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

Authors: Jean-Pierre Magnot
Journal: Journal of Mathematics
 (2017)

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## 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.


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## Body

## 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 (1)lim⁡1μUn∫Unfdμare well known since [3]; see, for example, [4, 5] on Rn and we find similar expressions in the theory of infinite dimensional oscillatory integrals; see, for example, [6–10]. 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, forexample,S∞; see, for example, [11–13]. 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 Section2 is then specialized to a restricted class of means, first to the means obtained with a σ-finite Radon measure, using a creasing sequence (Un)n∈N of Borel subsets with finite measure satisfying ⋃n∈NUn=X among other technical conditions, by (2)f¯=limn→+∞⁡1μUn∫Unfdμ,following definitions present in [3–5]. 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 U and on the measure μ, we do not adopt the notation f¯ but prefer WMVμU(f) or MVμ(f), 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 (f~∞g,f=O(g) and f=o(g)) to measures. As a particular case, the mean value with respect to the Lebesgue measure on R 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 Section4. 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 Section4. 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 f 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 H1 paths is of measure 0. 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(X,μ) be a measured space. Following [5, 13], let us fix a vector subspace F⊂L∞(X,μ) such that 1X∈F. Amean on F is a linear map ϕ:F→C such that ϕ(1X)=1. Alternately, if (X,d) is a metric space, given F⊂Cb0(X) (space of continuous bounded maps), amean on F is a linear map ϕ:F→C such that ϕ(1X)=1. These two terminologies come from the basic example where μ is a Borel probability measure on a metric space (X,d), for which the mean of a continuous integrable map f is its expectation value (3)∫Xfdμand can be approximated by sequences of barycenters of Dirac measures via Monte Carlo methods.LetF0 be a vector space of (bounded, measurable) maps that contains constant maps, with values in a complete locally convex topological vector space (clctvs) V. We shall call all along this paper M(X,F0) the space of means, that is, the space of linear maps ϕ:F0→R such that ϕ(1X)=1. The mean ϕ can be defined on another domain D⊃F0, but the space F0 will serve as reference domain. Moreover, we set K=R or C.


### 2.1. Means Spanned by Probability Measures

LetX be a complete metric space and let Cb0(X) be the space of bounded K-valued continuous maps on X. We note by P(X) the space of Borel probability measures on X. Let us first set V=K.Definition 1.
AK-probability mean is a linear map τ:Dτ⊂Cb0(X)→K which is defined as the limit of barycenters with K-weights of a sequence of Borel probability measures on X;thatis,(4)
∃
μ
n
,
α
n
n
∈
N
∈
P
X
×
K
N
,
∀
m
∈
N
∗
,
∑
n
=
0
m
α
n
≠
0
∧
∀
f
∈
C
b
0
X
,
τ
f
=
lim
m
→
+
∞
⁡
1
∑
n
=
0
m
α
n
∑
n
=
0
m
α
n
μ
n
f
.We note byPM~K(X) the space of K-probability means, by PMK(X) the set of probability means τ such that Dτ=Cb0(X), and by PM~R+(X) the means τ obtained by a sequence (αn)n∈N∈R+∗ and we set(5)PMR+X=PMCX∩PM~R+X.We have a special class spanned by the Dirac measures.Definition 2 (see [14]).
AK-Dirac mean is a linear map τ:Dτ⊂Cb0(X)→K which is defined as the limit of barycenters with K-weights of a sequence of Dirac measures on X, (6)
∑
n
=
0
m
α
n
≠
0
∧
∀
f
∈
C
b
0
X
,
τ
f
=
lim
m
→
+
∞
⁡
1
∑
n
=
0
m
α
n
∑
n
=
0
m
α
n
δ
x
n
f
,
∃
x
n
,
α
n
n
∈
N
∈
X
×
K
N
,
∀
m
∈
N
∗
.We note byDM~K(X),DMK(X),DM~R+(X),DMR+(X) the sets of Dirac means corresponding, respectively, to PM~K(X),PMK(X),PM~R+(X),PMR+(X).Proposition 3.
P M ~ K X ,   P M K X ,   D M ~ K ( X ), and DMK(X) are K-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.
IfX is moreover a locally compact manifold, one has the following inclusions:(i)
P(X)⊂DMR+(X).(ii)
IfX is compact, P(X)=DMR+(X)=PMR+(X).Proof.
(i) Letμ∈P(X) and let (xn)n∈N be a uniformly distributed sequence with respect to μ. Then, ∀f∈Cb0(X),limm→+∞1/n+1∑n=0mf(xn)=Eμ(f)=∫Xfdμ. Thus (7)μf=limm→+∞⁡1n+1∑n=0mδxnf.
(ii) IfX is compact, the space of (signed) finite measures on X coincides with Cb0(X)′. Since PMR(X)⊂Cb0(X)′, 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]).
Aspace with metric and measure, or ametric measured space (mm-space for short), is a triple (X,d,μ), where (X,d) is a metric space and μ is a probability measure on the Borel tribu on X.LetA⊂X, and let ε>0. We note (8)Aε=x∈X∣dA,x<ε.Definition 6 (see [12]).
ALevy family is a sequence X=(Xn,dn,μn)n∈N of mm-spaces if, for each sequence (An)n∈N,(9)Anis  a  Borel  subset  ofXn,∀n∈Nsatisfying (10)liminfn→+∞⁡μnAn>0,and then (11)limn→+∞⁡μnAnε=1,∀ε>0.In the sequel, we shall assume that(12)Xn⊂Xn+1,∀n∈Nwith continuous injection. Notice that we do not assume that dn is the restriction of dn+1 which allows us some freedom on metric requirements. The technical necessary condition is the following: let n∈N and let Bn+1 be a Borel subset of Xn+1. Then Bn+1∩Xn is a Borel subset of Xn. 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 312.19, 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(13)S1⊂S2⊂⋯⊂Sn⊂Sn+1⋯⊂S∞=⋃n=1∞Snequipped with the classical Euclide (or Hilbert) distance and (except for S∞) the normalized spherical measure μ (we drop the index for the measure in sake of clear notations). Then, for any R-valued 1-Lipschitz function on S∞, there exists a∈R such that (14)μx∈Sn∣fx-a>ϵ<2e-n-1ϵ2/2,∀ϵ>0.In a more intuitive formation, one can say that any 1-Lipschitz function concentrates around a real valuea with respect to μ. We leave the reader with [11] for more on the metric geometry of this example.Proposition 8.
LetX=(Sn;·;μ)n∈N∗. Then for any 1-Lipschitz function f defined on S∞, and with the notations used before,(15)LMXf=a.Example 9.
For Levy families induced by Lebesgue measures letm,n∈(N∗)2. Take Km⊂Rn. For each m∈N∗, we equip Km with the usual distance d induced by Rn and with the probability measure (16)μn=1KnλKnλ.Setting K=(Km,d,μm)m∈N∗, we get that K is a Levy family, but there is no concentration property. This example will be studied in the next sections of this article.Definition 10.
Letf:X→C be a map such that, for each n∈N, the restriction of f to Xn is μn-integrable. Then, themean value of f with respect to the family X is (17)WMVXf=limn→+∞⁡∫Xnfdμnif the limit exists.


