Abstract
We study, for any positive integer and for any subset of , the Banach space of the bounded real sequences and a measure over that generalizes the -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 . This change of variables is defined by some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms.
1. Introduction
The aim of this paper is to generalize the results of article [1], where, for any positive integer and for any subset of , we study a particular infinite-dimensional measure that, in the case , coincides with the -dimensional Lebesgue one on . The measure is a product indexed by of -finite measures on the Borel -algebra on (by using a generalization of the Jessen theorem), and it is defined over the measurable space , and in particular over , where is the product indexed by of the same -algebra , is the Banach space of the bounded real sequences , and is the restriction to of .
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 -finite measures on , and that one of Baker [4], which defines a measure on that is not -finite).
In paper [1], the main result is a change of variables’ formula for the integration of the measurable real functions on the space . This change of variables is defined by a particular class of linear functions over , called -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 Section 2, we construct the infinite-dimensional Banach space , 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 and the diffeomorphisms. Moreover, we introduce a class of functions, called -standard, that generalizes the set of the linear -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 ; this change of variables is defined by the biunique, and -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 by and the sets and , respectively. Let be a set and let ; indicate by , by , by , by , by , by , by , and by , respectively, the Euclidean topology on , the Euclidean topology on , the topology , the Borel -algebra on , the Borel -algebra on , the -algebra , the Lebesgue measure on , and the Lebesgue measure on . Moreover, for any set , indicate by the -algebra induced by on , and by the topology induced by on ; analogously, for any set , define the -algebra and the topology . Finally, if is a Cartesian product, for any and for any , define , and define the projection on as the function given by .
Theorem 1. Let be a set and, for any , let be a measure space such that is finite. Moreover, suppose that, for some countable set , is a probability measure for any and . Then, over the measurable space , there is a unique finite measure , indicated by , such that, for any such that and for any , where , we have In particular, if is countable, then for any .
Proof. See the proof of Corollary in Asci [1].
Henceforth, we will suppose that are sets such that ; moreover, for any , we will indicate by the set of the first elements of (with the natural order and with the convention if ) and analogously ; finally, for any , set .
Theorem 2. Let be a probability space, for any , let , and let . Moreover, for any such that , define the measurable function bywhere is the probability space . Then, -a.e. one has .
Proof. See, for example, Theorem , page 349, in Rao [6].
Corollary 3. Let be a measure space such that is finite, for any , and ; moreover, let , let , and let the measurable function defined by (1), for any . Then, -a.e. one has .
Proof. Suppose that , ; then, is a probability measure, and also , , and , ; then, define the function byFrom Theorem 2, -a.e., we haveConversely, if , for some , thenThus, since , we obtain the statement.
Definition 4. For any set , define the function byand define the vector spaceMoreover, indicate by the -algebra , by the topology induced on by the product topology , and by the topology induced on by the distance defined by , ; furthermore, for any set , indicate by the topology induced by on . Finally, for any and for any , indicate by the set .
Observe that , ; moreover, is a Banach space, with the norm (see, e.g., the proof of Remark in [1]).
Proposition 5. Let and be sets such that ; then, the function is continuous and open.
Proof. See, for example, the theory of the product spaces in Weidmann’s book [7].
Proposition 6. Let and be sets such that , and let be the function given by , for any ; then,(1) is continuous and open;(2) is continuous and open.
Proof of (1). , there exists such that , and so ; then, since by Proposition 5, we have , and so is a continuous function. Moreover, , there exists such that , and so ; then, since by Proposition 5, we have , and so is an open function.
Proof of (2). and , we have , and so there exists and such that ; let such that , , and , ; then, , and so is a continuous function. Moreover, and , let such that , ; since , there exists and such that , and so ; then, is an open function.
Definition 7. Let , let , let , and let be a function; we say that if, for any neighbourhood of , there exists a neighbourhood of such that, for any , we have .
Definition 8. Let and let be a function; we say that is continuous in if , and we say that is continuous in if is continuous in for any .
Remark 9. Let , let , and let be a function; then, is continuous if and only if is continuous in .
Remark 10. Let , , and be sets such that , let , and let be a function continuous in ; then, the function is continuous in .
Proof. The statement follows from Proposition 6.
Definition 11. Let and let ; a function is called homeomorphism if is biunique and the functions and are continuous.
