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Journal of Mathematics
Volume 2017, Article ID 9853672, 14 pages
https://doi.org/10.1155/2017/9853672
Research Article

The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures

Lycée Jeanne d’Arc, Avenue de Grande Bretagne, 63000 Clermont-Ferrand, France

Correspondence should be addressed to Jean-Pierre Magnot; moc.liamg@tongam.pj

Received 1 July 2016; Revised 16 September 2016; Accepted 19 September 2016; Published 30 January 2017

Academic Editor: Tepper L. Gill

Copyright © 2017 Jean-Pierre Magnot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of means. In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (e.g., on a Hilbert space) is defined. This last object, which is not the classical infinite dimensional Lebesgue measure but its “normalized” version, is shown to be invariant under translation, scaling, and restriction.

1. Introduction

The very early starting point of this work is the well-known lack of adequate definition of an infinite dimensional Lebesgue measure on a Hilbert space. Even if infinite dimensional version of the Lebesgue measure is well known on Hilbert space [1, 2], and translation invariant measures are already described in Banach spaces, these measures fail to have “enough” measurable sets with finite, nonzero measure. From another approach, some expressions of the type are well known since [3]; see, for example, [4, 5] on and we find similar expressions in the theory of infinite dimensional oscillatory integrals; see, for example, [610]. Yet in another setting, there is a property of concentration of measure in metric measured spaces which can coincide with the definition of a mean for uniform functions in, ; see, for example, [1113]. These approaches, except the ones involving infinite dimensional measures, appear as relevant of the same procedure: defining means from limits of measures. This is why, following [14], we suggest a setting in Section 2 for means defined by limits of finite measures. To our knowledge (and surprisingly also), these frameworks have not been gathered yet. We show how what we defined as Dirac means in [14], or their generalizations that we call probability means or limit means, describe a unified framework to deal with concentration properties in metric measured spaces on one hand and integrals of cylindrical functions on the other hand.

The theory developed in Section 2 is then specialized to a restricted class of means, first to the means obtained with a -finite Radon measure, using a creasing sequence of Borel subsets with finite measure satisfying among other technical conditions, by following definitions present in [35]. We give in a way as systematic as possible their basic properties in Section 3. Since this mean value depends (in general) on the sequence and on the measure , we do not adopt the notation but prefer or , abbreviations for “weak mean value” and for “mean value.” Formulas for changing of measure lead us to an extension of the asymptotic comparison of functions ( and to measures. As a particular case, the mean value with respect to the Lebesgue measure on appears as a linear extension of the limit at of functions. We know very few about the behaviour of the mean value of limit of functions: the mean value is not continuous for vague convergence, but continuous for uniform convergence. There is certainly an intermediate kind of convergence more adapted to mean values, to be determined. We also give an application of this notion: the homology map as a mean value of a function on the space of harmonic forms, using Hodge theory.

Secondly, we get to infinite products of measured spaces in Section 4. Recall that there is an induced measure on an infinite product of measured spaces only if we have spaces with finite measures. We consider cylindrical functions and define very easily their mean values as mean values of functions defined on a finite product of measured space. Then, we extend it to functions that are uniform limits of sequences of cylindrical functions. As an application, we give a definition of the mean value on infinite configuration spaces for Poisson measure.

Finally, we get to vector subspaces of Hilbert spaces in Section 4. This is where we decide to focus on the announced heuristic infinite dimensional Lebesgue mean, which is not the infinite dimensional Lebesgue measure described in [1, 2]. The mean value is developed and we study its invariance properties. It appears invariant by translation and by scaling and also by action of the unitary group. But the last one remains dependent on the choice of the orthonormal basis used for the definition, which is classical in the procedure of approximation by cylindrical functions [15]. As a concluding remark, we show that this approach has a technical difference with the approach by measures on infinite dimensional spaces. We show that the mean value of a bounded continuous function remains the same while restricting to a dense vector subspace. This exhibits a striking difference from, for example, the Wiener measure on continuous paths, for which the space of paths is of measure With all these elements, we can now explain where is the originality of our approach. Here, the total volume is not considered as a constant of the total space, but as a scale-like element to compare with the integral of a function.

