Integration over an Infinite-Dimensional Banach Space and Probabilistic Applications

Abstract

We study, for some subsets 𝐼 of 𝐍^∗, the Banach space 𝐸 of bounded real sequences {𝑥_𝑛}_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.

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 𝐑^𝐍∗ 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 𝐑^𝐍∗. 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 subsets 𝐼 of 𝐍^∗, and we suppose that 𝐑^𝐼 is endowed with the standard infinity-norm generalized to assume the values in [0,+∞]; then, the vector space 𝐸 of the elements of 𝐑^𝐼 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 𝐑^𝐼 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 Section 2 we use Corollary 4 (that generalizes the Jessen theorem) to define a measure 𝜆^(𝑘)_𝑁,𝑎 over (𝐸,ℬ(𝐸)), where 𝑘∈𝐍; consequently, we define also a theory of integration. In the case 𝐼={1,…,𝑘}, the measure 𝜆^(𝑘)_𝑁,𝑎 coincides with the 𝑘-dimensional Lebesgue measure on 𝐑^𝑘.

In Section 3, 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 that 𝑘∈𝐍, 𝑁∈𝐑⁺, and 𝐼={𝑛∈𝐍^∗∶𝑛<𝑀}, where 𝑘+1≤𝑀≤+∞ and 𝑎=(𝑎_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 such that there exists ∏_𝑛∈𝐼𝑎_𝑛∈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 𝐑^𝑘, the Lebesgue measure on 𝐑, and the Lebesgue measure on 𝐑^𝑘. 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 ‖⋅‖∶𝐑^𝐼→[0,+∞] by ‖𝑥‖=sup_𝑛∈𝐼𝑥_𝑛,∀𝑥=𝑥_𝑛∶𝑛∈𝐼∈𝐑^𝐼, (1) and define the following vector space on the field 𝐑: 𝐸=𝑥∈𝐑^𝐼∶‖𝑥‖<+∞. (2)

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|𝑥_𝑛−𝑦_𝑛|, ∀𝑥=(𝑥_𝑛∶𝑛∈𝐼)∈𝐸, and ∀𝑦=(𝑦_𝑛∶𝑛∈𝐼)∈𝐸. Moreover, let {𝑥_𝑖}_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 ‖𝑥_𝑖−𝑥_𝑗‖<𝜀, and so, ∀𝑛∈𝐼, |(𝑥_𝑖)_𝑛−(𝑥_𝑗)_𝑛|<𝜀. Since 𝐑 is complete, ∀𝑛∈𝐼, ∃𝑙_𝑛∈𝐑 such that lim_𝑗→+∞(𝑥_𝑗)_𝑛=𝑙_𝑛; then, by setting 𝑙=(𝑙_𝑛∶𝑛∈𝐼)∈𝐑^𝐼, we have 𝑥_𝑖𝑛−𝑙_𝑛=lim_𝑗→+∞𝑥_𝑖𝑛−𝑥_𝑗𝑛⟹𝑥≤𝜀_𝑖−𝑙=sup_𝑛∈𝐼𝑥_𝑖𝑛−𝑙_𝑛≤𝜀. (3) This implies that 𝑙∈𝐸 and lim_𝑖→+∞𝑥_𝑖=𝑙; then, 𝐸 is complete, and so it is a Banach space.

In order to develop the next arguments, for any set 𝐼 and for any 𝐻⊂𝐼 define the projection 𝜋_𝐻 on 𝐑^𝐻 as the function 𝜋_𝐻∶𝐑^𝐼→𝐑^𝐻 given by 𝜋_𝐻(𝑥_𝑛∶𝑛∈𝐼)=(𝑥_ℎ∶ℎ∈𝐻). We will use the following result, whose proof can be found, for example, in Rao [6, page 346].

Theorem 3 (Jessen theorem). Let 𝐼 be a set and, for any 𝑖∈𝐼, let (𝐸_𝑖,ℰ_𝑖,𝜇_𝑖) be a probability space. Then, over the measurable space (∏_𝑖∈𝐼𝐸_𝑖,⨂_𝑖∈𝐼ℰ_𝑖), there is a unique probability measure 𝜇, indicated by ⨂_𝑖∈𝐼𝜇_𝑖, such that, for any 𝐻⊂𝐼 such that |𝐻|<+∞ and for any 𝐴=∏_ℎ∈𝐻𝐴_ℎ×∏_{𝑖∈𝐼⧵𝐻}𝐸_𝑖∈⨂_𝑖∈𝐼ℰ_𝑖, where 𝐴_ℎ∈ℰ_ℎ, ∀ℎ∈𝐻, we have 𝜇(𝐴)=∏_ℎ∈𝐻𝜇_ℎ(𝐴_ℎ). In particular, if 𝐼 is countable, then 𝜇(𝐴)=∏_𝑖∈𝐼𝜇_𝑖(𝐴_𝑖) for any 𝐴=∏_𝑖∈𝐼𝐴_𝑖∈⨂_𝑖∈𝐼ℰ_𝑖.

Corollary 4. Let 𝐼 be a set and, for any 𝑖∈𝐼, let (𝐸_𝑖,ℰ_𝑖,𝜇_𝑖) be a measure space such that 𝜇_𝑖 is finite. Moreover, suppose that, for some countable set 𝐽⊂𝐼, 𝜇_𝑖 is a probability measure for any 𝑖∈𝐼⧵𝐽 and ∏_𝑗∈𝐽𝜇_𝑗(𝐸_𝑗)∈𝐑⁺. Then, over the measurable space (∏_𝑖∈𝐼𝐸_𝑖,⨂_𝑖∈𝐼ℰ_𝑖), there is a unique finite measure 𝜇, indicated by ⨂_𝑖∈𝐼𝜇_𝑖, such that, for any 𝐻⊂𝐼 such that |𝐻|<+∞ and for any 𝐴=∏_ℎ∈𝐻𝐴_ℎ×∏_{𝑖∈𝐼⧵𝐻}𝐸_𝑖∈⨂_𝑖∈𝐼ℰ_𝑖, where 𝐴_ℎ∈ℰ_ℎ, ∀ℎ∈𝐻, one has 𝜇(𝐴)=∏_ℎ∈𝐻𝜇_ℎ(𝐴_ℎ)∏_{𝑗∈𝐽⧵𝐻}𝜇_𝑗(𝐸_𝑗). In particular, if 𝐼 is countable, then 𝜇(𝐴)=∏_𝑖∈𝐼𝜇_𝑖(𝐴_𝑖) for any 𝐴=∏_𝑖∈𝐼𝐴_𝑖∈⨂_𝑖∈𝐼ℰ_𝑖.

Proof. For any 𝑖∈𝐼, 𝜇_𝑖=(𝜇_𝑖/𝜇_𝑖(𝐸_𝑖)) is a probability measure; then, if 𝜇=⨂_𝑖∈𝐼𝜇_𝑖 is the probability measure defined by Theorem 3, the finite measure 𝜇=(∏_𝑗∈𝐽𝜇_𝑗(𝐸_𝑗))𝜇 satisfies the statement.