3. Differentiation Theory in the Infinite-Dimensional Context
The following concept generalizes Definition in [1] (see also the theory in the Lang’s book [8]).
Definition 12. Let be a real matrix (eventually infinite); then, define the linear function , and write , in the following manner:on condition that, for any , the sum in (7) converges to a real number.
Definition 13. Let ; a function is called differentiable in if there exists a linear and continuous function defined by a real matrix , and one has
If is differentiable in for any , is called differentiable in . The function is called differential of the function in , and it is indicated by the symbol .
Remark 14. Let and let be differentiable functions in ; then, for any , the function is differentiable in , and .
Proof. Observe that
Remark 15. A linear and continuous function , defined byis differentiable and , for any .
Remark 16. Let and let be a function differentiable in ; then, for any , the component is differentiable in , and is the matrix given by the th row of . Moreover, if and is differentiable in , for any , then is differentiable in .
Remark 17. Let and let be a function differentiable in ; then, is continuous in .
Proof. , setFrom (8), we have ; moreover,from which .
Definition 18. Let and let such that ; a function is called derivable in in the direction if there exists the limitThis limit is indicated by , and it is called derivative of in in the direction . If for some one has , where , for any , indicate by , and call it partial derivative of in , with respect to . Moreover, if there exists the linear function defined by the matrix , where , for any and , then is called Jacobian matrix of the function in .
Remark 19. Let and suppose that a function is differentiable in ; then, for any such that and for any , the function is derivable in in the direction , and one has
Proof. , by setting , for some matrix we haveThen, since , we havefrom which
Corollary 20. Let and let be a function differentiable in ; then, the linear function is defined and continuous; moreover, for any , one has .
Proof. From Remark 19, and , we have .
Theorem 21. Let , let be a function differentiable in , let such that , and let be a function differentiable in . Then, the function is differentiable in , and one has .
Proof. The proof is analogous to that one true in the particular case , , (see, e.g., the Lang’s book [9]).
Definition 22. Let , let , and let be a function derivable in with respect to , such that the function is derivable in with respect to . Indicate by and call it second partial derivative of in with respect to and . If , it is indicated by . Analogously, for any and for any , define and call it th partial derivative of in with respect to .
Definition 23. Let and let ; a function is called in if, in a neighbourhood of , for any and for any , there exists the function defined by , and this function is continuous in ; is called in if, for any , is in . Moreover, is called strongly in if, in a neighbourhood of , there exists the function defined by , and this function is continuous in , with . Finally, is called strongly in if, for any , is strongly in .
Definition 24. Let and let ; a function is called diffeomorphism if is biunique and in , and the function is in .
Remark 25. Let and let be a function in , where , ; then, is strongly in .
Theorem 26. Let , let be a function in , let , and let be a permutation of . Then, one has
Proof. The proof is analogous to that one true in the particular case (see, e.g., the Lang’s book [9]).
Proposition 27. Let , where , for any , and let be a function in , such thatwhere , and ; moreover, suppose that there exists a neighbourhood of such that . Then, is continuous in ; in particular, if is strongly in and , is differentiable in .
Proof. Since is in , from (19) there exists a neighbourhood of that we can suppose to be , such that , where , , and such that, and , exists in . Let ; from the Lagrange theorem, and , there exists such that . Then,where . Thus, if , , we havefrom whichthen, is continuous in . Moreover, we haveThen, if is strongly in and , we obtainand so is differentiable in , with .
Definition 28. Let , let , where , is an open interval, for any , and let be an increasing function. A function is called -standard if(1), there exist some functions and such that, , one has ;(2) and , one has and , ;(3), the function is constant or injective derivable; moreover, , there exists , where .If , , is called -standard; moreover, if the sequence converges uniformly to , with respect to , then is called strongly -standard.
Furthermore, , define the function by , whereFinally, define the -standard function in the following manner:and indicate by .
Remark 29. Let be a -standard function. Then,(1)if is injective, for any , , is -standard;(2) is biunique if and only if , ;(3)if , there exist and , , such that, for any , one has .
Proof. Points and follow from the fact that is increasing. Moreover, the proof of the point is trivial.
Remark 30. Let be a -standard function such that , and is injective; then, there exists , , such that, for any , is bounded. In particular, if , is not surjective.
