(U) is a diffeomorphism, then, for any n∈N, n≥m, the functions φ(n,n):πI,InU→φ(n,n)πI,InU and φ(n):U→φ(n)U are diffeomorphisms.Proof of (1).
Sinceφ¯(U)∈τ·I, ∀H⊂I∖Im such that 0<H≤2, we have πI,H(φ(U))=πI,H(φ¯(U))∈τ·H=τ(H); then, from Corollary 34, ∀n∈N, n≥m, the function φ(n,n) is injective. Moreover, from Remark 33, the functions gi, ∀i∈I∖Im, and φ(m,m) are injective, and σ is biunique. Furthermore, from Proposition 35 and the continuity of the functions gi, ∀i∈In∖Im, we have that φ(n,n) is continuous in πI,InU. Finally, ∀y∈φ(n,n)πI,InU, let x=φ(n,n)-1(y)∈πI,InU; ∀i∈In∖Im, we have xi=gi-1(yi); then, ∀i∈Im,(55)yi=φim,mxi1,…,xim+∑j∈In∖Imφijgj-1yj,and so(56)xi1,…,xim=φm,m-1zi1,…,zim,xi=gi-1yi,∀i∈In∖Im,where(57)zi=yi-∑j∈In∖Imφijgj-1yj,∀i∈Im.Then, since the functions φ(m,m)-1, φij, and gj-1 are continuous, the function φ(n,n)-1 is continuous too, and so φ(n,n)πI,InU∈τ(n); then, φ(n,n):πI,InU→φ(n,n)πI,InU is a homeomorphism. Furthermore, since φ¯(U)∈τ·I, we have πI,I∖Inφ¯(U)∈τ·I∖In, and so(58)φnU=φn,nπI,InU×πI,I∖Inφ¯U∈τ·I.Proof of (2).
Ifφ¯:U→φ¯(U) is a diffeomorphism, from Proposition 38, the functions φ(m,m):U(m)→φ(m,m)U(m) and gi:Ai→gi(Ai), ∀i∈I∖Im, are diffeomorphisms; in particular, the functions φ(m,m)-1 and gi-1, ∀i∈In∖Im, are C1; then, since ∀i∈Im, ∀j∈In∖Im, the function φij is C1, from (56) and (57), we obtain that the function φ(n,n)-1 is C1 in φ(n,n)πI,InU. Moreover, from Proposition 35 and since, ∀i∈In∖Im, the functions gi are C1, φ(n,n) is C1 in πI,InU, and so φ(n,n) is a diffeomorphism. Finally, since φ¯(U)∈τ·I, ∀H⊂I∖In such that 0<H≤2, we have πI,H(φ(n)(U))=πI,H(φ¯(U))∈τ·H=τ(H); then, from Proposition 38, φ(n):U→φ(n)U is a diffeomorphism.Remark 40.
A linear functionA=aiji,j∈I:EI→EI is m,σ-standard, where m∈N∗ and σ:I∖Im→I∖Im is an increasing function, if(1)
aij=0, ∀(i,j)∉Im×I∪⋃n∈I∖Im{(n,σ(n))};(2)
there exists∏i∈I∖Im:λi≠0λi∈R∗, where λi=(-1)i+σiai,σ(i), ∀i∈I∖Im.Recall the following concept, defined in [1].Definition 41.
LetA=aiji,j∈I:EI→EI be a linear m,σ-standard function; define the determinant of A, and call it detm,σA, or det⁡A, the real number(59)detm,σA=det⁡Am,m∏i∈I∖Imλiifσis  biunique0ifσis  not  biunique,where A(m,m) is the m×m real matrix defined by A(m,m)(i,j)=aij, ∀i,j∈Im.Proposition 42.
LetA=(aij)i,j∈I:EI→EI be a linear and injective m,σ-standard function; then, A is biunique if and only if A is (τ·I,τ·I)-open.Proof.