2. The Space of Means Spanned by Sequences of Finite Measures

Let be a measured space. Following [5, 13], let us fix a vector subspace such that A mean on is a linear map such that Alternately, if is a metric space, given (space of continuous bounded maps), a mean on is a linear map such that These two terminologies come from the basic example where is a Borel probability measure on a metric space , for which the mean of a continuous integrable map is its expectation value and can be approximated by sequences of barycenters of Dirac measures via Monte Carlo methods.

Let be a vector space of (bounded, measurable) maps that contains constant maps, with values in a complete locally convex topological vector space (clctvs) . We shall call all along this paper the space of means, that is, the space of linear maps such that The mean can be defined on another domain , but the space will serve as reference domain. Moreover, we set or .

2.1. Means Spanned by Probability Measures

Let be a complete metric space and let be the space of bounded -valued continuous maps on We note by the space of Borel probability measures on Let us first set .

Definition 1. A -probability mean is a linear map which is defined as the limit of barycenters with -weights of a sequence of Borel probability measures on ,

We note by the space of -probability means, by the set of probability means such that , and by the means obtained by a sequence and we setWe have a special class spanned by the Dirac measures.

Definition 2 (see [14]). A -Dirac mean is a linear map which is defined as the limit of barycenters with -weights of a sequence of Dirac measures on ,

We note by the sets of Dirac means corresponding, respectively, to .

Proposition 3. , and are -affine spaces.

The proof is obvious adapting elementary proofs on (classical, finite) barycenters. We give also the following, in order to make quickly the link with the Monte Carlo method.

Proposition 4. If is moreover a locally compact manifold, one has the following inclusions:(i)(ii)If is compact,

Proof. (i) Let and let be a uniformly distributed sequence with respect to Then, Thus (ii) If is compact, the space of (signed) finite measures on coincides with Since , we get the result.

2.2. Probability Means in the mm-Space Setting

We use two handbooks for preliminaries on these notions: [11, 13].

Definition 5 (see [12]). A space with metric and measure, or a metric measured space (mm-space for short), is a triple , where is a metric space and is a probability measure on the Borel tribu on

Let , and let We note

Definition 6 (see [12]). A Levy family is a sequence of mm-spaces if, for each sequence ,satisfying and then

In the sequel, we shall assume that with continuous injection. Notice that we do not assume that is the restriction of which allows us some freedom on metric requirements. The technical necessary condition is the following: let and let be a Borel subset of Then is a Borel subset of We have here a priori a class of limit means following the terminology of Definition 11. Let us quote first the classical (and historical) example of a Levy family; see, for example, [11], section , which gives an example of mean value.

Example 7 (the Levy family of spheres and the concentration phenomenon). Let us consider the sequence of inclusions equipped with the classical Euclide (or Hilbert) distance and (except for ) the normalized spherical measure (we drop the index for the measure in sake of clear notations). Then, for any -valued 1-Lipschitz function on , there exists such that

In a more intuitive formation, one can say that any 1-Lipschitz function concentrates around a real value with respect to . We leave the reader with [11] for more on the metric geometry of this example.

Proposition 8. Let . Then for any 1-Lipschitz function defined on , and with the notations used before,

Example 9. For Levy families induced by Lebesgue measures let . Take For each , we equip with the usual distance induced by and with the probability measure Setting , we get that is a Levy family, but there is no concentration property. This example will be studied in the next sections of this article.

Definition 10. Let be a map such that, for each , the restriction of to is -integrable. Then, the mean value of with respect to the family is if the limit exists.

2.3. Limit Means and Infinite Dimensional Integrals

Definition 11. Let be a sequence of probability spaces such that(i) is a metric space;(ii), and the topology of restricted to coincides with the topology of ;(iii) Then, we define, for the maps defined on , if , and if the limit converges,called limit mean of with respect to .

This definition intends to fit with the procedure of integration of cylindrical functions in Hilbert spaces. Let us first describe the “toy” example of these infinite dimensional integrals, where such an approach is not needed: the limit mean considered is in fact a Dirac mean.