Since for any 𝑛∈𝐼⧵{1,…,𝑘} the measure (1/2𝑁)Leb(⋅∩[−𝑁𝑎_𝑛,𝑁𝑎_𝑛]) is a finite measure over (𝐑,ℬ), from Corollary 4 we can define the 𝜎-finite measure 𝜆^(𝑘)_𝑁,𝑎 over (𝐸,ℬ(𝐸)) in the following manner: 𝜆^(𝑘)_𝑁,𝑎=1(2𝑁)^𝑘Leb^(𝑘)⊗_{𝑛∈𝐼⧵{1,…,𝑘}}12𝑁Leb⋅∩−𝑁𝑎_𝑛,𝑁𝑎_𝑛. (4)

Remark 5. For any 𝑁∈𝐑⁺, we have 𝜆^(𝑘)_𝑁,𝑎(𝐸)=_𝑛∈𝐼𝑎_𝑛if𝑘=0+∞if𝑘∈𝐍^∗. (5)

Proof. If 𝑁∈𝐑⁺ and 𝑘=0, from Corollary 4, we have 𝜆^(𝑘)_𝑁,𝑎(𝐸)=_𝑛∈𝐼12𝑁Leb−𝑁𝑎_𝑛,𝑁𝑎_𝑛=_𝑛∈𝐼𝑎_𝑛. (6) Analogously, if 𝑁∈𝐑⁺ and 𝑘∈𝐍^∗: 𝜆^(𝑘)_𝑁,𝑎(𝐸)=1(2𝑁)^𝑘Leb^(𝑘)𝐑^𝑘_{𝑛∈𝐼⧵{1,…,𝑘}}𝑎_𝑛=+∞. (7)

3. Infinite-Dimensional Matrices

Definition 6. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼} be a real matrix 𝐼×𝐼  (eventually infinite, if 𝐼=𝐍^∗); then, define the linear function 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐑^𝐼, and write 𝑥→𝐴𝑥, in the following manner: (𝐴𝑥)_𝑖=_𝑗∈𝐼𝑎_𝑖𝑗𝑥_𝑗,∀𝑥∈𝐸,∀𝑖∈𝐼, (8) on condition that, for any 𝑖∈𝐼, the sum in (8) converges to a real number.

Proposition 7. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼} be a real matrix 𝐼×𝐼; then(1)the linear function 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐑^𝐼 given by (8) is defined if and only if, for any 𝑖∈𝐼, ∑_𝑗∈𝐼|𝑎_𝑖𝑗|<+∞;(2)𝐴(𝐸)⊂𝐸 and 𝐴 is continuous if and only if sup_𝑖∈𝐼∑_𝑗∈𝐼|𝑎_𝑖𝑗|<+∞; moreover, in this case, ‖𝐴‖=sup_𝑖∈𝐼∑_𝑗∈𝐼|𝑎_𝑖𝑗|.

Proof. (1) Suppose that the function 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐑^𝐼 is defined; then, ∀𝑖∈𝐼; let 𝑥=(𝑥_𝑛∶𝑛∈𝐼)∈𝐸 be such that 𝑥_𝑛=1 if 𝑎_𝑖𝑛≥0, and 𝑥_𝑛=−1 if 𝑎_𝑖𝑛<0; since 𝐴𝑥∈𝐑^𝐼, we have _𝑗∈𝐼𝑎_𝑖𝑗=(𝐴𝑥)_𝑖∈𝐑. (9) Conversely, suppose that ∑_𝑗∈𝐼|𝑎_𝑖𝑗|<+∞, ∀𝑖∈𝐼; then, ∀𝑥∈𝐸 and ∀𝑖∈𝐼, ∑_𝑗∈𝐼(𝑎_𝑖𝑗𝑥_𝑗)⁺≤∑_𝑗∈𝐼|𝑎_𝑖𝑗||𝑥_𝑗|≤∑_𝑗∈𝐼|𝑎_𝑖𝑗|‖𝑥‖<+∞; analogously, ∑_𝑗∈𝐼(𝑎_𝑖𝑗𝑥_𝑗)⁻<+∞, from which (𝐴𝑥)_𝑖=∑_𝑗∈𝐼(𝑎_𝑖𝑗𝑥_𝑗)⁺−∑_𝑗∈𝐼(𝑎_𝑖𝑗𝑥_𝑗)⁻∈𝐑, and so 𝐴𝑥∈𝐑^𝐼.
(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 _𝑗∈𝐼𝑎_𝑖𝑗=(𝐴𝑥)_𝑖≤‖𝐴𝑥‖≤‖𝐴‖<+∞⟹sup_{𝑖∈𝐼𝑗∈𝐼}𝑎_𝑖𝑗≤‖𝐴‖<+∞. (10) Conversely, if sup_𝑖∈𝐼∑_𝑗∈𝐼|𝑎_𝑖𝑗|<+∞, ∀𝑥∈𝐸, such that ‖𝑥‖=1, we have ‖𝐴𝑥‖=sup_𝑖∈𝐼(𝐴𝑥)_𝑖=sup_{𝑖∈𝐼𝑗∈𝐼}𝑎_𝑖𝑗𝑥_𝑗≤sup_{𝑖∈𝐼𝑗∈𝐼}𝑎_𝑖𝑗⟹‖𝐴‖<+∞=sup_{‖𝑥‖𝑥∈𝐸∶=1}‖𝐴𝑥‖≤sup_{𝑖∈𝐼𝑗∈𝐼}𝑎_𝑖𝑗<+∞. (11) Finally, if sup_𝑖∈𝐼∑_𝑗∈𝐼|𝑎_𝑖𝑗|<+∞, from (10) and (11) we have ‖𝐴‖=sup_{𝑖∈𝐼𝑗∈𝐼}𝑎_𝑖𝑗. (12)

Definition 8. A linear function 𝐴=(𝑎_𝑖𝑗)_{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)𝑎_𝑖𝑗=0, ∀(𝑖,𝑗)∉({1,…,𝑚}×𝐼)∪⋃_{𝑛∈𝐼⧵{1,…,𝑚}}{(𝑛,𝜎(𝑛))};(2)there exists ∏_{𝑛∈𝐼⧵{1,…,𝑚}∶𝜆𝑛≠0}𝜆_𝑛∈𝐑^∗, where 𝜆_𝑛=𝑎_{𝑛,𝜎(𝑛)}, ∀𝑛∈𝐼⧵{1,…,𝑚}.

Moreover, indicate by 𝐴_𝑚 the matrix (𝑎_𝑖𝑗)_{𝑖,𝑗∈{1,…,𝑚}}∈𝑀_𝑚(𝐑). Finally, indicate by ℳ_(𝑚,𝜎)(𝐸) the set of the linear (𝑚,𝜎)-standard functions from 𝐸 to 𝐸.

Remark 9. Let 𝐴=(𝑎_𝑖𝑗)_{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 Definition 8, sup_{𝑖∈𝐼𝑗∈𝐼}𝑎_𝑖𝑗=supsup_{𝑖∈{1,…,𝑚}𝑗∈𝐼}𝑎_𝑖𝑗,sup_{𝑛∈𝐼⧵{1,…,𝑚}∶𝜆𝑛≠0}𝜆_𝑛. (13)
We have sup_{𝑖∈{1,…,𝑚}}∑_𝑗∈𝐼|𝑎_𝑖𝑗|<+∞ from Proposition 7; moreover, if 𝜆_𝑛=0 for 𝑛 sufficiently large, obviously sup_{𝑛∈𝐼⧵{1,…,𝑚}∶𝜆𝑛≠0}|𝜆_𝑛|<+∞; otherwise, consider the subsequence {𝜆_𝑛𝑘}_𝑘∈𝐍={𝜆_𝑛≠0∶𝑛∈𝐼⧵{1,…,𝑚}}; from the point 2 of Definition 8, we obtain lim_𝑘→+∞𝜆_𝑛𝑘=1, and so sup_{𝑛∈𝐼⧵{1,…,𝑚}∶𝜆𝑛≠0}|𝜆_𝑛|<+∞ again. Then, sup_𝑖∈𝐼∑_𝑗∈𝐼|𝑎_𝑖𝑗|<+∞, 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. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 be a linear (𝑚,𝜎)-standard function; then, 𝐴 is biunique if and only if the matrix 𝐴_𝑚 is invertible, 𝑎_{𝑛,𝜎(𝑛)}≠0, ∀𝑛∈𝐼⧵{1,…,𝑚}, and 𝜎 is biunique.