### 2.3. Limit Means and Infinite Dimensional Integrals

Definition 11.
LetX=(Xn,τn)n∈N be a sequence of probability spaces such that(i)
∀n∈N,Xn is a metric space;(ii)
∀n∈N,Xn⊂Xn+1, and the topology of Xn+1 restricted to Xn coincides with the topology of Xn;(iii)
∀n∈N,τn∈PM~C(Xn). Then, we define, for the maps f defined on ⋃n∈NXn, if ∀n∈N, f∣Xn∈Dτn and if the limit converges,(18)LMXf=limn→+∞⁡τnfcalledlimit mean of f with respect to X.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 functionsf on [0;1]N. Let (19)Pk:x1,x2,…∈RN⟼x1,x2,…,xk,0,…be the k-dimensional projection. Then, there exists l∈N∗ such that(20)f=f∘Pl.
Then, adequate sequences for the Monte Carlo method are those whose push-forwards on[0;1]k are also adequate for this method. The projectors Pk converge (weakly) to identity, the condition on the sequence (xn) is that, for each k∈N, the push-forwards of the sequences (Pk(xn)) on [0;1]k fit with the desired conditions: the sequence (Pk(xn)) is a Monte Carlo sequence for the cube [0;1]k equipped with the (trace of) Lebesgue measure. It is well known that such a sequence (xn) exists, through, for example, the powers of π: (21)xn=nπl+1-intnπl+1l∈N∈0;1N,∀n∈N∗,where int⁡(x) is the integer part of the real number x. Thus, Daniell integral appears by its definition as a limit mean for the sequence (Xn)N∗ defined by Xn=[0;1]n, 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,PM~K(X) and DM~K(X).Example 13 (Fresnel-type integrals).
First, letτ∈PM~K(X). There exists a sequence (μn)N∈P(X)N satisfying Definition 1, and for each k∈N∗, there exists a sequence of Dirac measures (δn(k))n∈N which converge to μk with respect to the Monte Carlo method. Thus, finding a sequence of Dirac measures which can define a Dirac mean which coincides with τ becomes a problem of extracting a sequence of Dirac measures which converges to τ. The same is for a limit mean τ = “lim⁡τn”. Let us describe more precisely the open problems on the example of oscillatory and Fresnel integrals. Let Φ∈C∞(Rn,R) be a fixed function. Following [16] (see, e.g., [7, 8, 17, 18]), we define the following.Definition 14.
Letf be a measurable function on Rn. Let φ∈S(Rn) be a weight function such that φ(0)=1. If the limit (22)limϵ→0⁡∫RneiΦxfxφϵxdxexists and is independent of the fixed function φ, then this limit is called oscillatory integral of f with respect to Φ, noted by(23)∫RnoeiΦxfxdx.The choiceΦ(x)=i/2hx2 is of particular interest and is known under the name ofFresnel integral. This choice gives us a mean, up to normalization by a factor (2iπh)-d/2, and can be generalized to a Hilbert space H the following way.Definition 15.
A Borel measurable functionf:H→C is called h-integrable in the sense of Fresnel for each creasing sequence of projectors (Pn)n∈N such that limn→+∞Pn=IdH, and the finite dimensional approximations of the oscillatory integrals of f(24)∫ImPnoei/2hPnx2fPnxdPnx∫ImPnoei/2hPnx2dPnx-1are well defined and the limit as n→+∞ does not depend on the sequence (Pn)n∈N. In this case, it is called infinite dimensional Fresnel integral of f and noted by(25)∫Hoei/2hx2fxdx.The invariance under the choices of the mapφ and the projections Pn 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 functionsf defined on Rn, the map (26)f⟼∫RnoeiΦxfxdx∈PM~CRn,(ii)
the map(27)f⟼limn→+∞⁡∫RnoeiΦxfxdxis a limit mean through the sequence R⊂⋯⊂Rn⊂Rn+1⊂⋯⊂H.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 [0;1]N (Daniell integral) as described in the previous example. Let us consider now cylinder functions eiΦ·f on [0;1]N. Let P be a finite dimensional projection such that (28)f=f∘Pand let (29)SP=S∘P.Then, adequate sequences for the Monte Carlo method are those whose push-forwards on [0;1]dimImP are also adequate for this method. Taking now a creasing sequence of orthogonal projectors Pk converging (weakly) to identity, the condition on the sequence (xn) is that, for each k∈N, the push-forwards of the sequences (Pk(xn)) on [0;1]dimImPk fit with the desired conditions: the sequence (Pk(xn)) is a Monte Carlo sequence for the cube [0;1]dimImPk equipped with the (trace of) Lebesgue measure. It is well known that such a sequence (xn) exists, through, for example, the powers of π: (30)xn=nπl+1-int⁡nπl+1l∈N∈0;1N,∀n∈N∗,where int⁡(x) is the integer part of the real number x. Let us now fix d=dimImPk. The maps ψ defined by, for example, (31)x=x1,…,xd∈Rd⟼ψx=12π∫-∞x1et12dt1,…,∫-∞xdetd2dtnor (32)x=x1,…,xd∈Rd⟼ψx=Argthx1,…,Argthxdare diffeomorphisms Rd→]0;1[d⊂[0;1]d and we get (33)∫RneiΦxfxφϵxdx=∫RneiΦ∘ψ-1xf∘ψ-1xφ∘ψ-1ϵxdet⁡Dxψ-1dx;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.