From Remark9 in [1], A is continuous; then, if A is biunique, from the Banach theorem of the open function, A is τ·I,τ·I-open. Conversely, if A is τ·I,τ·I-open, ∀H⊂I∖Im such that 0<|H|≤2, we have πI,H(A(EI))∈τ·H=τ(H); moreover, since A is linear, the function Ai:EI→R is C1, ∀i∈Im; furthermore, by proceeding as in the proof of Proposition 31, we obtain that A(m,m) is injective, and so det⁡JA(m,m)(x)≠0, ∀x∈EI; then, from Proposition 31, the functions gi, ∀i∈I∖Im, are injective, and σ is biunique. Thus, since the functions A(m,m) and gi, ∀i∈I∖Im, are linear, they are biunique, and so A is biunique from Proposition 32.


## 4. Change of Variables’ Formula

Henceforth, we will supposeI=+∞.Definition 43.
Letk∈N∗, let M,N∈R+, let a=ai:i∈I∈[0,+∞)I such that ∏i∈I:ai≠0ai∈R+, and let v=vi:i∈I∈EI; define the following sets in BI:(60)EN,a,vk,I=Rk×∏i∈I∖Ikvi-Nai,vi+Nai;EM,N,a,vk,I=-M,Mk×∏i∈I∖Ikvi-Nai,vi+Nai.Moreover, define the σ-finite measure λN,a,v(k,I) over RI,B(I) in the following manner:(61)λN,a,vk,I=⨂i∈Ik12NLeb⊗⨂i∈I∖Ik12NLeb·∩vi-Nai,vi+Nai.Proposition 44.
Letφ:U⊂EI→EI be a m,σ-standard function such that φ¯ is biunique; moreover, let N∈R+, let a=ai:i∈I∈[0,+∞)I such that ∏i∈I:ai≠0ai∈R+, and let v∈EI; then,(1)
there existb=bi:i∈I∈[0,+∞)I and z∈EI such that ∏i∈I:bi≠0bi∈R+ and such that, for any n,k∈N, n≥m, k≥m, one has(62)φ-1EN,a,vk,I=φn-1EN,a,vk,I=EN,b,zk,I;(2)
suppose thatφi:U→R is C1, for any i∈Im, and the function φ(m,m):U(m),τ(m)(U(m)))→(Rm,τ(m) is open; then, for any M1∈R+, there exists M2∈R+ such that, for any n,k∈N, n≥m, k≥m, one has(63)φ-1EM1,N,a,vk,I⊂EM2,N,b,zk,I,φn-1EM1,N,a,vk,I⊂EM2,N,b,zk,I.Proof of (1).
Sinceφ¯ is biunique, from Remark 33, the functions gi, ∀i∈I∖Im, and σ are biunique. Set b=bi:i∈I∈[0,+∞)I, z=zi:i∈I∈EI such that bi=zi=1, ∀i∈Im; moreover, define(64)bi=gi-1vi+Nai-gi-1vi-Nai2N,∀i∈I∖Im,zi=gi-1vi+Nai+gi-1vi-Nai2,∀i∈I∖Im.Observe that, ∀i∈I∖Im, we have bi≠0 if and only if ai≠0; then, by definition of b=bi:i∈I, we have(65)∏i∈I:bi≠0bi=∏i∈I:ai≠0bi=∏i∈I∖Im:ai≠0gi-1vi+Nai-gi-1vi-Nai2Nai∏i∈I∖Im:ai≠0ai.Moreover, since ∀i∈I∖Im the function gi-1 is derivable on R, if ai≠0, the Lagrange theorem implies that, for some ξi∈(vi-Nai,vi+Nai), we have(66)gi-1vi+Nai-gi-1vi-Nai2Nai=gi-1′ξi=1gi′gi-1ξi,from which ∏i∈I:bi≠0bi∈R+. Furthermore, ∀i∈I∖Im, we have(67)gi-1vi-Nai,vi+Nai=gi-1vi-Nai,gi-1vi+Naiifgiis  increasinggi-1vi+Nai,gi-1vi-Naiifgiis  decreasing=zi-Nbi,zi+Nbi,from which(68)gizi-Nbi,zi+Nbi=vi-Nai,vi+Nai.Then, from (67) and (68), ∀k∈N, k≥m, we obtain(69)φ-1EN,a,vk,I⊂EN,b,zk,I,φEN,b,zk,I⊂EN,a,vk,I⟹EN,b,zk,I⊂φ-1EN,a,vk,I⟹φ-1EN,a,vk,I=EN,b,zk,I.Analogously, ∀n∈N, n≥m, from (67) and (68), we have φ(n)-1EN,a,v(k,I)=EN,b,z(k,I).Proof of (2).