Proof. If 𝐴_𝑚 is invertible and 𝑎_{𝑛,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 𝑎_{𝑛,𝜎(𝑛)}𝑥_𝜎(𝑛)=𝑎_{𝑛,𝜎(𝑛)}𝑦_𝜎(𝑛), from which 𝑥_𝜎(𝑛)=𝑦_𝜎(𝑛); then, if 𝜎 is biunique, we have 𝜎(𝑛)=𝑛, and so (𝑥_𝑛∶𝑛>𝑚)=(𝑦_𝑛∶𝑛>𝑚). This implies that 𝐴_𝑚^𝑡(𝑥₁,…,𝑥_𝑚)=𝐴_𝑚^𝑡(𝑦₁,…,𝑦_𝑚), and so (𝑥₁,…,𝑥_𝑚)=(𝑦₁,…,𝑦_𝑚); then, 𝑥=𝑦; that is, 𝐴 is injective. Moreover, ∀𝑦∈𝐸, define 𝑥∈𝐸 in the following manner: 𝑥_𝑛=𝑦_𝑛𝑎_𝑛𝑛,∀𝑛∈𝐼⧵{1,…,𝑚_𝑡},𝑥₁,…,𝑥_𝑚=𝐴_𝑚^−1𝑡𝑧₁,…,𝑧_𝑚, (14) where 𝑧_𝑖=𝑦_𝑖−_𝑛>𝑚𝑎_𝑖𝑛𝑥_𝑛,∀𝑖∈{1,…,𝑚}. (15) It is easy to prove that 𝐴𝑥=𝑦; that is, 𝐴 is surjective.
Conversely, if 𝐴 is biunique, let 𝐱, 𝐲∈𝐑^𝑚 be such that 𝐴_𝑚𝐱=𝐴_𝑚𝐲, and let 𝑥, 𝑦∈𝐸 be such that 𝑥_𝑛=𝑥_𝑛, 𝑦_𝑛=𝑦_𝑛, ∀𝑛∈{1,…,𝑚}, and 𝑥_𝑛=𝑦_𝑛=0, ∀𝑛∈𝐼⧵{1,…,𝑚}. We have 𝐴_𝑚𝐱=𝝅_{1,…,𝑚}(𝐴𝑥), 𝐴_𝑚𝐲=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→𝐴_𝑚𝐱 is injective; that is, 𝐴_𝑚 is invertible. Moreover, we have 𝑎_{𝑛,𝜎(𝑛)}≠0, ∀𝑛∈𝐼⧵{1,…,𝑚}; in fact, by supposing by contradiction that 𝑎_{𝑛,𝜎(𝑛)}=0, for some 𝑛>𝑚, then 𝐴(𝐸)⊂{𝑥∈𝐸∶𝑥_𝑛=0}⊆_/𝑞𝐸, 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∶𝑥_𝑛1𝑎_{𝑛2,𝜎(𝑛2)}=𝑥_𝑛2𝑎_{𝑛1,𝜎(𝑛1)}}⊆_/𝑞𝐸 (a contradiction). Finally, 𝜎 must be surjective, because otherwise, ∀𝑦∈𝐸 and ∀𝑛∈(𝐼⧵{1,…,𝑚})⧵𝜎(𝐼⧵{1,…,𝑚}), we could choose arbitrarily 𝑥_𝑛∈𝐑 in order to determine 𝑥=(𝑥_𝑛∶𝑛∈𝐼)∈𝐸 such that 𝐴𝑥=𝑦. Then, 𝐴 should not be injective (again a contradiction).

In order to study the inverse of 𝐴, 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. Let 𝐴=(𝑎_𝑖𝑗)_{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 det𝐴, the real number: det_(𝑚,𝜎)𝐴=det𝐴_{𝑚𝑛∈𝐼⧵{1,…,𝑚}}𝜆_𝑛if𝜎isbiunique0if𝜎isnotbiunique. (16)

Remark 12. If 𝐴∈ℳ_{(𝑚1,𝜎1)}(𝐸)∩ℳ_{(𝑚2,𝜎2)}(𝐸), then det_{(𝑚1,𝜎1)}𝐴=det_{(𝑚2,𝜎2)}𝐴.

Proof. Suppose that 𝑚₁≤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 det_{(𝑚1,𝜎1)}𝐴=det𝐴_{𝑚1𝑛∈𝐼⧵{1,…,𝑚1}}𝜆_𝑛=det𝐴_{𝑚1𝑝∈{𝑚1+1,…,𝑚2}}𝜆_{𝑝𝑛∈𝐼⧵{1,…,𝑚2}}𝜆_𝑛=det𝐴_{𝑚2𝑛∈𝐼⧵{1,…,𝑚2}}𝜆_𝑛=det_{(𝑚2,𝜎2)}𝐴. (17)
Instead, if 𝜎₁ 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 det_{(𝑚1,𝜎1)}𝐴=0=det_{(𝑚2,𝜎2)}𝐴. (18) In the second case, we have det𝐴_𝑚2=0, and so det_{(𝑚1,𝜎1)}𝐴=0=det𝐴_{𝑚2𝑛∈𝐼⧵{1,…,𝑚2}}𝜆_𝑛=det_{(𝑚2,𝜎2)}𝐴. (19)

Proposition 13. Let 𝐴=(𝑎_𝑖𝑗)_{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 exist 𝑢=(𝑢_𝑛∶𝑛∈𝐼)∈𝐸, 𝑣=(𝑣_𝑛∶𝑛∈𝐼)∈𝐸, and 𝑐₁,𝑐₂∈𝐑 such that ∑_𝑛∈𝐼|𝑢_𝑛|<+∞, ∑_𝑛∈𝐼|𝑣_𝑛|<+∞, 𝑎_𝑡𝑗=𝑐₁𝑢_𝑗+𝑐₂𝑣_𝑗, ∀𝑗∈𝐼, by indicating by 𝑈 and 𝑉 the linear functions obtained by substituting the 𝑡th row of 𝐴 for 𝑢 and 𝑣, respectively, then 𝑈 and 𝑉 are (𝑝,𝜏)-standard and det𝐴=𝑐₁det𝑈+𝑐₂det𝑉;(2)if  𝐵=(𝑏_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 is the linear function obtained by exchanging the 𝑠th row of 𝐴 for the 𝑡th row of 𝐴, then 𝐵 is (𝑝,𝜏)-standard and det𝐵=−det𝐴;(3)if  𝐶=(𝑐_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 is the linear function obtained by substituting the 𝑡th row of 𝐴 for the 𝑠th row of 𝐴, then 𝐶 is (𝑝,𝜏)-standard and det𝐶=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, det𝐴=det𝐴_𝑝∏_{𝑛∈𝐼⧵{1,…,𝑝}}𝜆_𝑛 and det𝐴_𝑝=𝑐₁det𝑈_𝑝+𝑐₂det𝑉_𝑝; then det𝐴=𝑐₁det𝑈_𝑝+𝑐₂det𝑉_{𝑝𝑛∈𝐼⧵{1,…,𝑝}}𝜆_𝑛=𝑐₁det𝑈_{𝑝𝑛∈𝐼⧵{1,…,𝑝}}𝜆_𝑛+𝑐₂det𝑉_{𝑝𝑛∈𝐼⧵{1,…,𝑝}}𝜆_𝑛=𝑐₁det𝑈+𝑐₂det𝑉. (20)(2) As we observed in the proof of the point 1, 𝐵 is (𝑝,𝜏)-standard; moreover, det𝐵=det𝐵_𝑝∏_{𝑛∈𝐼⧵{1,...,𝑝}}𝜆_𝑛, where 𝐵_𝑝 is the matrix obtained by exchanging the 𝑠th row of 𝐴_𝑝 for the 𝑡th row of 𝐴_𝑝; then, det𝐵_𝑝=−det𝐴_𝑝, from which det𝐵=−det𝐴_{𝑝𝑛∈𝐼⧵{1,…,𝑝}}𝜆_𝑛=−det𝐴. (21)(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 det𝐶=−det𝐶, from which det𝐶=0.