## 2.1. Means Spanned by Probability Measures

LetX be a complete metric space and let Cb0(X) be the space of bounded K-valued continuous maps on X. We note by P(X) the space of Borel probability measures on X. Let us first set V=K.Definition 1.
AK-probability mean is a linear map τ:Dτ⊂Cb0(X)→K which is defined as the limit of barycenters with K-weights of a sequence of Borel probability measures on X;thatis,(4)
∃
μ
n
,
α
n
n
∈
N
∈
P
X
×
K
N
,
∀
m
∈
N
∗
,
∑
n
=
0
m
α
n
≠
0
∧
∀
f
∈
C
b
0
X
,
τ
f
=
lim
m
→
+
∞
⁡
1
∑
n
=
0
m
α
n
∑
n
=
0
m
α
n
μ
n
f
.We note byPM~K(X) the space of K-probability means, by PMK(X) the set of probability means τ such that Dτ=Cb0(X), and by PM~R+(X) the means τ obtained by a sequence (αn)n∈N∈R+∗ and we set(5)PMR+X=PMCX∩PM~R+X.We have a special class spanned by the Dirac measures.Definition 2 (see [14]).
AK-Dirac mean is a linear map τ:Dτ⊂Cb0(X)→K which is defined as the limit of barycenters with K-weights of a sequence of Dirac measures on X, (6)
∑
n
=
0
m
α
n
≠
0
∧
∀
f
∈
C
b
0
X
,
τ
f
=
lim
m
→
+
∞
⁡
1
∑
n
=
0
m
α
n
∑
n
=
0
m
α
n
δ
x
n
f
,
∃
x
n
,
α
n
n
∈
N
∈
X
×
K
N
,
∀
m
∈
N
∗
.We note byDM~K(X),DMK(X),DM~R+(X),DMR+(X) the sets of Dirac means corresponding, respectively, to PM~K(X),PMK(X),PM~R+(X),PMR+(X).Proposition 3.
P M ~ K X ,   P M K X ,   D M ~ K ( X ), and DMK(X) are K-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.
IfX is moreover a locally compact manifold, one has the following inclusions:(i)
P(X)⊂DMR+(X).(ii)
IfX is compact, P(X)=DMR+(X)=PMR+(X).Proof.
(i) Letμ∈P(X) and let (xn)n∈N be a uniformly distributed sequence with respect to μ. Then, ∀f∈Cb0(X),limm→+∞1/n+1∑n=0mf(xn)=Eμ(f)=∫Xfdμ. Thus (7)μf=limm→+∞⁡1n+1∑n=0mδxnf.
(ii) IfX is compact, the space of (signed) finite measures on X coincides with Cb0(X)′. Since PMR(X)⊂Cb0(X)′, 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]).
Aspace with metric and measure, or ametric measured space (mm-space for short), is a triple (X,d,μ), where (X,d) is a metric space and μ is a probability measure on the Borel tribu on X.LetA⊂X, and let ε>0. We note (8)Aε=x∈X∣dA,x<ε.Definition 6 (see [12]).
ALevy family is a sequence X=(Xn,dn,μn)n∈N of mm-spaces if, for each sequence (An)n∈N,(9)Anis  a  Borel  subset  ofXn,∀n∈Nsatisfying (10)liminfn→+∞⁡μnAn>0,and then (11)limn→+∞⁡μnAnε=1,∀ε>0.In the sequel, we shall assume that(12)Xn⊂Xn+1,∀n∈Nwith continuous injection. Notice that we do not assume that dn is the restriction of dn+1 which allows us some freedom on metric requirements. The technical necessary condition is the following: let n∈N and let Bn+1 be a Borel subset of Xn+1. Then Bn+1∩Xn is a Borel subset of Xn. 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 312.19, 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(13)S1⊂S2⊂⋯⊂Sn⊂Sn+1⋯⊂S∞=⋃n=1∞Snequipped with the classical Euclide (or Hilbert) distance and (except for S∞) the normalized spherical measure μ (we drop the index for the measure in sake of clear notations). Then, for any R-valued 1-Lipschitz function on S∞, there exists a∈R such that (14)μx∈Sn∣fx-a>ϵ<2e-n-1ϵ2/2,∀ϵ>0.In a more intuitive formation, one can say that any 1-Lipschitz function concentrates around a real valuea with respect to μ. We leave the reader with [11] for more on the metric geometry of this example.Proposition 8.
LetX=(Sn;·;μ)n∈N∗. Then for any 1-Lipschitz function f defined on S∞, and with the notations used before,(15)LMXf=a.Example 9.
For Levy families induced by Lebesgue measures letm,n∈(N∗)2. Take Km⊂Rn. For each m∈N∗, we equip Km with the usual distance d induced by Rn and with the probability measure (16)μn=1KnλKnλ.Setting K=(Km,d,μm)m∈N∗, we get that K is a Levy family, but there is no concentration property. This example will be studied in the next sections of this article.Definition 10.
Letf:X→C be a map such that, for each n∈N, the restriction of f to Xn is μn-integrable. Then, themean value of f with respect to the family X is (17)WMVXf=limn→+∞⁡∫Xnfdμnif the limit exists.


## 2.3. Limit Means and Infinite Dimensional Integrals

Definition 11.
LetX=(Xn,τn)n∈N be a sequence of probability spaces such that(i)
∀n∈N,Xn is a metric space;(ii)
∀n∈N,Xn⊂Xn+1, and the topology of Xn+1 restricted to Xn coincides with the topology of Xn;(iii)
∀n∈N,τn∈PM~C(Xn). Then, we define, for the maps f defined on ⋃n∈NXn, if ∀n∈N, f∣Xn∈Dτn and if the limit converges,(18)LMXf=limn→+∞⁡τnfcalledlimit mean of f with respect to X.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 functionsf on [0;1]N. Let (19)Pk:x1,x2,…∈RN⟼x1,x2,…,xk,0,…be the k-dimensional projection. Then, there exists l∈N∗ such that(20)f=f∘Pl.
Then, adequate sequences for the Monte Carlo method are those whose push-forwards on[0;1]k are also adequate for this method. The projectors Pk converge (weakly) to identity, the condition on the sequence (xn) is that, for each k∈N, the push-forwards of the sequences (Pk(xn)) on [0;1]k fit with the desired conditions: the sequence (Pk(xn)) is a Monte Carlo sequence for the cube [0;1]k equipped with the (trace of) Lebesgue measure. It is well known that such a sequence (xn) exists, through, for example, the powers of π: (21)xn=nπl+1-intnπl+1l∈N∈0;1N,∀n∈N∗,where int⁡(x) is the integer part of the real number x. Thus, Daniell integral appears by its definition as a limit mean for the sequence (Xn)N∗ defined by Xn=[0;1]n, 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,PM~K(X) and DM~K(X).Example 13 (Fresnel-type integrals).
First, letτ∈PM~K(X). There exists a sequence (μn)N∈P(X)N satisfying Definition 1, and for each k∈N∗, there exists a sequence of Dirac measures (δn(k))n∈N which converge to μk with respect to the Monte Carlo method. Thus, finding a sequence of Dirac measures which can define a Dirac mean which coincides with τ becomes a problem of extracting a sequence of Dirac measures which converges to τ. The same is for a limit mean τ = “lim⁡τn”. Let us describe more precisely the open problems on the example of oscillatory and Fresnel integrals. Let Φ∈C∞(Rn,R) be a fixed function. Following [16] (see, e.g., [7, 8, 17, 18]), we define the following.Definition 14.
Letf be a measurable function on Rn. Let φ∈S(Rn) be a weight function such that φ(0)=1. If the limit (22)limϵ→0⁡∫RneiΦxfxφϵxdxexists and is independent of the fixed function φ, then this limit is called oscillatory integral of f with respect to Φ, noted by(23)∫RnoeiΦxfxdx.The choiceΦ(x)=i/2hx2 is of particular interest and is known under the name ofFresnel integral. This choice gives us a mean, up to normalization by a factor (2iπh)-d/2, and can be generalized to a Hilbert space H the following way.Definition 15.
A Borel measurable functionf:H→C is called h-integrable in the sense of Fresnel for each creasing sequence of projectors (Pn)n∈N such that limn→+∞Pn=IdH, and the finite dimensional approximations of the oscillatory integrals of f(24)∫ImPnoei/2hPnx2fPnxdPnx∫ImPnoei/2hPnx2dPnx-1are well defined and the limit as n→+∞ does not depend on the sequence (Pn)n∈N. In this case, it is called infinite dimensional Fresnel integral of f and noted by(25)∫Hoei/2hx2fxdx.The invariance under the choices of the mapφ and the projections Pn 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 functionsf defined on Rn, the map (26)f⟼∫RnoeiΦxfxdx∈PM~CRn,(ii)
the map(27)f⟼limn→+∞⁡∫RnoeiΦxfxdxis a limit mean through the sequence R⊂⋯⊂Rn⊂Rn+1⊂⋯⊂H.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 [0;1]N (Daniell integral) as described in the previous example. Let us consider now cylinder functions eiΦ·f on [0;1]N. Let P be a finite dimensional projection such that (28)f=f∘Pand let (29)SP=S∘P.Then, adequate sequences for the Monte Carlo method are those whose push-forwards on [0;1]dimImP are also adequate for this method. Taking now a creasing sequence of orthogonal projectors Pk converging (weakly) to identity, the condition on the sequence (xn) is that, for each k∈N, the push-forwards of the sequences (Pk(xn)) on [0;1]dimImPk fit with the desired conditions: the sequence (Pk(xn)) is a Monte Carlo sequence for the cube [0;1]dimImPk equipped with the (trace of) Lebesgue measure. It is well known that such a sequence (xn) exists, through, for example, the powers of π: (30)xn=nπl+1-int⁡nπl+1l∈N∈0;1N,∀n∈N∗,where int⁡(x) is the integer part of the real number x. Let us now fix d=dimImPk. The maps ψ defined by, for example, (31)x=x1,…,xd∈Rd⟼ψx=12π∫-∞x1et12dt1,…,∫-∞xdetd2dtnor (32)x=x1,…,xd∈Rd⟼ψx=Argthx1,…,Argthxdare diffeomorphisms Rd→]0;1[d⊂[0;1]d and we get (33)∫RneiΦxfxφϵxdx=∫RneiΦ∘ψ-1xf∘ψ-1xφ∘ψ-1ϵxdet⁡Dxψ-1dx;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 [3–5].