Suppose thatφi:U→R is C1, for any i∈Im, and φ(m,m):U(m),τ(m)(U(m)))→(Rm,τ(m) is an open function; thus, ∀k∈N, k≥m, and ∀x=xi:i∈I∈φ-1EM1,N,a,v(k,I), let y=φ(x)∈EM1,N,a,v(k,I); ∀i∈I∖Im, we have xi=gi-1(yi); then, ∀i∈Im,(70)yi=φim,mxi1,…,xim+∑j∈I∖Imφijgj-1yj,and so(71)xi1,…,xim=φm,m-1zi1,…,zim,xi=gi-1yi,∀i∈I∖Im,where(72)zi=yi-∑j∈I∖Imφijgj-1yj,∀i∈Im.Moreover, since ∀i∈Im, ∀j∈I∖Im, the functions φij and gj-1 are continuous and yj≤maxM1,vI∖Im+NaI∖Im, there exists M¯∈R+ such that(73)supi∈Im⁡∑j∈I∖Imφijgj-1yj≤M¯,and so (zi1,…,zim)Im≤M1+M¯; then, since the function φ(m,m)-1 is continuous, there exists M~∈R+ such that (xi1,…,xim)Im<M~; finally, if b, z are the sequences defined by the point (1), by setting M2=maxM~,zI∖Im+NbI∖Im, from (67), we have φ-1EM1,N,a,v(k,I)⊂EM2,N,b,z(k,I). Analogously, ∀n∈N, n≥m, we have(74)supi∈Im⁡∑j∈In∖Imφijgj-1yj≤M¯,from which φ(n)-1EM1,N,a,v(k,I)⊂EM2,N,b,z(k,I).Proposition 45.
LetS,Σ be a measurable space, let I be a π-system on S, and let μ1 and μ2 be two measures on S,Σ, σ-finite on I; if σ(I)=Σ and μ1 and μ2 coincide on I, then μ1 and μ2 coincide on Σ.Proof.
See, for example, Theorem10.3 in Billingsley [10].Lemma 46.
Letk∈N∗, let N∈R+, let a=ai:i∈I∈[0,+∞)I such that ∏i∈I:ai≠0ai∈R+, and let v=vi:i∈I∈EI; then, for any measurable function f:RI,B(I)→(R,B) such that f+ (or f-) is λN,a,v(k,I)-integrable, one has(75)∫RIfdλN,a,vk,I=∫EN,a,vk,IfdλN,a,vk,I.Proof.