Proof. Since , from Remark 29, there exists , , such that, , the set is bounded; then, let ; since is injective and increasing, we have . Moreover, we have ; indeed, , the set is bounded, and so there exists such that ; by supposing by contradiction , such that , , we would obtain (a contradiction). Then, there exists , , such that, , is bounded. In particular, , the function is not surjective; then, if , is not surjective.
Proposition 31. Let be a -standard function; then,(1)suppose that is injective, , for any such that , the function is , for any , and , ; then, the functions , for any , and are injective, and is biunique.(2)if is biunique, then the functions , for any , , and are biunique.
Proof of (1). Suppose that is injective, , for any such that , and let ; we have , since otherwise we would obtain , , for some , from which , and this should contradict the assumption; then, is injective. Moreover, must be injective; in fact, by supposing by contradiction that , for some , then(a contradiction). Moreover, is surjective; in fact, suppose by contradiction that there exists ; since is injective, , there is a unique such that , and soThen, consider the function defined byFrom (28) and by assumption, we havethen, there exist a neighbourhood of and a neighbourhood of such that, , there exists a unique such that (a contradiction with the uniqueness of such that ). Finally, let be such that , and let be such that and , , , . We havefrom which ; then, since is injective, we have , and so . Then, the function is injective.
Proof of (2). Suppose that is injective, , and let be such that ; then, we have , from which ; then, if is biunique, from Remark 29, we have , and so . This implies thatand so, if is injective, we have ; then, ; that is, is injective.
Proposition 32. Let be a -standard function; then:(1)Suppose that is biunique, the function is , for any , and , for any . Then, the functions , for any , and are biunique.(2)If the functions , for any , and are biunique, then is biunique.
Proof of (1). If is biunique, such that ; we have ; then, from Proposition 31, the functions , , and are injective, and is biunique; thus, , we have . Moreover, , is surjective, since is surjective. Furthermore, , let and let be such thatmoreover, let , from which , . Since is injective, we have , ; then, , we havewhere ; then, is surjective.
Proof of (2). If the functions , , and are biunique, from Proposition 31, we obtain that is injective. Moreover, , define in the following manner:whereLet ; , we havemoreover, the function is derivable, andthen, the Lagrange theorem implies that, for some , we havethus, from (37) and (38), we obtainMoreover, we have , from which there exists such that , , we have ; then, there exists such that , and so formula (40) impliesthen, we have , from which . Finally, it is easy to prove that , and so is surjective.
Remark 33. Let be a -standard function such that , for any , for any , and for any ; then,(1)if is injective, and , for any such that , then the functions , for any , and are injective, and is biunique.(2)if is biunique, then the functions , for any , and are biunique.
Proof of (1). Suppose that is injective; by proceeding as in the proof of Proposition 31, we have that the functions , , , and are injective. Moreover, is surjective; in fact, suppose by contradiction that there exists , and let ; moreover, , , let be the sequence defined by , , and ; we havethus, we have , and so is not injective (a contradiction).
Proof of (2). Since is surjective, the functions , , and are surjective; moreover, such that , we have ; then, since is injective, from point 1, the functions , , and are injective, and is biunique.
Corollary 34. Let be a -standard function such that , for any such that , and is injective; then, the functions , , and , for any , , are injective.
Proof. Observe that is -standard, and , such that ; then, from Remark 33, we have that the functions , , and are injective, and is biunique; then, from Proposition 31, is injective; analogously, since , , the function is -standard, from Proposition 31 is injective, Moreover, we have , such that ; then, from Remark 33, is injective.
Proposition 35. Let be a function in and -standard. Then, for any , the function is in , the function is in , there exists the function , and it is continuous. Moreover, if is in and strongly -standard, then is differentiable in . Finally, if is strongly in and strongly -standard, then is differentiable in .
Proof. By assumption, there exists a neighbourhood of such that, , there exists the function on , and this function is continuous in ; then, , let such that ; since is a -standard function, , we havefrom which is in . Moreover, , we haveand so is in . Furthermore, , the function is differentiable in , since depends only on a finite number of variables, and so there exists the function ; moreover, since we have and since the sequence is convergent, this implies that , and so ; then, there exists the function , and it is continuous.