### 3.1. Definitions

On measured spaces, the definitions are those given in classical mathematical literature, see for example, [3]. Let (X,μ) be a topological space equipped with a σ-additive, positive measure μ. Let T(X) be a Borel σ-algebra on X. We note by Renμ(X) the set of sequences U=(Un)n∈N∈T(X)N such that(1)
⋃n∈NUn=X;(2)
∀n∈N,0<μ(Un)<+∞ and Un⊂Un+1.Remark 16.
We have in particularlimn→+∞μ(Un)=μ(X).
In what follows we assume the natural conditionRenμ≠∅.Definition 17.
LetU∈Renμ. Let V be a separable complete locally convex topological vector space (sclctvs). Let f:X→V be a measurable map. We define, if the limit exists, the weak mean value of f with respect to U as(34)WMVμUf=limn→+∞⁡1μUn∫Unfdμ.Moreover, if WMVμU(f) does not depend on U, we call it mean value of f, noted by MVμ(f).Notice that(i)
ifV=R, setting f+=1/2(f+f) and f-=1/2(f-f),WMVμU(f)=WMVμU(f+)+WMVμU(f-) for each U∈Renμ, if f,f+, and f- have a finite mean value.(ii)
The same way ifV=C,WMVμU(f)=WMVμU(Rf)+iWMVμU(If) for each U∈Renμ.(iii)
We denote byFμU the set of functions f such that WMVμU(f) exists in V and by Fμ the set of functions f such that MVμ(f) is well defined.Examples 1.
(1) Let(X,μ) be an arbitrary measured space. Let f=1X. Let U∈Renμ.∀n∈N,1/μ(Un)∫Unfdμ=μ(Un)/μ(Un)=1 so that(35)MVμ1X=1.
(2) Let(X,δx) be a space X equipped with the Dirac measure at x∈X. Let f be an arbitrary map to an arbitrary sclctvs. U∈Renδx⇔∀n∈N,δx(Un)>0⇔∀n∈N,x∈Un. Thus, if U∈Renδx∀n∈N,1/δx(Un)∫Unfdδx=f(x) so that (36)MVδxf=fx.
(3) Let(X,μ) be a measured space with μ(X)<+∞. Let f be an arbitrary bounded measurable map. Then one can show very easily that we recover the classical mean value of f:(37)MVμf=1μX∫Xfdμ.
(4) LetX=R equipped with the classical Lebesgue measure λ. Let g∈L1(R,R+) (integrable R+-valued function). Let U∈Renλ. We have that limn→+∞⁡∫Ungdλ≤∫Rgdλ<+∞ so that(38)MVλg=0.
(5) LetX=R equipped with the Lebesgue measure λ. Let f(x)=sin(x) and let Un=[-(n+1);(n+1)]. The map sin is odd so that WMVλU(sin)=0. Now, let Un′=[-2πn;2πn]∪⋃j=0n[2(n+j)π;(2(n+j)+1)π]. Then WMVλU′(sin)=1/5π. This shows that sin has no (strong) mean value for the Lebesgue measure.
(6) LetX=N equipped with γ, the counting measure. Let n∈N and set Un=[0;n]∩N. Let (un)∈RN and U=(Un)n∈N. Then, (39)WMVγUun=limn→+∞⁡1n+1∑k=0nukis the Cesàro limit.


### 3.2. Basic Properties

In what follows and till the end of this paper we assume the natural conditionRenμ≠∅ for the measures μ we consider.Proposition 18.
Let(X,μ) be a measured space. Let U∈Renμ. Then(1)
FμU is a vector space and WMVμU is linear;(2)
Fμ is a vector space and MVμ 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.
Letμ and ν be Radon measures. Let U=(Un)n∈N∈Renμ∩Renν. Assume that Θ(μ,ν)=limn→+∞μ(Un)/(μ+ν)(Un)∈[0;1] exists. Then f∈Fμ+νU and (40)WMVμ+νUf=Θμ,νWMVμUf+Θν,μWMFνUf.Proof.
(41) 1 μ + ν U n ∫ U n f d μ + ν = μ U n μ + ν U n 1 μ U n ∫ U n f d μ + ν U n μ + ν U n 1 ν U n ∫ U n f d ν .Thus, we get the result taking the limit.Proposition 20.
Letμ be a Radon measure and let k∈R+∗. Then Renkμ=Renμ,Fkμ=Fμ; moreover ∀U∈Renμ,FkμU=FμU and WMVkμU=WMFμU.Proof.
The proof is  obvious.Theorem 21.
Letμ be a measure on X, let U∈Renμ, and f∈FμU. Let (42)Mμ,U,f=ν∣U∈Renν,WMVνUf=WMVμUf.M(μ,U,F) is a convex cone.Proof.
Letk>0 and let ν∈M(μ,U,F). Setting ν′=kν, we get WMVν′U(f)=WMVνU(f) by Proposition 20; thus M(μ,U,F) is a cone.
Now, let(ν,ν′)∈M(μ,U,F)2. Let t∈[0;1] and let ν′′=tν+1-tν′.(i)
Let us show thatU∈Renν′′. Let n∈N. We have ν′′(Un)=tν(Un)+1-tν′(Un), so that ν′′(Un)∈R+∗.(ii)
Let us show thatWMVν′′U(f)=WMVμU(f). We already know that WMVν′U(f)=WMVνU(f)=WMVμU(f). Let n∈N,(43)1ν′′Un∫Unfdν′′=tνUntν+1-tν′Un1tνUn∫Unfdtν+1-tν′Untν+1-tν′Un11-tν′Un∫Unfd1-tν′=tνUntν+1-tν′Un+1-tν′Untν+1-tν′UnWMVμUf+tνUntν+1-tν′Un1νUn∫Unfdν-WMVμUf+1-tν′Untν+1-tν′Un1ν′Un∫Unfdν′-WMVμUf.
Now, we remark that(44)1=tνUntν+1-tν′Un+1-tν′Untν+1-tν′Unand that (45)0=limn→+∞⁡tνUntν+1-tν′Un1νUn∫Unfdν-WMVμUf,since tν(Un)/(tν+1-tν′)(Un)∈[0;1] and limn→+∞1/ν(Un)∫Unfd(ν)=WMVνU(f)=WMVμU(f), and finally that limn→+∞tν(Un)/(tν+1-tν′)(Un)1/ν(Un)∫Unfd(ν)-WMVμU(f)=0 the same way. Thus, (46)WMVν′′Uf=limn→+∞⁡1ν′′Un∫Unfdν′′=WMVμUf.ν′′∈M(μ,U,F), thus M(μ,U,F) is a convex cone.