Remark 14. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 be a linear (𝑚,𝜎)-standard function; then, 𝐴 is biunique if and only if det𝐴≠0.

Proof. If 𝐴 is biunique, from Proposition 10  𝜎 is biunique, and so det𝐴=det𝐴_𝑚∏_{𝑛∈𝐼⧵{1,…,𝑚}}𝜆_𝑛; moreover, we have det𝐴_𝑚≠0 and 𝜆_𝑛≠0, ∀𝑛∈𝐼⧵{1,…,𝑚}, from which ∏_{𝑛∈𝐼⧵{1,…,𝑚}}𝜆_𝑛=∏_{𝑛∈𝐼⧵{1,…,𝑚}∶𝜆𝑛≠0}𝜆_𝑛≠0; then, det𝐴≠0.
Conversely, if det𝐴≠0, then 𝜎 is biunique by definition of det𝐴, and so 0≠det𝐴=det𝐴_𝑚∏_{𝑛∈𝐼⧵{1,…,𝑚}}𝜆_𝑛; this implies that det𝐴_𝑚≠0 and 𝜆_𝑛≠0, ∀𝑛∈𝐼⧵{1,…,𝑚}; then, from Proposition 10, 𝐴 is biunique.

Definition 15. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 be a linear (𝑚,𝜎)-standard function; define the 𝐼×𝐼 matrix cof𝐴=(𝐴_𝑖𝑗)_{𝑖,𝑗∈𝐼} by 𝐴_𝑖𝑗=(−1)^𝑖+𝑗det𝐴1⋯𝑖̂⋯∣1⋯𝑗̂⋯, (22) where 𝐴(1⋯𝑖̂⋯∣1⋯𝑗̂⋯) is the (𝐼⧵{𝑖})×(𝐼⧵{𝑗}) matrix obtained by deleting the 𝑖th row and the 𝑗th column of 𝐴.

Proposition 16. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 be a linear (𝑚,𝜎)-standard function; then, for any 𝑖∈𝐼, one has det𝐴=_𝑗∈𝐼𝑎_𝑖𝑗𝐴_𝑖𝑗. (23)

Proof. Suppose that 𝜎 is biunique; then, ∀𝑖∈{1,…,𝑚}, we have det𝐴=det𝐴_{𝑚𝑛∈𝐼⧵{1,…,𝑚}}𝜆_𝑛=^𝑚_𝑗=1𝑎_𝑖𝑗𝐴_{𝑚𝑖𝑗𝑛∈𝐼⧵{1,…,𝑚}}𝜆_𝑛=^𝑚_𝑗=1𝑎_𝑖𝑗𝐴_𝑖𝑗. (24) Moreover, ∀𝑖∈{1,…,𝑚} and ∀𝑗>𝑚, the matrix 𝐴(1⋯𝑖̂⋯∣1⋯𝑗̂⋯) is (𝑚−1,𝜎)-standard, where 𝜎∶𝐼⧵{1,…,𝑚−1}⟶𝐼⧵{1,…,𝑚−1} (25) is not surjective because 𝑚∉𝜎(𝐼⧵{1,…,𝑚−1}), and so 𝐴_𝑖𝑗=0; then, det𝐴=∑_𝑗∈𝐼𝑎_𝑖𝑗𝐴_𝑖𝑗. Finally, ∀𝑖>𝑚, we have 𝑎_𝑖𝑗=0, ∀𝑗≠𝑖; then _𝑗∈𝐼𝑎_𝑖𝑗𝐴_𝑖𝑗=𝑎_𝑖𝑖𝐴_𝑖𝑖=𝑎_𝑖𝑖(−1)^2𝑖det𝐴_{𝑚{}𝑛∈𝐼⧵1,…,𝑚,𝑖}𝜆_𝑛=det𝐴_{𝑚𝑛∈𝐼⧵{1,…,𝑚}}𝜆_𝑛=det𝐴. (26)
Instead, if 𝜎 is not biunique, ∀𝑖,𝑗∈{1,…,𝑚}, the matrix 𝐴(1⋯𝑖̂⋯∣1⋯𝑗̂⋯) is (𝑚−1,𝜎)-standard, where 𝜎(𝑛)=𝜎(𝑛+1), ∀𝑛>𝑚−1; then, 𝜎 is not biunique, from which 𝐴_𝑖𝑗=0. Moreover, ∀𝑖∈{1,…,𝑚} and ∀𝑗>𝑚, as in the case 𝜎 being biunique, we have 𝐴_𝑖𝑗=0. Finally, ∀𝑖>𝑚, we have 𝑎_𝑖𝑗=0, ∀𝑗≠𝜎(𝑖); then _𝑗∈𝐼𝑎_𝑖𝑗𝐴_𝑖𝑗=𝑎_{𝑖,𝜎(𝑖)}𝐴_{𝑖,𝜎(𝑖)}. (27) Moreover, the matrix 𝐴(1…𝑖̂…∣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 𝐴_{𝑖,𝜎(𝑖)}=0, from which det𝐴=0=_𝑗∈𝐼𝑎_𝑖𝑗𝐴_𝑖𝑗. (28)

Corollary 17. Let 𝐴=(𝑎_𝑖𝑗)_{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=(𝑏_𝑖𝑗)_{𝑖,𝑗∈𝐼}; moreover 𝐴⁻¹=1det𝐴^𝑡(cof𝐴). (29)

Proof. From Proposition 16, we have _𝑛∈𝐼𝑎_𝑖𝑛𝐴_𝑖𝑛=det𝐴. (30) Moreover, we have _𝑛∈𝐼𝑎_𝑖𝑛𝐴_𝑗𝑛=0,∀𝑖,𝑗∈𝐼,𝑖≠𝑗; (31) in fact, from Proposition 16, the left side of (31) is equal to det𝐶, 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 det𝐶=0. This implies that _𝑛∈𝐼𝑎_𝑖𝑛𝐴_𝑗𝑛=(det𝐴)𝛿_𝑖𝑗,∀𝑖,𝑗∈𝐼, (32) where 𝛿_𝑖𝑗 is the Kronecker symbol, and so 𝐴^𝑡(cof𝐴)_𝑖𝑗=(det𝐴)𝛿_𝑖𝑗,∀𝑖,𝑗∈𝐼, (33) from which the formula (29) follows. Moreover, as we observed in the proof of Proposition 16, ∀𝑖∈{1,…,𝑚} and ∀𝑗>𝑚, we have 𝐴_𝑖𝑗=0; finally, ∀𝑖,𝑗>𝑚 such that 𝑖≠𝑗, the matrix 𝐴(1⋯𝑖̂⋯∣1⋯𝑗̂⋯) 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 𝐴_𝑖𝑗=0 again; from formula (29), this implies that 𝐴⁻¹ is (𝑚,𝜎)-standard.

Definition 18. Define the function ‖⋅‖∶𝐂^𝐼→[0,+∞] by ‖𝑥‖=sup_𝑛∈𝐼𝑥_𝑛,∀𝑥=𝑥_𝑛∶𝑛∈𝐼∈𝐂^𝐼, (34) and define the following vector space on the field 𝐂, with the norm ‖⋅‖: 𝐹=𝑥∈𝐂^𝐼∶‖𝑥‖<+∞⊃𝐸. (35)

Definition 19. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼} be a real matrix 𝐼×𝐼; then, define the linear function 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐹→𝐂^𝐼 and write 𝑥→𝐴𝑥, in the following manner: (𝐴𝑥)_𝑖=_𝑗∈𝐼𝑎_𝑖𝑗𝑥_𝑗,∀𝑥∈𝐹,∀𝑖∈𝐼, (36) on condition that, for any 𝑖∈𝐼, the sum in (36) converges to a complex number.