LetB=∏i∈IBi∈B(I); by definition of λN,a,v(k,I), we have(76)∫RI1BdλN,a,vk,I=∫EN,a,vk,I1BdλN,a,vk,I;then, consider the measures μ1=λN,a,v(k,I) and μ2=λN,a,v(k,I)·∩EN,a,v(k,I) on RI,B(I); from (76), μ1 and μ2 coincide on the set I=B∈B(I):B=∏i∈IBi; moreover, we have RI=⋃n∈N[-n,n]k×RI∖Ik, where [-n,n]k×RI∖Ik∈I, μ1[-n,n]k×RI∖Ik=μ2[-n,n]k×RI∖Ik<+∞, ∀n∈N, and so μ1 and μ2 are σ-finite on I. Then, since I is a π-system on RI such that σ(I)=B(I), from Proposition 45, formula (76) is true ∀B∈B(I). This implies that, if ψ:RI,B(I)→([0,+∞),B[0,+∞)) is a simple function, we have(77)∫RIψdλN,a,vk,I=∫EN,a,vk,IψdλN,a,vk,I.Then, if f:RI,B(I)→([0,+∞),B[0,+∞)) is a measurable function, and ψnn∈N is a sequence of increasing positive simple functions over RI,B(I) such that limn→+∞⁡ψn=f, from Beppo Levi theorem we have(78)∫RIfdλN,a,vk,I=limn→+∞⁡∫RIψndλN,a,vk,I=limn→+∞⁡∫EN,a,vk,IψndλN,a,vk,I=∫EN,a,vk,IfdλN,a,vk,I.Then, for any measurable function f:RI,B(I)→(R,B) such that f+ (or f-) is λN,a,v(k,I)-integrable,(79)∫RIfdλN,a,vk,I=∫RIf+dλN,a,vk,I-∫RIf-dλN,a,vk,I=∫EN,a,vk,If+dλN,a,vk,I-∫EN,a,vk,If-dλN,a,vk,I=∫EN,a,vk,IfdλN,a,vk,I.Now, we can prove the main result of our paper, that improves Theorem29 in [1] and generalizes the change of variables’ formula for the integration of a measurable function on Rm with values in R (see, e.g., the Lang’s book [9]).Theorem 47 (change of variables’ formula).
Letφ:U⊂EI→EI be a C1 and m,σ-standard function, such that the function φ¯:U→EI is a diffeomorphism; moreover, let N∈R+, let a=ai:i∈I∈[0,+∞)I such that ∏i∈I:ai≠0ai∈R+, let v∈EI, and let b∈[0,+∞)I and z∈EI defined by Proposition 44. Then, for any k∈N, k≥m, for any B∈B(I), and for any measurable function f:RI,B(I)→(R,B) such that f+ (or f-) is λN,a,v(k,I)-integrable, one has(80)∫BfdλN,a,vk,I=∫φ-1Bfφdet⁡Jφ¯dλN,b,zk,I.Proof.
Letn,k∈N, n≥k≥m; moreover, since φ¯ is biunique, ∀M1,N∈R+, ∀a=ai:i∈I∈[0,+∞)I such that ∏i∈I:ai≠0ai∈R+, and ∀v∈EI, let M2∈R+ and let b and z be the sequences defined by Proposition 44. Then, ∀B=∏i∈IBi∈B(I)EM1,N,a,v(k,I), we have Bi∈B(vi-Nai,vi+Nai) and gi-1(Bi)∈B(zi-Nbi,zi+Nbi), ∀i∈I∖Ik, and so(81)∫BdλN,a,vk,I=∫Bi1×⋯×Bik×∏q∈I∖IkBqd⨂p=1k12NLeb⊗⨂q∈I∖Ik12NLebBvq-Naq,vq+Naq=∫Bi1×⋯×Bin×∏q∈I∖InBqd⨂p=1n12NLeb⊗⨂q∈I∖In12NLebBvq-Naq,vq+Naq=∫Bi1×⋯×Bind⨂p=1n12NLeb·∫∏q∈I∖InBqd⨂q∈I∖In12NLebBvq-Naq,vq+Naq.