### 3.3. Asymptotic Comparison of Radon Measures

We now turn to the numberΘ that appeared in Proposition 19. In this section, μ and ν are fixed Radon measures and U is a fixed sequence in Renμ∩Renν.Proposition 22.
One has the following.(1)
Θ(μ,ν)∈[0;1].(2)
Θ(μ,ν)=1-Θ(ν,μ).Proof.
The proof is obvious.Definition 23.
One has the following.(1)
ν=oU(μ) if Θ(μ,ν)=1.(2)
ν=OU(μ) if Θ(μ,ν)>0.(3)
ν~Uμ if Θ(μ,ν)=1/2.Let us now compare three measuresμ,ν, and ρ. U is here a fixed arbitrary sequence.Lemma 24.
Letμ and ν be two measures and let U∈Renμ∩Renν,(47)Θμ,ν=11+θμ,ν,where θ(μ,ν)=limn→+∞ν(Un)/μ(Un)∈R¯=[0;+∞].Proof.
The  proof is obvious.Proposition 25.
LetU∈Renμ∩Renν∩Renρ.(1)
θ(μ,ρ)=θ(μ,ν)θ(ν,ρ) if (θ(μ,ν),θ(ν,ρ))∉0;+∞,+∞;0.(2)
Θ(μ,ρ)=Θ(μ,ν)Θ(ν,ρ)/2Θ(μ,ν)Θ(ν,ρ)-Θ(μ,ν)-Θ(ν,ρ)+1 if (Θ(μ,ν),Θ(ν,ρ))∉1;0,0;1.Proof.
Letn∈N. We have μ(Un)/(μ+ν)(Un)=1/1+ν(Un)/μ(Un),ν(Un)/(ν+ρ)(Un)=1/1+ρ(Un)/ν(Un) and μ(Un)/(μ+ρ)(Un)=1/1+ρ(Un)/μ(Un). For the first part of the statement, (48)ρUnμUn=ρUnνUnνUnμUn(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 (49)ρUnμUn=μ+ρUnμUn-1,ρUnνUn=ν+ρUnνUn-1,νUnμUn=μ+νUnμUn-1, and we get (50)μUnμ+ρUn=μUn/μ+νUnνUn/ν+ρUn2μUn/μ+νUnνUn/ν+ρUn-μUn/μ+νUn-νUn/ν+ρUn+1.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 Definition23 shows this correspondence. Through easy calculations of θ or Θ, one can easily see that if μ, ν, and ν′ are comparable measures,(1)
(μ~ν)∧(ν~ν′)⇒(μ~ν′);(2)
(μ~ν)⇔(ν~μ);(3)
(μ=o(ν))⇒(μ=O(ν));(4)
(μ=O(ν))∧(ν=O(ν′))⇒(μ=O(ν′));(5)
(μ=o(ν))∧(ν=o(ν′))⇒(μ=o(ν′));(6)
(μ=O(ν′))∧(ν=O(ν′))⇒(μ+ν=O(ν′));and other easy relations can be deduced in the same spirit.


### 3.4. Limits and Mean Value

IfX is, for example, a connected locally compact, paracompact, and not compact manifold, equipped with a Radon measure μ such that μ(X)=+∞, any exhaustive sequence K=(Kn)n∈N of compact subsets of X is such that K∈Renμ. In this setting, it is natural to consider X¯=X∪∞ the Alexandroff compactification of X.Theorem 26.
Letf:X→R be a bounded measurable map which extends to f¯:X¯→R, a continuous map at ∞. Then WMVμK(f)=f¯(∞) for each exhaustive sequence K of compact subsets of X.Proof.
We can assume thatf¯∞=0; in other words (51)limx→∞⁡fx=0.The sequence (Knc)n∈N gives a basis of neighborhood of ∞; thus (52)supx∈Knc⁡fx<ϵ,∀ϵ′>0,∃N′∈N,∀n≥N′.Moreover, since limn→+∞μ(Kn)=+∞, (53)μKn0<ϵ′′μKn,∀n0∈N,∀ϵ′′>0,∃N′′∈N,∀n≥N′′.Let ϵ>0. Let ϵ′=ϵ/2. We set n0=N′ and ϵ′′=ϵ/2supXf. Then, ∀n≥N=max⁡(n0,N′′), (54)∫Knfdμ≤∫Knfdμ=∫Kn0fdμ+∫Kn-Kn0fdμ≤supX⁡fμKn0+ϵ′μKn-Kn0.The second term is bounded by ϵ′μ(Kn)=ϵμ(Kn)/2 and we majorate the first term by ϵ′′(supXf)μ(Kn)=ϵμ(Kn)/2. Thus (55)1μKn∫Knfdμ≤ϵ,∀ϵ>0,∃N>0,∀n≥N,and hence WMVμK(f)=0.As mentioned in the Introduction, we found no straightforward Beppo-Levy type theorem for mean values. The first counterexample we find is, forX=R and μ=λ the Lebesgue measure, an increasing sequence of L1(λ) which converges to 1R (uniformly on each compact subset of R),forexample, the sequence (e-x2/n)n∈N∗. Let Kn=[-n-1;n+1] and K=(Kn)n∈N. We have K∈Renλ,WMVλK(1R)=1, and WMVλK(e-x2/n)=0 by Theorem 26. We can only state the following theorem on uniform convergence.Lemma 27.
Letμ be a measure on X and let U∈Renμ. Let f1 and f2 be two functions in FμU(X,V), where V in a sclctvs.
Letp be a norm on V. If there exists ϵ∈R+∗ such that supx∈Xpf1x-f2x<ϵ, then (56)pWMVμUf1-ϵ≤pWMVμUf2≤pWMVμUf1+ϵ.Proof.
Letn∈N, and(57)p1μUn∫Unf2dμ≤1μUn∫Unpf1-f2dμ+p1μUn∫Unf1dμ≤ϵ+p1μUn∫Unf1dμ.We get in the same way (58)p1μUn∫Unf1dμ-ϵ≤p1μUn∫Unf2dμ.The result is obtained by taking the limit.Theorem 28.
Let(fn)n∈N∈FμUN be a sequence which converges uniformly on X to a μ-measurable map f. Then (1)
f∈FμU.(2)
WMVμU(f)=limn→+∞WMVμU(fn).Proof.
Letun=WMVμU(fn).
(i) Let us prove that(un) has a limit u∈V. Let p be a norm on V. Let ϵ∈R+∗. There exists N∈N such that, for each (n,m)∈N2, (59)supx∈X⁡pfn-fm<ϵ.Thus, by Lemma 27 with f1=0 and f2=fn-fm, (60)pun-um=pWMVμUfn-fm≤ϵ.Thus, the sequence (un) is a Cauchy sequence. Since V is complete, the sequence (un) has a limit u∈V.
(ii) Moreover, we remember that∀ϵ>0,∀(n,m)∈N2, (61)supx∈X⁡p⁡fn-f<ϵ∧supx∈X⁡pfm-f<ϵ⟹supx∈X⁡pfn-fm<2ϵ⟹pWMVμUfn-fm<2ϵ⟹pun-u<2ϵ.
(iii) Let us prove thatu=limn→+∞1/μ(Un)∫Unfdμ. Let n,k∈N2,(62)p1μUn∫Unfdμ-u≤p1μUn∫Unfdμ-1μUn∫Unfkdμ+p1μUn∫Unfkdμ-uk+puk-u.Let ϵ∈R+∗. Let K such that ∀k>K,supx∈Xp(f-fk)<ϵ/8. Then (63)p1μUn∫Unfdμ-1μUn∫Unfkdμ<ϵ8,puk-u<ϵ4.Let N such that, for each n>N, (64)p1μUn∫UnfK+1dμ-uK+1<ϵ8.Then, by the same arguments, for each k>K, (65)p1μUn∫Unfkdμ-uk<3ϵ8.Gathering these inequalities, we get (66)p1μUn∫Unfdμ-u<ϵ8+3ϵ8+ϵ4<ϵ.