Proposition 20. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼} be a real matrix 𝐼×𝐼; then(1)the linear function 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐹→𝐂^𝐼 given by (36) is defined if and only if, for any 𝑖∈𝐼, ∑_𝑗∈𝐼|𝑎_𝑖𝑗|<+∞.(2)𝐴(𝐹)⊂𝐹 and 𝐴 is continuous if and only if sup_𝑖∈𝐼∑_𝑗∈𝐼|𝑎_𝑖𝑗|<+∞; moreover, in this case ‖𝐴‖=sup_𝑖∈𝐼∑_𝑗∈𝐼|𝑎_𝑖𝑗|.

Proof. The proof is analogous to that one of Proposition 7.

Definition 21. Let 𝑉 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. Let 𝐴=(𝑎_𝑖𝑗)_{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 𝑉𝑃(𝐴)=𝑉𝑃𝐴_𝑚∪𝜆_𝑛∶𝑛∈𝐼⧵{1,…,𝑚}. (37) Moreover det𝐴=_{𝜆∈𝑉𝑃(𝐴)}𝜆. (38)

Proof. Let 𝜆∈𝐂 be an eigenvalue of 𝐴_𝑚, let 𝐱∈𝐂^𝑚⧵{𝟎} be the corresponding eigenvector, and let 𝑦∈𝐂^𝐼⧵{0} be such that 𝑦_𝑛=𝑥_𝑛, ∀𝑛∈{1,…,𝑚}, and 𝑦_𝑛=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 𝑎_𝑖𝑛=0, ∀𝑖∈{1,…,𝑚}, let 𝑥∈𝐑^𝐼⧵{0} be such that 𝑥_𝑖=𝛿_𝑖𝑛, ∀𝑖∈𝐼; we have 𝐴𝑥=𝜆_𝑛𝑥, and so 𝜆_𝑛∈𝑉𝑃(𝐴). Otherwise, suppose that 𝑎_𝑖𝑛≠0 for some 𝑖∈{1,…,𝑚}; if 𝜆_𝑛∈𝑉𝑃(𝐴_𝑚), then 𝜆_𝑛∈𝑉𝑃(𝐴) by the previous arguments; conversely, if (𝐴_𝑚−𝜆_𝑛𝐼_𝑚)𝐱≠𝟎, ∀𝐱∈𝐂^𝑚⧵{𝟎}, the matrix (𝐴_𝑚−𝜆_𝑛𝐼_𝑚) is invertible and so there exists 𝐱∈𝐑^𝑚⧵{𝟎} such that 𝐴_𝑚𝐱−𝜆_𝑛𝐱=^𝑡(−𝑎_1𝑛,…,−𝑎_𝑖𝑛,…,−𝑎_𝑚𝑛); then, by considering 𝑦∈𝐑^𝐼⧵{0} such that 𝑦_𝑖=𝑥_𝑖, ∀𝑖∈{1,…,𝑚}, 𝑦_𝑖=𝛿_𝑖𝑛, ∀𝑖∈𝐼⧵{1,…,𝑚}, we have 𝐴𝑦=𝜆_𝑛𝑦, and so 𝜆_𝑛∈𝑉𝑃(𝐴). Then 𝑉𝑃𝐴_𝑚∪𝜆_𝑛∶𝑛∈𝐼⧵{1,…,𝑚}⊂𝑉𝑃(𝐴). (39)
Conversely, if 𝜆∈𝑉𝑃(𝐴), we have 𝐴𝑥=𝜆𝑥, for some 𝑥∈𝐂^𝐼⧵{0}, and so, ∀𝑛∈𝐼⧵{1,…,𝑚}, 𝜆_𝑛𝑥_𝑛=(𝐴𝑥)_𝑛=𝜆𝑥_𝑛; then, by supposing 𝜆∉{𝜆_𝑛∶𝑛∈𝐼⧵{1,…,𝑚}}, we have 𝑥_𝑛=0, from which 𝑥_𝑛≠0 for some 𝑛∈{1,…,𝑚}. Moreover, we have 𝐴_𝑚^𝑡𝑥₁,…,𝑥_𝑚=^𝑡(𝐴𝑥)₁,…,(𝐴𝑥)_𝑚=𝜆^𝑡𝑥₁,…,𝑥_𝑚, (40) and so 𝜆∈𝑉𝑃(𝐴_𝑚). Then, we have 𝑉𝑃(𝐴)⊂𝑉𝑃𝐴_𝑚∪𝜆_𝑛∶𝑛∈𝐼⧵{1,…,𝑚}, (41) from which (37) follows. Moreover, since 𝜎 is biunique, from (37), we have det𝐴=det𝐴_{𝑚𝑛∈𝐼⧵{1,…,𝑚}}𝜆_𝑛=_{𝜆∈𝑉𝑃(𝐴)}𝜆. (42)

4. Change of Variables’ Formula

Definition 23. Let 𝑘∈𝐍, let 𝑀,𝑁∈𝐑⁺, and let 𝑎=(𝑎_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 such that ∏_𝑛∈𝐼𝑎_𝑛∈𝐑⁺; define the following sets in ℬ(𝐸): 𝐸^(𝑘)_𝑁,𝑎=𝐑^𝑘×_{𝑛∈𝐼⧵{1,…,𝑘}}−𝑁𝑎_𝑛,𝑁𝑎_𝑛;𝐸^(𝑘)_{𝑀,𝑁,𝑎}=[−𝑀,𝑀]^𝑘×_{𝑛∈𝐼⧵{1,…,𝑘}}−𝑁𝑎_𝑛,𝑁𝑎_𝑛. (43)

Definition 24. Let 𝑎=(𝑎_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 and 𝑏=(𝑏_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 be such that ∏_𝑛∈𝐼𝑎_𝑛∈𝐑⁺, ∏_𝑛∈𝐼𝑏_𝑛∈𝐑⁺; define 𝑎𝑏∈(𝐑⁺)^𝐼 in the following manner: 𝑎𝑏=𝑎_𝑛𝑏_𝑛∶𝑛∈𝐼. (44)

Proposition 25. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 be a biunique and linear (𝑚,𝜎)-standard function; then, for any 𝑎=(𝑎_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 such that ∏_𝑛∈𝐼𝑎_𝑛∈𝐑⁺, there exists 𝑏=(𝑏_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 such that ∏_𝑛∈𝐼𝑏_𝑛∈𝐑⁺ and such that, for any 𝑘∈𝐍, 𝑘≥𝑚, and for any 𝑁∈𝐑⁺, one has 𝐴⁻¹𝐸^(𝑘)_𝑁,𝑎=𝐸^(𝑘)_𝑁,𝑏. (45)

Proof. From Corollary 17, 𝐴⁻¹=(𝑏_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 is a linear (𝑚,𝜎)-standard function. By setting 𝜌_𝑛=𝑏_𝑛𝑛, ∀𝑛>𝑚, from (29), we have 𝜌_𝑛=1𝑎_𝑛𝑛=1𝜆_𝑛⟹_{𝑛∈𝐼⧵{1,…,𝑚}}𝜌_𝑛=_{𝑛∈𝐼⧵{1,…,𝑚}}1𝜆_𝑛∈𝐑^∗. (46) Set 𝑏=(𝑏_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 such that 𝑏_𝑛{},𝑏=1,∀𝑛∈1,…,𝑚_𝑛∶𝑛>𝑚=𝑎_𝑛∶𝑛>𝑚𝜌_𝑛∶𝑛>𝑚. (47) By definition of 𝑏, we have _𝑛∈𝐼𝑏_𝑛=_{𝑛∈𝐼⧵{1,…,𝑚}}𝑎_{𝑛𝑛∈𝐼⧵{1,…,𝑚}}1𝜆_𝑛∈𝐑⁺; (48) moreover, for any 𝑘∈𝐍, 𝑘≥𝑚, and for any 𝑁∈𝐑⁺, we have 𝐴⁻¹(𝐸^(𝑘)_𝑁,𝑎)⊂𝐸^(𝑘)_𝑁,𝑏. Analogously, it is possible to prove that 𝐴(𝐸^(𝑘)_𝑁,𝑏)⊂𝐸^(𝑘)_𝑁,𝑐, where 𝑐_𝑛∶𝑛>𝑚=𝑏_𝑛∶𝑛>𝑚𝜆_𝑛∶𝑛>𝑚=𝑎_𝑛∶𝑛>𝑚. (49) Moreover, since 𝑘≥𝑚, we have 𝐸^(𝑘)_𝑁,𝑐=𝐸^(𝑘)_𝑁,𝑎, and so 𝐸^(𝑘)_𝑁,𝑏⊂𝐴⁻¹(𝐸^(𝑘)_𝑁,𝑎), from which (45) follows.