Moreover, we have(82)∫∏q∈I∖InBqd⨂q∈I∖In12NLebBvq-Naq,vq+Naq=∫∏q∈I∖InBqd⨂q∈I∖In12NLebBBq=limp→+∞⁡∫∏q∈Ip∖InBqd⨂q∈Ip∖In12NLebBBqby  Theorem  1=limp→+∞⁡∫∏q∈Ip∖Ingq-1Bq∏q∈Ip∖Indet⁡Jgqxq·d⨂q∈Ip∖In12NLebBgq-1BqxIp∖Inby  Proposition  38=limp→+∞⁡1∏q∈I∖Ipdet⁡Jgqxq∫∏q∈Ip∖Ingq-1Bq∏q∈I∖Indet⁡Jgqxq·d⨂q∈Ip∖In12NLebBgq-1BqxIp∖In=∫∏q∈I∖Ingq-1Bq∏q∈I∖Indet⁡Jgqxq·d⨂q∈I∖In12NLebBgq-1BqxI∖Inby  Corollary  3=∫∏q∈I∖Ingq-1Bq∏q∈I∖Indet⁡Jgqxq·d⨂q∈I∖In12NLebBzq-Nbq,zq+NbqxI∖In.Then, from Proposition 39, formula (81) implies(83)∫BdλN,a,vk,I=∫φn,n-1Bii×⋯×Bindet⁡Jφn,nd⨂p=1n12NLeb·∫∏q∈I∖Ingq-1Bq∏q∈I∖Indet⁡Jgqxq·d⨂q∈I∖In12NLebBzq-Nbq,zq+NbqxI∖In=∫φn-1Bdet⁡Jφnd⨂p=1n12NLeb⊗⨂q∈I∖In12NLebBzq-Nbq,zq+Nbq=∫φn-1Bdet⁡Jφnd⨂p=1k12NLeb⊗⨂q∈I∖Ik12NLebBzq-Nbq,zq+Nbqsinceφn-1B⊂EN,b,zk,I=∫φn-1Bdet⁡Jφ¯dλN,b,zk,I.Consider the measures μ1 and μ2 on Σ≡B(I)EM1,N,a,v(k,I) defined by(84)μ1B=∫BdλN,a,vk,I,μ2B=∫φn-1Bdet⁡Jφ¯dλN,b,zk,I;from (83), μ1 and μ2 coincide on the set(85)I=B∈Σ:B=∏i∈IBi;moreover, we have μ1EM1,N,a,v(k,I)=μ2EM1,N,a,v(k,I)<+∞, EM1,N,a,v(k,I)∈I, and so μ1 and μ2 are σ-finite on I. Then, since I is a π-system on EM1,N,a,v(k,I) such that σ(I)=Σ, from Proposition 45, ∀B∈B(I)EM1,N,a,v(k,I), we have(86)∫BdλN,a,vk,I=∫φn-1Bdet⁡Jφ¯dλN,b,zk,I;moreover, since φ(n)-1(B)⊂EM2,N,b,z(k,I) from Proposition 44, we have(87)∫EM1,N,a,vk,I1BdλN,a,vk,I=∫EM2,N,b,zk,I1Bφndet⁡Jφ¯dλN,b,zk,I.This implies that, if ψ:RI,B(I)→([0,+∞),B[0,+∞)) is a simple function such that ψ(x)=0,∀x∉EM1,N,a,v(k,I), we have(88)∫EM1,N,a,vk,IψdλN,a,vk,I=∫EM2,N,b,zk,Iψφndet⁡Jφ¯dλN,b,zk,I.Then, if l:RI,B(I)→([0,+∞),B[0,+∞)) is a measurable function such that l(x)=0, ∀x∉EM1,N,a,v(k,I), and ψii∈N is a sequence of increasing positive simple functions over RI,B(I) such that limi→+∞⁡ψi=l, ψi(x)=0,∀x∉EM1,N,a,v(k,I), ∀i∈N, from Beppo Levi theorem we have (89)∫EM1,N,a,vk,IldλN,a,vk,I=limi→+∞⁡∫EM1,N,a,vk,IψidλN,a,vk,I=limi→+∞⁡∫EM2,N,b,zk,Iψiφndet⁡Jφ¯dλN,b,zk,I=∫EM2,N,b,zk,Ilφndet⁡Jφ¯dλN,b,zk,I=limn→+∞⁡∫EM2,N,b,zk,Ilφndet⁡Jφ¯dλN,b,zk,I.