Example 12 (Daniell integral). Let us consider cylinder functions on Let be the -dimensional projection. Then, there exists such thatThen, adequate sequences for the Monte Carlo method are those whose push-forwards on are also adequate for this method. The projectors converge (weakly) to identity, the condition on the sequence is that, for each , the push-forwards of the sequences on fit with the desired conditions: the sequence is a Monte Carlo sequence for the cube equipped with the (trace of) Lebesgue measure. It is well known that such a sequence exists, through, for example, the powers of : where is the integer part of the real number Thus, Daniell integral appears by its definition as a limit mean for the sequence defined by , equipped with the classical Lebesque measure. But Daniell integral appears also as a Dirac mean whose domain contains cylindrical functions.

This example motivates the comparison between limit means, and

Example 13 (Fresnel-type integrals). First, let There exists a sequence satisfying Definition 1, and for each , there exists a sequence of Dirac measures which converge to with respect to the Monte Carlo method. Thus, finding a sequence of Dirac measures which can define a Dirac mean which coincides with becomes a problem of extracting a sequence of Dirac measures which converges to The same is for a limit mean = “”. Let us describe more precisely the open problems on the example of oscillatory and Fresnel integrals. Let be a fixed function. Following [16] (see, e.g., [7, 8, 17, 18]), we define the following.

Definition 14. Let be a measurable function on Let be a weight function such that If the limit exists and is independent of the fixed function , then this limit is called oscillatory integral of with respect to , noted by

The choice is of particular interest and is known under the name of Fresnel integral. This choice gives us a mean, up to normalization by a factor , and can be generalized to a Hilbert space the following way.

Definition 15. A Borel measurable function is called -integrable in the sense of Fresnel for each creasing sequence of projectors such that , and the finite dimensional approximations of the oscillatory integrals of are well defined and the limit as does not depend on the sequence . In this case, it is called infinite dimensional Fresnel integral of and noted by

The invariance under the choices of the map and the projections is assumed mostly to enable stronger analysis on these objects, which intend to be useful to describe physical quantities and hence can be manipulated in applications where one sometimes works “with no fear on the mathematical rigor” in calculations. But we can also remark that(i)for functions defined on , the map (ii)the map is a limit mean through the sequence The limit mean obtained is got through the classical trick of cylindrical functions, which we shall also use in the sequel. But we have no way to define some adequate sequence of Dirac means which could approximate the oscillatory integral, even in the finite dimensional case.

Following another approach, from [14], one can try to give another definition to the oscillatory integral straightway. Let us begin with integration of cylinder functions on the infinite cube (Daniell integral) as described in the previous example. Let us consider now cylinder functions on Let be a finite dimensional projection such that and let Then, adequate sequences for the Monte Carlo method are those whose push-forwards on are also adequate for this method. Taking now a creasing sequence of orthogonal projectors converging (weakly) to identity, the condition on the sequence is that, for each , the push-forwards of the sequences on fit with the desired conditions: the sequence is a Monte Carlo sequence for the cube equipped with the (trace of) Lebesgue measure. It is well known that such a sequence exists, through, for example, the powers of : where is the integer part of the real number Let us now fix The maps defined by, for example, or are diffeomorphisms and we get thus a pull-back of the Monte Carlo method can be performed this way to get a Dirac mean which could be considered as an oscillatory integral. But nothing can ensure, to our knowledge, that this approach defines the same oscillatory integral as Definition 14.

3. Example: Mean Value on a Measured Space

On measured spaces, the definitions are those given in classical mathematical literature; see for example [35].

3.1. Definitions

On measured spaces, the definitions are those given in classical mathematical literature, see for example, [3]. Let be a topological space equipped with a -additive, positive measure Let be a Borel -algebra on . We note by the set of sequences such that(1);(2) and

Remark 16. We have in particular
In what follows we assume the natural condition

Definition 17. Let Let be a separable complete locally convex topological vector space (sclctvs). Let be a measurable map. We define, if the limit exists, the weak mean value of with respect to asMoreover, if does not depend on , we call it mean value of , noted by

Notice that(i)if , setting and for each , if , and have a finite mean value.(ii)The same way if for each (iii)We denote by the set of functions such that exists in and by the set of functions such that is well defined.