Lemma 26. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 be a biunique and linear (𝑚,𝜎)-standard function; then, for any 𝑀₁∈𝐑⁺ and for any 𝑎=(𝑎_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 such that ∏_𝑛∈𝐼𝑎_𝑛∈𝐑⁺, there exist 𝑀₂,𝑀₃∈𝐑⁺ and 𝑏=(𝑏_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼, 𝑐=(𝑐_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 such that ∏_𝑛∈𝐼𝑏_𝑛∈𝐑⁺, ∏_𝑛∈𝐼𝑐_𝑛∈𝐑⁺, and such that, for any 𝑘∈𝐍 and for any 𝑁∈𝐑⁺, one has 𝐴⁻¹𝐸_𝑀^(𝑘)_1,𝑁,𝑎⊂𝐸_𝑀^(𝑘)_2,𝑁,𝑏;𝐴𝐸_𝑀^(𝑘)_2,𝑁,𝑏⊂𝐸_𝑀^(𝑘)_3,𝑁,𝑐. (50)(51) Moreover, (𝑐_𝑛∶𝑛>𝑚)=(𝑎_𝑛∶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∈𝐸_𝑀^(𝑘)_1,𝑁,𝑎, we have 𝐴⁻¹(𝑥)≤𝐴⁻¹‖𝑥‖≤𝐴⁻¹max𝑀₁,𝑁,‖𝑎‖. (52) Set 𝑀₂=‖𝐴⁻¹‖max{𝑀₁,𝑁,‖𝑎‖} and 𝑏=(𝑏_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 such that 𝑏_𝑛=𝑀₂𝑁{},𝑏,∀𝑛∈1,…,𝑚_𝑛∶𝑛>𝑚=𝑎_𝑛∶𝑛>𝑚𝜌_𝑛∶𝑛>𝑚, (53) where 𝜌_𝑛, ∀𝑛∈𝐼, is defined as in the proof of Proposition 25. By definition of 𝑏, we have _𝑛∈𝐼𝑏_𝑛=𝑀₂𝑁^𝑚_{𝑛∈𝐼⧵{1,…,𝑚}}𝑎_{𝑛𝑛∈𝐼⧵{1,…,𝑚}}1𝜆_𝑛∈𝐑⁺, (54) and (50) holds. Analogously, it is possible to prove (51); moreover 𝑐_𝑛∶𝑛>𝑚=𝑏_𝑛∶𝑛>𝑚𝜆_𝑛∶𝑛>𝑚=𝑎_𝑛∶𝑛>𝑚. (55)

Remark 27. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 be a linear (𝑚,𝜎)-standard function; then, 𝐴 is ℬ(𝐸)/ℬ(𝐸)-measurable.

Proof. Let 𝜏 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. Let 𝜇₁ 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 on 𝐑^𝑚 with values in 𝐑^𝑚 (see, e.g., Lang's book [10]).

Theorem 29 (change of variables’ formula). Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 be a biunique and linear (𝑚,𝜎)-standard function, let 𝑎=(𝑎_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 be such that ∏_𝑛∈𝐼𝑎_𝑛∈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 𝜆^(𝑘)_𝑁,𝑎-integrable, one has _𝐵𝑓𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐴−1(𝐵)}𝑓(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏. (56)

Proof. ∀𝑛∈𝐍, let ℎ_𝑛∶𝐸→𝐸 be the biunique and linear (𝑚,𝜎)-standard function given by ℎ_𝑛(𝑥)_𝑖=𝐴_𝑛𝜋_{1,…,𝑛}(𝑥)_𝑖{};ℎ,∀𝑥∈𝐸,∀𝑖∈1,…,𝑛_𝑛(𝑥)_𝑖=𝜆_𝑖𝑥_𝑖,∀𝑥∈𝐸,∀𝑖∈𝐼⧵{1,…,𝑛}. (57) Moreover, ∀𝑀₁∈𝐑⁺ and ∀𝑎=(𝑎_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 such that ∏_𝑛∈𝐼𝑎_𝑛∈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 ℎ_𝑛; finally, consider the analogous constants 𝑀₂, 𝑀₃, and the sequences 𝑏, 𝑐 defined by 𝐴. Observe that 𝑀₂(𝑛)≤𝑀₂, (𝑏(𝑛))_𝑖≤𝑏_𝑖, ∀𝑖∈𝐼, ∀𝑛∈𝐍. Suppose that 𝑛≥𝑘≥𝑚 and 𝑁∈𝐑⁺; then, ∀𝐵=∏_𝑝∈𝐼𝐵_𝑝, where 𝐵_𝑝∈ℬ([−𝑀₁,𝑀₁]), ∀𝑝∈{1,…,𝑘}, 𝐵_𝑝∈ℬ([−𝑁𝑎_𝑝,𝑁𝑎_𝑝]), ∀𝑝>𝑘, we have _𝐵𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐵1×⋯×𝐵𝑘×∏𝑞>𝑘𝐵𝑞}𝑑^𝑘_𝑝=11⊗2𝑁Leb_𝑞>𝑘12𝑁Leb_{ℬ([−𝑁𝑎𝑞,𝑁𝑎𝑞])}=_{𝐵1×⋯×𝐵𝑛×∏𝑞>𝑛𝐵𝑞}𝑑^𝑛_𝑝=11⊗2𝑁Leb_𝑞>𝑛12𝑁Leb_{ℬ([−𝑁𝑎𝑞,𝑁𝑎𝑞])}=_{𝐵1×⋯×𝐵𝑛}𝑑^𝑛_𝑝=11×2𝑁Leb_{∏𝑞>𝑛𝐵𝑞}𝑑_𝑞>𝑛12𝑁Leb_{ℬ([−𝑁𝑎𝑞,𝑁𝑎𝑞])}=_{𝐴𝑛−1𝐵1×⋯×𝐵𝑛}det𝐴_𝑛𝑑^𝑛_𝑝=11×2𝑁Leb_{∏𝑞𝑞>𝑛1/𝜆𝐵𝑞𝑞>𝑛}𝜆_𝑞𝑑_𝑞>𝑛12𝑁Leb_{ℬ([−𝑁𝑏𝑞,𝑁𝑏𝑞])}=_{ℎ𝑛−1(𝐵)}detℎ_𝑛𝑑^𝑛_𝑝=11⊗2𝑁Leb_𝑞>𝑛12𝑁Leb_{ℬ([−𝑁𝑏𝑞,𝑁𝑏𝑞])}=_{ℎ𝑛−1(𝐵)}detℎ_𝑛𝑑^𝑘_𝑝=11⊗2𝑁Leb_𝑞>𝑘12𝑁Leb_{ℬ([−𝑁𝑏𝑞,𝑁𝑏𝑞])}=_{ℎ𝑛−1(𝐵)}detℎ_𝑛𝑑𝜆^(𝑘)_𝑁,𝑏. (58) Consider the measures 𝜇₁ and 𝜇₂ on ℬ(𝐸_𝑀^(𝑘)_1,𝑁,𝑎) defined by 𝜇₁(𝐵)=_𝐵𝑑𝜆^(𝑘)_𝑁,𝑎;𝜇₂(𝐵)=_{ℎ𝑛−1(𝐵)}detℎ_𝑛𝑑𝜆^(𝑘)_𝑁,𝑏. (59) From (58), 𝜇₁ and 𝜇₂ coincide on the set ℐ= {𝐵∈ℬ(𝐸_𝑀^(𝑘)_1,𝑁,𝑎)∶𝐵=∏_𝑝∈𝐼𝐵_𝑝}; since ℐ is a 𝜋-system on 𝐸_𝑀^(𝑘)_1,𝑁,𝑎 such that 𝜎(ℐ)=ℬ(𝐸_𝑀^(𝑘)_1,𝑁,𝑎) and since 𝜇₁(𝐸_𝑀^(𝑘)_1,𝑁,𝑎)=𝜇₂(𝐸_𝑀^(𝑘)_1,𝑁,𝑎)=(𝑀₁/𝑁)^𝑘∏_𝑝>𝑘𝑎_𝑝<+∞, from Proposition 28, we have that ∀𝐵∈ℬ(𝐸_𝑀^(𝑘)_1,𝑁,𝑎): _{𝐸𝑀1(𝑘),𝑁,𝑎}1_𝐵𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐸𝑀2(𝑘),𝑁,𝑏}1_𝐵ℎ_𝑛detℎ_𝑛𝑑𝜆^(𝑘)_𝑁,𝑏. (60) This implies that if 𝜑∶(𝐸_𝑀^(𝑘)_3,𝑁,𝑎,ℬ(𝐸_𝑀^(𝑘)_3,𝑁,𝑎))→([0,+∞),ℬ([0,+∞))) is a simple function such that 𝜑(𝑥)=0, ∀𝑥∉𝐸_𝑀^(𝑘)_1,𝑁,𝑎, we have _{𝐸𝑀1(𝑘),𝑁,𝑎}𝜑𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐸𝑀2(𝑘),𝑁,𝑏}𝜑ℎ_𝑛detℎ_𝑛𝑑𝜆^(𝑘)_𝑁,𝑏. (61) Then, if 𝑙∶(𝐸_𝑀^(𝑘)_3,𝑁,𝑎,ℬ(𝐸_𝑀^(𝑘)_3,𝑁,𝑎))→([0,+∞),ℬ([0,+∞))) is a measurable function such that 𝜑(𝑥)=0, ∀𝑥∉𝐸_𝑀^(𝑘)_1,𝑁,𝑎, and {𝜑_𝑖}_𝑖∈𝐍 is a sequence of increasing positive simple functions over 𝐸_𝑀^(𝑘)_3,𝑁,𝑎 such that lim_𝑖→+∞𝜑_𝑖=𝑙, 𝜑_𝑖(𝑥)=0, ∀𝑥∉𝐸_𝑀^(𝑘)_1,𝑁,𝑎, ∀𝑖∈𝐍, from Beppo Levi theorem we have _{𝐸𝑀1(𝑘),𝑁,𝑎}𝑙𝑑𝜆^(𝑘)_𝑁,𝑎=lim_{𝑖→+∞𝐸𝑀1(𝑘),𝑁,𝑎}𝜑_𝑖𝑑𝜆^(𝑘)_𝑁,𝑎=lim_{𝑖→+∞𝐸𝑀2(𝑘),𝑁,𝑏}𝜑_𝑖ℎ_𝑛detℎ_𝑛𝑑𝜆^(𝑘)_𝑁,𝑏=_{𝐸𝑀2(𝑘),𝑁,𝑏}𝑙ℎ_𝑛detℎ_𝑛𝑑𝜆^(𝑘)_𝑁,𝑏=lim_{𝑛→+∞𝐸𝑀2(𝑘),𝑁,𝑏}𝑙ℎ_𝑛detℎ_𝑛𝑑𝜆^(𝑘)_𝑁,𝑏. (62)
In particular, the formula (62) is true for any continuous and bounded function 𝑙∶𝐸_𝑀^(𝑘)_3,𝑁,𝑎→[0,1]. In this case, let {𝑓_𝑛}_𝑛∈𝐍 be the sequence of the measurable functions 𝑓_𝑛∶(𝐸_𝑀^(𝑘)_2,𝑁,𝑏,ℬ(𝐸_𝑀^(𝑘)_2,𝑁,𝑎))→(𝐑,ℬ) given by 𝑓_𝑛(𝑥)=𝑙ℎ_𝑛(𝑥)detℎ_𝑛,∀𝑥∈𝐸_𝑀^(𝑘)_2,𝑁,𝑏,∀𝑛∈𝐍. (63) Since detℎ_𝑛=det𝐴, ∀𝑛≥𝑚, we have |𝑓_𝑛|≤𝑔, where 𝑔∶(𝐸_𝑀^(𝑘)_2,𝑁,𝑏,ℬ(𝐸_𝑀^(𝑘)_2,𝑁,𝑎))→([0,+∞),ℬ([0,+∞))) is the measurable function defined by 𝑔(𝑥)=|det𝐴|,∀𝑥∈𝐸_𝑀^(𝑘)_2,𝑁,𝑏. (64) Moreover _{𝐸𝑀2(𝑘),𝑁,𝑏}𝑔𝑑𝜆^(𝑘)_𝑁,𝑏=|det𝐴|𝜆^(𝑘)_𝑁,𝑏𝐸_𝑀^(𝑘)_2,𝑁,𝑏=|det𝐴|2𝑀₂^𝑘(2𝑁)^𝑘_𝑝>𝑘12𝑁Leb−𝑁𝑏_𝑝,𝑁𝑏_𝑝=|det𝐴|𝑀^𝑘₂𝑁^𝑘_𝑝>𝑘𝑏_𝑝<+∞. (65) Moreover, we have lim_𝑛→+∞ℎ_𝑛=𝐴, and so lim_𝑛→+∞𝑓_𝑛=𝑙(𝐴)|det𝐴|; then, from the dominated convergence theorem, lim_{𝑛→+∞𝐸𝑀2(𝑘),𝑁,𝑏}𝑙ℎ_𝑛detℎ_𝑛𝑑𝜆^(𝑘)_𝑁,𝑏=_{𝐸𝑀2(𝑘),𝑁,𝑏}𝑙(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏. (66) Then, from (62) we have _{𝐸𝑀1(𝑘),𝑁,𝑎}𝑙𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐸𝑀2(𝑘),𝑁,𝑏}𝑙(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏. (67)
Let 𝐵=∏_𝑝∈𝐼𝐵_𝑝∈ℬ(𝐸_𝑀^(𝑘)_1,𝑁,𝑎), where 𝐵_𝑝=(𝑎_𝑝,𝑏_𝑝), ∀𝑝∈𝐼; moreover, ∀𝑛∈𝐍^∗, consider the continuous function 𝑙_𝑛∶𝐸_𝑀^(𝑘)_3,𝑁,𝑎→[0,1] defined by 𝑙_𝑛(𝑥)=1if𝑥∈_𝑝∈𝐼𝑎_𝑝+𝛿_𝑝𝑛,𝑏_𝑝−𝛿_𝑝𝑛𝑥−𝑥₂𝑥₁−𝑥₂if𝑥∈𝐵⧵_𝑝∈𝐼𝑎_𝑝+𝛿_𝑝𝑛,𝑏_𝑝−𝛿_𝑝𝑛0if𝑥∉𝐵, (68) where 𝛿_𝑝=(𝑏_𝑝−𝑎_𝑝)/2, ∀𝑝∈𝐼, 𝑥₁=𝑟∩𝜕(∏_𝑝∈𝐼(𝑎_𝑝+(𝛿_𝑝/𝑛),𝑏_𝑝−(𝛿_𝑝/𝑛))), 𝑥₂=𝑟∩𝜕𝐵, where 𝑟 is the half-line with initial point ∏_𝑝∈𝐼((𝑎_𝑝+𝑏_𝑝)/2) and containing 𝑥. Since {𝑙_𝑛}_𝑛∈𝐍 is an increasing positive sequence such that lim_𝑛→+∞𝑙_𝑛=1_𝐵, from Beppo Levi theorem and (67), we have _𝐵𝑑𝜆^(𝑘)_𝑁,𝑎=lim_{𝑛→+∞𝐸𝑀1(𝑘),𝑁,𝑎}𝑙_𝑛𝑑𝜆^(𝑘)_𝑁,𝑎=lim_{𝑛→+∞𝐸𝑀2(𝑘),𝑁,𝑏}𝑙_𝑛(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏=_{𝐴−1(𝐵)}|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏. (69) Moreover, Proposition 28 again implies that the formula (69) is true ∀𝐵∈ℬ(𝐸_𝑀^(𝑘)_1,𝑁,𝑎). Consider the measures 𝜇 and 𝜐 on ℬ(𝐸^(𝑘)_𝑁,𝑎) defined by 𝜇(𝐵)=_𝐵𝑑𝜆^(𝑘)_𝑁,𝑎,𝜐(𝐵)=_{𝐴−1(𝐵)}|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏, (70) and set 𝐵_𝑛=𝐵∩𝐸^(𝑘)_{𝑛,𝑁,𝑎}, ∀𝑛∈𝐍^∗, ∀𝐵∈ℬ(𝐸^(𝑘)_𝑁,𝑎). Since 𝐵_𝑛⊂𝐵_𝑛+1, 𝐴⁻¹(𝐵_𝑛)⊂𝐴⁻¹(𝐵_𝑛+1), ⋃_𝑛∈𝐍^∗𝐵_𝑛=𝐵, and ⋃_𝑛∈𝐍^∗𝐴⁻¹(𝐵_𝑛)=𝐴⁻¹(𝐵), from the continuity property of 𝜇 and 𝜐 and (69), we have _𝐵𝑑𝜆^(𝑘)_𝑁,𝑎=lim_{𝑛→+∞𝐵𝑛}𝑑𝜆^(𝑘)_𝑁,𝑎=lim_{𝑛→+∞𝐴−1𝐵𝑛}|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏=_{𝐴−1(𝐵)}|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏. (71)
Then, ∀𝐷∈ℬ(𝐸^(𝑘)_𝑁,𝑎), _𝐵1_𝐷𝑑𝜆^(𝑘)_𝑁,𝑎=_𝐵∩𝐷𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐴−1()𝐵∩𝐷}|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏=_{𝐴−1(𝐵)}1_𝐴⁻¹_(𝐷)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏=_{𝐴−1(𝐵)}1_𝐷(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏. (72) Thus, by proceeding as in the proof of the formula (62), for any measurable function 𝑓∶(𝐸^(𝑘)_𝑁,𝑎,ℬ(𝐸^(𝑘)_𝑁,𝑎))→([0,+∞),ℬ([0,+∞))), we obtain _𝐵𝑓𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐴−1(𝐵)}𝑓(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏. (73) Then, if 𝑓∶(𝐸^(𝑘)_𝑁,𝑎,ℬ(𝐸^(𝑘)_𝑁,𝑎))→(𝐑,ℬ) is a measurable function such that 𝑓⁺ (or 𝑓⁻) is 𝜆^(𝑘)_𝑁,𝑎-integrable: _𝐵𝑓𝑑𝜆^(𝑘)_𝑁,𝑎=_𝐵𝑓⁺𝑑𝜆^(𝑘)_𝑁,𝑎−_𝐵𝑓⁻𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐴−1(𝐵)}𝑓⁺(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏−_{𝐴−1(𝐵)}𝑓⁻(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏=_{𝐴−1(𝐵)}𝑓(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏. (74)
Finally, suppose that 𝐵∈ℬ(𝐸) and 𝑓∶(𝐸,ℬ(𝐸))→(𝐑,ℬ) is a measurable function such that 𝑓⁺ (or 𝑓⁻) is 𝜆^(𝑘)_𝑁,𝑎-integrable; from formula (74), Proposition 25 and definitions of 𝜆^(𝑘)_𝑁,𝑎 and 𝜆^(𝑘)_𝑁,𝑏 given by (4), we have _𝐵𝑓𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐵∩𝐸(𝑘)𝑁,𝑎}𝑓𝑑𝜆^(𝑘)_𝑁,𝑎=_{𝐴−1𝐵∩𝐸(𝑘)𝑁,𝑎}𝑓(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏=_{𝐴−1(𝐵)}𝑓(𝐴)|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏. (75)