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

In this section,X=Rm with n∈N∗,λ is the Lebesgue measure, K=(Kn)n∈N is the renormalization procedure defined by (67)Kn=-n-1;n+1n.and L=(Ln)n∈N is the renormalization procedure defined by (68)Ln=x∈Rn;x≤n+1,where · is the Euclidian norm. We denote by ·∞ the sup norm and d∞ its associated distance. Let v∈Rn. We use the obvious notations K+v=(Kn+v)n∈N and L+v=(Ln+v)n∈N for the translated sequences. Let (A,B)∈P(X)2. We denote by AΔB=(A-B)∪(B-A) the symmetric difference of subsets.Proposition 29.
Letv∈Rm. Let f∈FλK (resp., f∈FλL) be a bounded function. Let U∈Renλ and v∈Rn. If (69)limn→+∞⁡λUnΔUn+vλUn=0,then (1)
f∈FλU+v (resp., f∈FλU+v) and WMVλUf=WMVλU+vf;(2)
forfv:x↦f(x-v) we have f∈FλU and WMVλUf=WMVλUfv,Proof.
We first notice that the second item is a reformulation of the first item: by change of variablesx↦x-v,WMVλK+v(f)=WMVλK(fv).
Let us now prove the first item. Letn∈N, and(70)1λUn∫Unfdλ-1λUn+v∫Un+vfdλ=1λUn∫Unfdλ-1λUn∫Un+vfdλ=1λUn∫Un-Un+vfdλ-∫Un+v-Unfdλ=1λKn∫UnΔUn+v1Un-Un+v-1Un+v-Unfdλ.Let M=supRm(f). Then (71)1λUn∫Unfdλ-1λUn+v∫Un+vfdλ≤MλUnΔUn+vλUn.Thus, we get the result.Lemma 30.
One has the following.(72)limn→+∞⁡λKnΔKn+vλKn=0,limn→+∞⁡λLnΔLn+vλLn=0.Proof.
We prove it for the sequenceK, and the proof is the same for the sequence L. We have Kn=(n+1)K0; thus λ(Kn)=(n+1)mλ(K0) and (73)λKnΔKn+v=n+1mλK0ΔK0+1n+1v.Let (74)An=x∈Rm∣d∞x,∂K0<2v∞n+1.We have K0ΔK0+1/n+1v⊂An and limn→+∞λ(An)=0. Thus, (75)limn→+∞⁡λKnΔKn+vλKn=0.Proposition 31.
Letv∈Rm. Let f∈FλK (resp., f∈FλL) be a bounded function.(1)
Thenf∈FλK+v (resp., f∈FλL+v) and WMVλK(f)=WMVλK+v(f) (resp., WMVλL(f)=WMVλL+v(f)).(2)
Letfv:x↦f(x-v). Then f∈FλK (resp., f∈FλL) and WMVλK(f)=WMVλK(fv) (resp., WMVλL(f)=WMVλL(fv)).Proof.
The proof forK and L 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,X is a smooth, locally compact, paracompact, connected, oriented, and noncompact manifold of dimension n≥1 equipped with a measure μ induced by a volume form ω and a Morse function F:X→R such that (76)μF≤a<+∞,∀a∈R.For the theory of Morse functions we refer to [19]. Notice that there exists some value A∈R such that μ(F<A)>0. Notice that we can have μ(X)∈]0;+∞].Definition 32.
Letf:X→V be a smooth function into a sclctvs V. Let t∈[A;+∞[. We define (77)Iμff,t=1μF≤t∫F≤tfxdμxand, if the limit exists, (78)WMVμFf=limt→+∞⁡IμFf,t.Of course this definition is the “continuum” version of the “sequential” Definition17. If V is metrizable, for any increasing sequence (αn)n∈N∈[A;+∞[N such that limn→+∞μF≤αn≥μ(X), setting Un=F≤αn, (79)WMVμUf=WMVμFfand conversely WMFμF(f) exists if WMFμU(f) exists and does not depend on the choice of the sequence (αn)n∈N.Moreover, sinceF is a Morse function, it has isolated critical points and changing X into X-C, where C is the set of critical points of F, for each t∈[A;+∞[, (80)F=t=F-1tis an (n-1)-dimensional manifold (disconnected or not). The first examples that we can give are definite positive quadratic forms on a vector space in which X is embedded.


### 3.7. Application: Homology as a Mean Value

LetM be a finite dimensional manifold equipped with a Riemannian metric g and the corresponding Laplace-Beltrami operator Δ and with finite dimensional de Rham cohomology space H∗(M,R). One of the standard results of Hodge theory is the onto and one-to-one map between H∗(M,R) and the space of L2-harmonic forms H made by integration over simplexes: (81)I:H⟶H∗M,R,α⟼Iα,where (82)Iα:ssimplex⟼Iαs=∫sα.We have assumed here that the simplex has the order of the harmonic form. This is mathematically coherent stating ∫sα=0 if s and α do not have the same order. Let λ be the Lebesgue measure on H with respect to the scalar product induced by the L2-scalar product. Let U=(Un)n∈N be the sequence of Euclidian balls centered at 0 such that, for each n∈N, the ball Un is of radius n.Proposition 33.
Assume thatH∗(M,R) is finite dimensional. Let s be a simplex. Let (83)φs=I·s1+I·s.The cohomology class of s is null if and only if (84)WMVλUφs=0.Proof.
(i) If the cohomology class ofs is null then, ∀α∈H,∫sα=0; thus φs(α)=0. Finally, (85)WMVλUφs=0.
(ii) If the cohomology class ofs is not null, let αs be the corresponding element in H. We have ∫sαs=1. Let πs be the projection onto the 1-dimensional vector space spanned by αs. Let n∈N∗. Let (86)Vn=α∈Unsuch  that∫sα>12.Then, (87)Vn=Un∩πs-1-1;1·αs.Moreover, (88)infα∈Vnφsα=12,λVn>λUn-1.Then (89)∫Unφsdλ≥∫Vnφsdλ≥λVn2.Thus (90)WMVλUφs=limn→+∞⁡1λUn∫Unφsdλ≥limn→+∞⁡λVn2λUn≥limn→+∞⁡λUn-12λUn=12≠0.Remark 34.
A very easy application of Theorem26 also shows that the map s↦WMVλU(φs) is a 0;1-valued map.