Moreover, suppose that is in and strongly -standard; then, the sequence converges uniformly to , with respect to ; thus, , there exists such that, , , and , we haveObserve that, since the function is differentiable in , there exists a neighbourhood of such that, , we haveMoreover, such that , is derivable in and so, from the Lagrange theorem, there exists such thatfrom whichThen, from (46) and (48), we haveand so is differentiable in .
Finally, if is strongly in and strongly -standard, the function given byis strongly in , and so it is differentiable in from Proposition 27; then, since is differentiable in , this is true for too, from Remark 14.
Proposition 36. Let be a function and -standard; then, is -measurable.
Proof. Let , and let , ; from Proposition 35, the function is . Then, , we have ; moreover, if we consider as a function from to , we have , and so ; then, since , , we have , and so . Moreover, since , the function is -measurable. Let, we haveFinally, since , , we have .
Proposition 37. Let be a -standard function, such that is a homeomorphism. Then, the functions and , , are homeomorphisms, and is biunique.
Proof. Since is a homeomorphism, we have ; then, from Proposition 6, such that , we have ; thus, since is injective, from Remark 33, the functions , , and are injective, and is biunique; then, the functions and , , are derivable, and so they are continuous. Moreover, we have , , . Finally, from Remark 10, the function is continuous; then, , we have , from which ; then, is continuous; analogously, we can prove that the function is continuous.
Proposition 38. Let be a -standard function such that , for any such that . Then, is a diffeomorphism if and only if the functions and , , are diffeomorphisms, and is biunique.
Proof. We have , for any such that . If is a diffeomorphism, then is injective, and so, from Remark 33, the functions , , and are injective, and is biunique. Moreover, is in , and so, from Proposition 35, is in ; analogously, since is in , then is in ; then, is a diffeomorphism. Moreover, , , let such that ; we have , and so is in ; analogously, , , let such that ; we have , and so is in ; then, is a diffeomorphism.
Conversely, if the functions and , , are diffeomorphisms and is biunique, then is injective from Remark 33; moreover, , set ; we haveand so is in ; analogously, , set ; we haveand so is in ; then, is a diffeomorphism.
Proposition 39. Let be a -standard function such that is , for any ; moreover, suppose that is injective and ; then,(1)if is an open function, then, for any , , the function is a homeomorphism, and one has .(2)if is a diffeomorphism, then, for any , , the functions and are diffeomorphisms.
Proof of (1). Since , such that , we have ; then, from Corollary 34, , , the function is injective. Moreover, from Remark 33, the functions , , and are injective, and is biunique. Furthermore, from Proposition 35 and the continuity of the functions , , we have that is continuous in . Finally, , let ; , we have ; then, ,and sowhereThen, since the functions , , and are continuous, the function is continuous too, and so ; then, is a homeomorphism. Furthermore, since , we have , and so
Proof of (2). If is a diffeomorphism, from Proposition 38, the functions and , , are diffeomorphisms; in particular, the functions and , , are ; then, since , , the function is , from (56) and (57), we obtain that the function is in . Moreover, from Proposition 35 and since, , the functions are , is in , and so is a diffeomorphism. Finally, since , such that , we have ; then, from Proposition 38, is a diffeomorphism.
Remark 40. A linear function is -standard, where and is an increasing function, if(1), ;(2)there exists , where , .
Recall the following concept, defined in [1].
Definition 41. Let be a linear -standard function; define the determinant of , and call it , or , the real numberwhere is the real matrix defined by , .
Proposition 42. Let be a linear and injective -standard function; then, is biunique if and only if is ()-open.
Proof. From Remark in [1], is continuous; then, if is biunique, from the Banach theorem of the open function, is -open. Conversely, if is -open, such that , we have ; moreover, since is linear, the function is , ; furthermore, by proceeding as in the proof of Proposition 31, we obtain that is injective, and so , ; then, from Proposition 31, the functions , , are injective, and is biunique. Thus, since the functions and , , are linear, they are biunique, and so is biunique from Proposition 32.
4. Change of Variables’ Formula
Henceforth, we will suppose .