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 𝜆^(𝑘)_𝑁,𝑎-continuous if there exists a measurable function 𝑓_𝑋∶(𝐸,ℬ(𝐸))→([0,+∞),ℬ([0,+∞))) such that, for any 𝐴∈ℬ(𝐸), 𝑃(𝑋∈𝐴)=_𝐴𝑓_𝑋𝑑𝜆^(𝑘)_𝑁,𝑎. (76) The function 𝑓_𝑋 is called infinite-dimensional probability density of 𝑋.

Theorem 31. Let 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗∈𝐼}∶𝐸→𝐸 be a biunique and linear (𝑚,𝜎)-standard function, let 𝑎=(𝑎_𝑛∶𝑛∈𝐼)∈(𝐑⁺)^𝐼 be such that ∏_𝑛∈𝐼𝑎_𝑛∈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 𝜆^(𝑘)_𝑁,𝑏-continuous random element 𝑋∶(Ω,ℱ,𝑃)→(𝐸,ℬ(𝐸)), the random element 𝑇=𝐴∘𝑋∶(Ω,ℱ,𝑃)→(𝐸,ℬ(𝐸)) is 𝜆^(𝑘)_𝑁,𝑎-continuous and one has 𝑓_𝑇(𝑡)=𝑓_𝑋𝐴⁻¹(𝑡)det𝐴⁻¹,∀𝑡∈𝐸. (77)

Proof. ∀𝐵∈ℬ(𝐸), we have 𝑃()1𝑇∈𝐵=𝐸_𝐵(𝑇)=𝐸1_𝐵(𝐴(𝑋=))_𝐸1_𝐵(𝐴(𝑥))𝑓_𝑋(𝑥)𝑑𝜆^(𝑘)_𝑁,𝑏(𝑥)=_{𝐴−1(𝐵)}𝑓_𝑋𝐴⁻¹(𝐴(𝑥))det𝐴⁻¹|det𝐴|𝑑𝜆^(𝑘)_𝑁,𝑏(𝑥)=_𝐵𝑓_𝑋𝐴⁻¹(𝑡)det𝐴⁻¹𝑑𝜆^(𝑘)_𝑁,𝑎(𝑡)(fromTheorem29). (78)

6. Problems for Further Study

A natural extension of this paper is the generalization of Theorem 29 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 Definition 30 and Theorem 31, in the probabilistic context it is possible to introduce many random elements that generalize the well-known continuous random vectors in 𝐑^𝑚 (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 {𝑋_𝑛}_𝑛∈𝐍 on ∏_𝑖∈𝐍^∗[0,𝑝) defined by 𝑋_𝑛+1=𝐴𝑋_𝑛+𝐵_𝑛mod𝑝, (79) where 𝑋₀=𝑥₀∈𝐸, 𝐴 is a (𝑚,𝜎)-standard matrix, 𝑝∈𝐑⁺, and {𝐵_𝑛}_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 𝐵_𝑛, the sequence {𝑋_𝑛}_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 𝐵_𝑛 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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is grateful to Professor Aljosa Volcic for his fruitful suggestions in writing this paper.