## 3.1. Definitions

On measured spaces, the definitions are those given in classical mathematical literature, see for example, [3]. Let (X,μ) be a topological space equipped with a σ-additive, positive measure μ. Let T(X) be a Borel σ-algebra on X. We note by Renμ(X) the set of sequences U=(Un)n∈N∈T(X)N such that(1)
⋃n∈NUn=X;(2)
∀n∈N,0<μ(Un)<+∞ and Un⊂Un+1.Remark 16.
We have in particularlimn→+∞μ(Un)=μ(X).
In what follows we assume the natural conditionRenμ≠∅.Definition 17.
LetU∈Renμ. Let V be a separable complete locally convex topological vector space (sclctvs). Let f:X→V be a measurable map. We define, if the limit exists, the weak mean value of f with respect to U as(34)WMVμUf=limn→+∞⁡1μUn∫Unfdμ.Moreover, if WMVμU(f) does not depend on U, we call it mean value of f, noted by MVμ(f).Notice that(i)
ifV=R, setting f+=1/2(f+f) and f-=1/2(f-f),WMVμU(f)=WMVμU(f+)+WMVμU(f-) for each U∈Renμ, if f,f+, and f- have a finite mean value.(ii)
The same way ifV=C,WMVμU(f)=WMVμU(Rf)+iWMVμU(If) for each U∈Renμ.(iii)
We denote byFμU the set of functions f such that WMVμU(f) exists in V and by Fμ the set of functions f such that MVμ(f) is well defined.Examples 1.
(1) Let(X,μ) be an arbitrary measured space. Let f=1X. Let U∈Renμ.∀n∈N,1/μ(Un)∫Unfdμ=μ(Un)/μ(Un)=1 so that(35)MVμ1X=1.
(2) Let(X,δx) be a space X equipped with the Dirac measure at x∈X. Let f be an arbitrary map to an arbitrary sclctvs. U∈Renδx⇔∀n∈N,δx(Un)>0⇔∀n∈N,x∈Un. Thus, if U∈Renδx∀n∈N,1/δx(Un)∫Unfdδx=f(x) so that (36)MVδxf=fx.
(3) Let(X,μ) be a measured space with μ(X)<+∞. Let f be an arbitrary bounded measurable map. Then one can show very easily that we recover the classical mean value of f:(37)MVμf=1μX∫Xfdμ.
(4) LetX=R equipped with the classical Lebesgue measure λ. Let g∈L1(R,R+) (integrable R+-valued function). Let U∈Renλ. We have that limn→+∞⁡∫Ungdλ≤∫Rgdλ<+∞ so that(38)MVλg=0.
(5) LetX=R equipped with the Lebesgue measure λ. Let f(x)=sin(x) and let Un=[-(n+1);(n+1)]. The map sin is odd so that WMVλU(sin)=0. Now, let Un′=[-2πn;2πn]∪⋃j=0n[2(n+j)π;(2(n+j)+1)π]. Then WMVλU′(sin)=1/5π. This shows that sin has no (strong) mean value for the Lebesgue measure.
(6) LetX=N equipped with γ, the counting measure. Let n∈N and set Un=[0;n]∩N. Let (un)∈RN and U=(Un)n∈N. Then, (39)WMVγUun=limn→+∞⁡1n+1∑k=0nukis the Cesàro limit.


## 3.2. Basic Properties

In what follows and till the end of this paper we assume the natural conditionRenμ≠∅ for the measures μ we consider.Proposition 18.
Let(X,μ) be a measured space. Let U∈Renμ. Then(1)
FμU is a vector space and WMVμU is linear;(2)
Fμ is a vector space and MVμ 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.
Letμ and ν be Radon measures. Let U=(Un)n∈N∈Renμ∩Renν. Assume that Θ(μ,ν)=limn→+∞μ(Un)/(μ+ν)(Un)∈[0;1] exists. Then f∈Fμ+νU and (40)WMVμ+νUf=Θμ,νWMVμUf+Θν,μWMFνUf.Proof.
(41) 1 μ + ν U n ∫ U n f d μ + ν = μ U n μ + ν U n 1 μ U n ∫ U n f d μ + ν U n μ + ν U n 1 ν U n ∫ U n f d ν .Thus, we get the result taking the limit.Proposition 20.
Letμ be a Radon measure and let k∈R+∗. Then Renkμ=Renμ,Fkμ=Fμ; moreover ∀U∈Renμ,FkμU=FμU and WMVkμU=WMFμU.Proof.
The proof is  obvious.Theorem 21.
Letμ be a measure on X, let U∈Renμ, and f∈FμU. Let (42)Mμ,U,f=ν∣U∈Renν,WMVνUf=WMVμUf.M(μ,U,F) is a convex cone.Proof.
Letk>0 and let ν∈M(μ,U,F). Setting ν′=kν, we get WMVν′U(f)=WMVνU(f) by Proposition 20; thus M(μ,U,F) is a cone.
Now, let(ν,ν′)∈M(μ,U,F)2. Let t∈[0;1] and let ν′′=tν+1-tν′.(i)
Let us show thatU∈Renν′′. Let n∈N. We have ν′′(Un)=tν(Un)+1-tν′(Un), so that ν′′(Un)∈R+∗.(ii)
Let us show thatWMVν′′U(f)=WMVμU(f). We already know that WMVν′U(f)=WMVνU(f)=WMVμU(f). Let n∈N,(43)1ν′′Un∫Unfdν′′=tνUntν+1-tν′Un1tνUn∫Unfdtν+1-tν′Untν+1-tν′Un11-tν′Un∫Unfd1-tν′=tνUntν+1-tν′Un+1-tν′Untν+1-tν′UnWMVμUf+tνUntν+1-tν′Un1νUn∫Unfdν-WMVμUf+1-tν′Untν+1-tν′Un1ν′Un∫Unfdν′-WMVμUf.
Now, we remark that(44)1=tνUntν+1-tν′Un+1-tν′Untν+1-tν′Unand that (45)0=limn→+∞⁡tνUntν+1-tν′Un1νUn∫Unfdν-WMVμUf,since tν(Un)/(tν+1-tν′)(Un)∈[0;1] and limn→+∞1/ν(Un)∫Unfd(ν)=WMVνU(f)=WMVμU(f), and finally that limn→+∞tν(Un)/(tν+1-tν′)(Un)1/ν(Un)∫Unfd(ν)-WMVμU(f)=0 the same way. Thus, (46)WMVν′′Uf=limn→+∞⁡1ν′′Un∫Unfdν′′=WMVμUf.ν′′∈M(μ,U,F), thus M(μ,U,F) is a convex cone.