In particular, formula (89) is true if l is a function of the form(90)lx=lmxi1,…,xik∏j∈I∖Ik1Bjxj,∀x=xj:j∈I∈RI,where l(k):Rk→[0,1] is a continuous function such that lk(x)=0,∀x∉(-M1,M1)k, and Bj∈B(vj-Naj,vj+Naj), ∀j∈I∖Ik. In this case, let fnn≥k be the sequence of the measurable functions fn:EM2,N,b,z(k,I),B(I)(EM2,N,b,z(k,I))→([0,+∞),B[0,+∞)) given by(91)fnx=lφnxdet⁡Jφ¯x,∀x∈EM2,N,b,zk,I,∀n≥k;∀n≥k, we have fn≤g, where g:EM2,N,b,z(k,I),B(I)(EM2,N,b,z(k,I))→([0,+∞),B[0,+∞)) is the measurable function defined by(92)gx=det⁡Jφ¯x,∀x∈EM2,N,b,zk,I.Moreover, we have(93)gx=det⁡Jφm,mxi1,…,xik∏j∈I∖Ikgj′xj,∀x∈EM2,N,b,zk,I;then, by definition of EM2,N,b,z(k,I) and since the functions Jφ(m,m) and gj′, ∀j∈I∖Im, are continuous, there exists β∈R+ such that g(x)≤β, ∀x∈EM2,N,b,z(k,I), and so(94)∫EM2,N,b,zk,IgdλN,b,zk,I≤βλN,b,zk,IEM2,N,b,zk,I=β∏p∈Ik12NLeb-M2,M2∏q∈I∖Ik12NLebzq-Nbq,zq+Nbq=βM2kNk∏q∈I∖Ikbq<+∞.Moreover, we have limn→+∞⁡φ(n)=φ, and so limn→+∞⁡fn(x)=lφxdet⁡Jφ¯x, ∀x∈EM2,N,b,z(k,I); then, from the dominated convergence theorem:(95)limn→+∞⁡∫EM2,N,b,zk,Ilφndet⁡Jφ¯dλN,b,zk,I=∫EM2,N,b,zk,Ilφdet⁡Jφ¯dλN,b,zk,I;consequently, from (89), we have(96)∫EM1,N,a,vk,IldλN,a,vk,I=∫EM2,N,b,zk,Ilφdet⁡Jφ¯dλN,b,zk,I.
LetB=∏j∈IBj∈B(I)EM1,N,a,v(k,I), where Bj=(aj,bj), ∀j∈Ik; moreover, ∀n∈N∗, consider the function ln:RI→[0,1] defined by (97)lnx=lnkxj:j∈Ik∏j∈I∖Ik1Bjxj,∀x=xj:j∈I∈RI,where(98)lnkx=1ifx∈∏j∈Ikaj+δjn,bj-δjnx-x2Imx1-x2Imifx∈∏j∈IkBj∖∏j∈Ikaj+δjn,bj-δjn0ifx∉∏j∈IkBj,δj=(bj-aj)/2, ∀j∈Ik, x=xj:j∈Ik, x1=r∩∂∏j∈Ikaj+δj/n,bj-δj/n, x2=r∩∂∏j∈IkBj, where r is the half-line with initial point ∏j∈Ik(aj+bj/2) and containing x. Since lnn∈N∗ is an increasing positive sequence such that limn→+∞⁡ln=1B, and since the function ln(k):Rm→[0,1] is continuous, from Beppo Levi theorem and (96), we have (99)∫EM1,N,a,vk,I1BdλN,a,vk,I=limn→+∞⁡∫EM1,N,a,vk,IlndλN,a,vk,I=limn→+∞⁡∫EM2,N,b,zk,Ilnφdet⁡Jφ¯dλN,b,zk,I=∫EM2,N,b,zk,I1Bφdet⁡Jφ¯dλN,b,zk,I.Moreover, Proposition 45 implies that formula (99) is true ∀B∈B(I)EM1,N,a,v(k,I). Consider the measures μ and υ on EN,a,v(k,I),B(I)EN,a,v(k,I) defined by(100)μB=∫BdλN,a,vk,I,υB=∫φ-1Bdet⁡Jφ¯dλN,b,zk,I,and set Bn=B∩En,N,a,v(k,I), ∀n∈N∗ and ∀B∈B(I)EN,a,v(k,I). Since Bn⊂Bn+1, A-1Bn⊂A-1Bn+1, ⋃n∈N∗Bn=B and ⋃n∈N∗A-1(Bn)=A-1(B), from the continuity property of μ and υ and (99), we