Examples 1. (1) Let be an arbitrary measured space. Let Let so that(2) Let be a space equipped with the Dirac measure at . Let be an arbitrary map to an arbitrary sclctvs. Thus, if so that (3) Let be a measured space with . Let be an arbitrary bounded measurable map. Then one can show very easily that we recover the classical mean value of :(4) Let equipped with the classical Lebesgue measure Let (integrable -valued function). Let We have that so that(5) Let equipped with the Lebesgue measure Let and let The map is odd so that Now, let Then This shows that has no (strong) mean value for the Lebesgue measure.
(6) Let equipped with , the counting measure. Let and set Let and Then, is the Cesàro limit.

3.2. Basic Properties

In what follows and till the end of this paper we assume the natural condition for the measures we consider.

Proposition 18. Let be a measured space. Let Then(1) is a vector space and is linear;(2) is a vector space and is linear.

Proof. The proof is obvious.

We now clarify the preliminaries that are necessary to study the perturbations of the mean value of a fixed function with respect to perturbations of the measure.

Proposition 19. Let and be Radon measures. Let Assume that exists. Then and

Proof. Thus, we get the result taking the limit.

Proposition 20. Let be a Radon measure and let Then ; moreover and

Proof. The proof is  obvious.

Theorem 21. Let be a measure on , let , and Let is a convex cone.

Proof. Let and let . Setting , we get by Proposition 20; thus is a cone.
Now, let Let and let (i)Let us show that . Let We have , so that (ii)Let us show that . We already know that Let , Now, we remark that and that since and , and finally that the same way. Thus, , thus is a convex cone.

3.3. Asymptotic Comparison of Radon Measures

We now turn to the number that appeared in Proposition 19. In this section, and are fixed Radon measures and is a fixed sequence in .

Proposition 22. One has the following.(1)(2)

Proof. The proof is obvious.

Definition 23. One has the following.(1) if (2) if (3) if

Let us now compare three measures , and . is here a fixed arbitrary sequence.

Lemma 24. Let and be two measures and let ,where

Proof. The  proof is obvious.

Proposition 25. Let (1) if .(2) if .

Proof. Let We have and For the first part of the statement, (since these numbers are positive, the equality makes sense). Thus, if the limits are compatible, we get (1) taking the limits of both parts. Then, we express each part as and we get Taking the limit, we get (2).

We recover by these results a straightforward extension of the comparison of the asymptotic behaviour of functions. The notation chosen in Definition 23 shows this correspondence. Through easy calculations of or , one can easily see that if , , and are comparable measures,(1);(2);(3);(4);(5);(6);

and other easy relations can be deduced in the same spirit.

3.4. Limits and Mean Value

If is, for example, a connected locally compact, paracompact, and not compact manifold, equipped with a Radon measure such that , any exhaustive sequence of compact subsets of is such that . In this setting, it is natural to consider the Alexandroff compactification of

Theorem 26. Let be a bounded measurable map which extends to , a continuous map at Then for each exhaustive sequence of compact subsets of

Proof. We can assume that ; in other words The sequence gives a basis of neighborhood of ; thus Moreover, since , Let Let We set and Then, , The second term is bounded by and we majorate the first term by Thus and hence

As mentioned in the Introduction, we found no straightforward Beppo-Levy type theorem for mean values. The first counterexample we find is, for and the Lebesgue measure, an increasing sequence of which converges to (uniformly on each compact subset of , the sequence Let and We have , and by Theorem 26. We can only state the following theorem on uniform convergence.

Lemma 27. Let be a measure on and let Let and be two functions in , where in a sclctvs.
Let be a norm on If there exists such that , then

Proof. Let , andWe get in the same way The result is obtained by taking the limit.