Definition 43. Let , let , let such that , and let ; define the following sets in :Moreover, define the -finite measure over in the following manner:
Proposition 44. Let be a -standard function such that is biunique; moreover, let , let such that , and let ; then,(1)there exist and such that and such that, for any , , , one has(2)suppose that is , for any , and the function is open; then, for any , there exists such that, for any , , , one has
Proof of (1). Since is biunique, from Remark 33, the functions , , and are biunique. Set , such that , ; moreover, defineObserve that, , we have if and only if ; then, by definition of , we haveMoreover, since the function is derivable on , if , the Lagrange theorem implies that, for some , we havefrom which . Furthermore, , we havefrom whichThen, from (67) and (68), , , we obtainAnalogously, , , from (67) and (68), we have .
Proof of (2). Suppose that is , for any , and is an open function; thus, , , and , let ; , we have ; then, ,and sowhereMoreover, since , , the functions and are continuous and , there exists such thatand so ; then, since the function is continuous, there exists such that ; finally, if , are the sequences defined by the point , by setting , from (67), we have . Analogously, , , we havefrom which .
Proposition 45. Let be a measurable space, let be a -system on , and let and be two measures on , -finite on ; if and and coincide on , then and coincide on .
Proof. See, for example, Theorem in Billingsley [10].
Lemma 46. Let , let , let such that , and let ; then, for any measurable function such that (or ) is -integrable, one has
Proof. Let ; by definition of , we havethen, consider the measures and on ; from (76), and coincide on the set ; moreover, we have , where , , , and so and are -finite on . Then, since is a -system on such that , from Proposition 45, formula (76) is true . This implies that, if is a simple function, we haveThen, if is a measurable function, and is a sequence of increasing positive simple functions over such that , from Beppo Levi theorem we haveThen, for any measurable function such that (or ) is -integrable,
Now, we can prove the main result of our paper, that improves Theorem in [1] and generalizes the change of variables’ formula for the integration of a measurable function on with values in (see, e.g., the Lang’s book [9]).
Theorem 47 (change of variables’ formula). Let be a and -standard function, such that the function is a diffeomorphism; moreover, let , let such that , let , and let and defined by Proposition 44. Then, for any , , for any , and for any measurable function such that (or ) is -integrable, one has
Proof. Let , ; moreover, since is biunique, , such that , and , let and let and be the sequences defined by Proposition 44. Then, , we have and , , and soMoreover, we haveThen, from Proposition 39, formula (81) impliesConsider the measures and on defined byfrom (83), and coincide on the setmoreover, we have , , and so and are -finite on . Then, since is a -system on such that , from Proposition 45, , we havemoreover, since from Proposition 44, we haveThis implies that, if is a simple function such that , we haveThen, if is a measurable function such that , , and is a sequence of increasing positive simple functions over such that , , , from Beppo Levi theorem we have In particular, formula (89) is true if is a function of the formwhere is a continuous function such that , and , . In this case, let be the sequence of the measurable functions given by, we have , where is the measurable function defined byMoreover, we havethen, by definition of and since the functions and , , are continuous, there exists such that , , and soMoreover, we have , and so , ; then, from the dominated convergence theorem:consequently, from (89), we haveLet , where , ; moreover, , consider the function defined by where, , , , , where is the half-line with initial point and containing . Since is an increasing positive sequence such that , and since the function is continuous, from Beppo Levi theorem and (96), we have Moreover, Proposition 45 implies that formula (99) is true . Consider the measures and on defined byand set , and . Since , , and , from the continuity property of and and (99), we haveThen, suppose that ; from Lemma 46, formula (101), and Proposition 44, we have thus, by proceeding as in the proof of formula (89), for any measurable function , we obtainThen, for any measurable function such that (or ) is -integrable and for any , we have
5. Problems for Further Study
A natural extension of this paper is the generalization of Theorem 47, by substituting the -standard functions for more general functions such that, for any , the function 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 -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 (e.g., the Gaussian random elements in defined by the -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 on defined by where , is a biunique, linear, integer and -standard function, , and is a sequence of independent and identically distributed random elements on . Our target is to prove that, with some assumptions on the law of , the sequence converges with geometric rate to a random element with law . Moreover, we wish to quantify the rate of convergence in terms of , , , and the law of , and to prove that, if has an eigenvalue that is a root of , then steps are necessary to achieve randomness.
Competing Interests
The author declares that there is no conflict of interests regarding the publication of this paper.