have(101)∫BdλN,a,vk,I=limn→+∞⁡∫BndλN,a,vk,I=limn→+∞⁡∫φ-1Bndet⁡Jφ¯dλN,b,zk,I=∫φ-1Bdet⁡Jφ¯dλN,b,zk,I.Then, suppose that B∈B(I); from Lemma 46, formula (101), and Proposition 44, we have (102)∫RI1BdλN,a,vk,I=∫B∩EN,a,vk,IdλN,a,vk,I=∫φ-1B∩EN,a,vk,Idet⁡Jφ¯dλN,b,zk,I=∫φ-1B∩EN,b,zk,Idet⁡Jφ¯dλN,b,zk,I=∫RI1Bφdet⁡Jφ¯dλN,b,zk,I;thus, by proceeding as in the proof of formula (89), for any measurable function f:RI,B(I)→([0,+∞),B[0,+∞)), we obtain(103)∫RIfdλN,a,vk,I=∫RIfφdet⁡Jφ¯dλN,b,zk,I.Then, for any measurable function f:RI,B(I)→(R,B) such that f+ (or f-) is λN,a,v(k,I)-integrable and for any B∈B(I), we have(104)∫BfdλN,a,vk,I=∫RIf+1BdλN,a,vk,I-∫RIf-1BdλN,a,vk,I=∫RIf+1Bφdet⁡Jφ¯dλN,b,zk,I-∫RIf-1Bφdet⁡Jφ¯dλN,b,zk,I=∫RIf1Bφdet⁡Jφ¯dλN,b,zk,I=∫φ-1Bfφdet⁡Jφ¯dλN,b,zk,I.


## 5. Problems for Further Study

A natural extension of this paper is the generalization of Theorem47, by substituting the m,σ-standard functions for more general functions φ:U⊂EI→EI such that, for any i∈I∖Im, the function φi:U⊂EI→R depends on a finite number of variables.Moreover, a natural application of this paper, in the probabilistic framework, is the development of the theory of the infinite-dimensional continuous random elements, defined in the paper [1]. In particular, we can prove the formula of the density of such random elements composed with the m,σ-standard functions given in the change of variables’ formula in Theorem 47. Consequently, 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 EI defined by the m,σ-standard matrices) and to develop some theoretical results and some applications in the statistical inference. It is possible also to define a convolution between the laws of two independent and infinite-dimensional continuous random elements, as in the finite case.Furthermore, we can generalize paper [11] by considering the recursion Xnn∈N on [0,p)N∗ defined by (105)Xn+1=AXn+Bnmodp,where X0=x0∈EI, A is a biunique, linear, integer and m,σ-standard function, p∈R+, and Bnn∈N is a sequence of independent and identically distributed random elements on EI. Our target is to prove that, with some assumptions on the law of Bn, the sequence Xnn∈N converges with geometric rate to a random element with law ⨂i∈N∗1/pLebB([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 Op2 steps are necessary to achieve randomness.



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*Source: 2619087-2016-12-08.xml*