Theorem 28. Let be a sequence which converges uniformly on to a -measurable map Then (1)(2)

Proof. Let
(i) Let us prove that has a limit . Let be a norm on Let There exists such that, for each , Thus, by Lemma 27 with and , Thus, the sequence is a Cauchy sequence. Since is complete, the sequence has a limit
(ii) Moreover, we remember that , (iii) Let us prove that Let ,Let Let such that Then Let such that, for each , Then, by the same arguments, for each , Gathering these inequalities, we get

3.5. Invariance of the Mean Value with respect to the Lebesgue Measure

In this section, with is the Lebesgue measure, is the renormalization procedure defined by and is the renormalization procedure defined by where is the Euclidian norm. We denote by the sup norm and its associated distance. Let We use the obvious notations and for the translated sequences. Let We denote by the symmetric difference of subsets.

Proposition 29. Let Let (resp., ) be a bounded function. Let and If then (1) (resp., ) and ;(2)for we have and ,

Proof. We first notice that the second item is a reformulation of the first item: by change of variables
Let us now prove the first item. Let , andLet Then Thus, we get the result.

Lemma 30. One has the following.

Proof. We prove it for the sequence , and the proof is the same for the sequence We have ; thus and Let We have and . Thus,

Proposition 31. Let Let (resp., ) be a bounded function.(1)Then (resp., ) and (resp., ).(2)Let Then (resp., ) and (resp., ).

Proof. The proof for and is a straightforward application of Proposition 29 which is valid thanks to the previous lemma.

3.6. Example: The Mean Value Induced by a Smooth Morse Function

In this example, is a smooth, locally compact, paracompact, connected, oriented, and noncompact manifold of dimension equipped with a measure induced by a volume form and a Morse function such that For the theory of Morse functions we refer to [19]. Notice that there exists some value such that Notice that we can have

Definition 32. Let be a smooth function into a sclctvs Let We define and, if the limit exists,

Of course this definition is the “continuum” version of the “sequential” Definition 17. If is metrizable, for any increasing sequence such that , setting , and conversely exists if exists and does not depend on the choice of the sequence

Moreover, since is a Morse function, it has isolated critical points and changing into , where is the set of critical points of , for each , is an -dimensional manifold (disconnected or not). The first examples that we can give are definite positive quadratic forms on a vector space in which is embedded.

3.7. Application: Homology as a Mean Value

Let be a finite dimensional manifold equipped with a Riemannian metric and the corresponding Laplace-Beltrami operator and with finite dimensional de Rham cohomology space One of the standard results of Hodge theory is the onto and one-to-one map between and the space of -harmonic forms made by integration over simplexes: where We have assumed here that the simplex has the order of the harmonic form. This is mathematically coherent stating if and do not have the same order. Let be the Lebesgue measure on with respect to the scalar product induced by the -scalar product. Let be the sequence of Euclidian balls centered at such that, for each , the ball is of radius

Proposition 33. Assume that is finite dimensional. Let be a simplex. Let The cohomology class of is null if and only if

Proof. (i) If the cohomology class of is null then, ; thus Finally, (ii) If the cohomology class of is not null, let be the corresponding element in We have Let be the projection onto the 1-dimensional vector space spanned by Let . Let Then, Moreover, Then Thus

Remark 34. A very easy application of Theorem 26 also shows that the map is a -valued map.

4. The Mean Value on Infinite Products

4.1. Mean Value on an Infinite Product of Measured Spaces

Let be an infinite (countable, continuous, or other) set of indexes. Let or for short be a family of measured spaces as before. We assume that, on each space , we have fixed a sequence Let be the Cartesian product of the sequence

Definition 35. Let for the product topology. is called cylindrical if and only if there exists , a finite subset of , and a map such that Then, we set, if , In this case we use the notation: (here, subsidiary notations are omitted since the sequence of measures and the sequences of renormalization are fixed in this section).

Notice that if we have with the notations used in the definition, and since is constant with respect to the variables indexed by , the definition of does not depend on the choice of , which makes it coherent.

Theorem 36. Let be a cylindrical function associated with the finite set of indexes