## 3.3. Asymptotic Comparison of Radon Measures

We now turn to the numberΘ that appeared in Proposition 19. In this section, μ and ν are fixed Radon measures and U is a fixed sequence in Renμ∩Renν.Proposition 22.
One has the following.(1)
Θ(μ,ν)∈[0;1].(2)
Θ(μ,ν)=1-Θ(ν,μ).Proof.
The proof is obvious.Definition 23.
One has the following.(1)
ν=oU(μ) if Θ(μ,ν)=1.(2)
ν=OU(μ) if Θ(μ,ν)>0.(3)
ν~Uμ if Θ(μ,ν)=1/2.Let us now compare three measuresμ,ν, and ρ. U is here a fixed arbitrary sequence.Lemma 24.
Letμ and ν be two measures and let U∈Renμ∩Renν,(47)Θμ,ν=11+θμ,ν,where θ(μ,ν)=limn→+∞ν(Un)/μ(Un)∈R¯=[0;+∞].Proof.
The  proof is obvious.Proposition 25.
LetU∈Renμ∩Renν∩Renρ.(1)
θ(μ,ρ)=θ(μ,ν)θ(ν,ρ) if (θ(μ,ν),θ(ν,ρ))∉0;+∞,+∞;0.(2)
Θ(μ,ρ)=Θ(μ,ν)Θ(ν,ρ)/2Θ(μ,ν)Θ(ν,ρ)-Θ(μ,ν)-Θ(ν,ρ)+1 if (Θ(μ,ν),Θ(ν,ρ))∉1;0,0;1.Proof.
Letn∈N. We have μ(Un)/(μ+ν)(Un)=1/1+ν(Un)/μ(Un),ν(Un)/(ν+ρ)(Un)=1/1+ρ(Un)/ν(Un) and μ(Un)/(μ+ρ)(Un)=1/1+ρ(Un)/μ(Un). For the first part of the statement, (48)ρUnμUn=ρUnνUnνUnμUn(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 (49)ρUnμUn=μ+ρUnμUn-1,ρUnνUn=ν+ρUnνUn-1,νUnμUn=μ+νUnμUn-1, and we get (50)μUnμ+ρUn=μUn/μ+νUnνUn/ν+ρUn2μUn/μ+νUnνUn/ν+ρUn-μUn/μ+νUn-νUn/ν+ρUn+1.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 Definition23 shows this correspondence. Through easy calculations of θ or Θ, one can easily see that if μ, ν, and ν′ are comparable measures,(1)
(μ~ν)∧(ν~ν′)⇒(μ~ν′);(2)
(μ~ν)⇔(ν~μ);(3)
(μ=o(ν))⇒(μ=O(ν));(4)
(μ=O(ν))∧(ν=O(ν′))⇒(μ=O(ν′));(5)
(μ=o(ν))∧(ν=o(ν′))⇒(μ=o(ν′));(6)
(μ=O(ν′))∧(ν=O(ν′))⇒(μ+ν=O(ν′));and other easy relations can be deduced in the same spirit.


## 3.4. Limits and Mean Value

IfX is, for example, a connected locally compact, paracompact, and not compact manifold, equipped with a Radon measure μ such that μ(X)=+∞, any exhaustive sequence K=(Kn)n∈N of compact subsets of X is such that K∈Renμ. In this setting, it is natural to consider X¯=X∪∞ the Alexandroff compactification of X.Theorem 26.
Letf:X→R be a bounded measurable map which extends to f¯:X¯→R, a continuous map at ∞. Then WMVμK(f)=f¯(∞) for each exhaustive sequence K of compact subsets of X.Proof.
We can assume thatf¯∞=0; in other words (51)limx→∞⁡fx=0.The sequence (Knc)n∈N gives a basis of neighborhood of ∞; thus (52)supx∈Knc⁡fx<ϵ,∀ϵ′>0,∃N′∈N,∀n≥N′.Moreover, since limn→+∞μ(Kn)=+∞, (53)μKn0<ϵ′′μKn,∀n0∈N,∀ϵ′′>0,∃N′′∈N,∀n≥N′′.Let ϵ>0. Let ϵ′=ϵ/2. We set n0=N′ and ϵ′′=ϵ/2supXf. Then, ∀n≥N=max⁡(n0,N′′), (54)∫Knfdμ≤∫Knfdμ=∫Kn0fdμ+∫Kn-Kn0fdμ≤supX⁡fμKn0+ϵ′μKn-Kn0.The second term is bounded by ϵ′μ(Kn)=ϵμ(Kn)/2 and we majorate the first term by ϵ′′(supXf)μ(Kn)=ϵμ(Kn)/2. Thus (55)1μKn∫Knfdμ≤ϵ,∀ϵ>0,∃N>0,∀n≥N,and hence WMVμK(f)=0.As mentioned in the Introduction, we found no straightforward Beppo-Levy type theorem for mean values. The first counterexample we find is, forX=R and μ=λ the Lebesgue measure, an increasing sequence of L1(λ) which converges to 1R (uniformly on each compact subset of R),forexample, the sequence (e-x2/n)n∈N∗. Let Kn=[-n-1;n+1] and K=(Kn)n∈N. We have K∈Renλ,WMVλK(1R)=1, and WMVλK(e-x2/n)=0 by Theorem 26. We can only state the following theorem on uniform convergence.Lemma 27.
Letμ be a measure on X and let U∈Renμ. Let f1 and f2 be two functions in FμU(X,V), where V in a sclctvs.
Letp be a norm on V. If there exists ϵ∈R+∗ such that supx∈Xpf1x-f2x<ϵ, then (56)pWMVμUf1-ϵ≤pWMVμUf2≤pWMVμUf1+ϵ.Proof.
Letn∈N, and(57)p1μUn∫Unf2dμ≤1μUn∫Unpf1-f2dμ+p1μUn∫Unf1dμ≤ϵ+p1μUn∫Unf1dμ.We get in the same way (58)p1μUn∫Unf1dμ-ϵ≤p1μUn∫Unf2dμ.The result is obtained by taking the limit.Theorem 28.
Let(fn)n∈N∈FμUN be a sequence which converges uniformly on X to a μ-measurable map f. Then (1)
f∈FμU.(2)
WMVμU(f)=limn→+∞WMVμU(fn).Proof.
Letun=WMVμU(fn).
(i) Let us prove that(un) has a limit u∈V. Let p be a norm on V. Let ϵ∈R+∗. There exists N∈N such that, for each (n,m)∈N2, (59)supx∈X⁡pfn-fm<ϵ.Thus, by Lemma 27 with f1=0 and f2=fn-fm, (60)pun-um=pWMVμUfn-fm≤ϵ.Thus, the sequence (un) is a Cauchy sequence. Since V is complete, the sequence (un) has a limit u∈V.
(ii) Moreover, we remember that∀ϵ>0,∀(n,m)∈N2, (61)supx∈X⁡p⁡fn-f<ϵ∧supx∈X⁡pfm-f<ϵ⟹supx∈X⁡pfn-fm<2ϵ⟹pWMVμUfn-fm<2ϵ⟹pun-u<2ϵ.
(iii) Let us prove thatu=limn→+∞1/μ(Un)∫Unfdμ. Let n,k∈N2,(62)p1μUn∫Unfdμ-u≤p1μUn∫Unfdμ-1μUn∫Unfkdμ+p1μUn∫Unfkdμ-uk+puk-u.Let ϵ∈R+∗. Let K such that ∀k>K,supx∈Xp(f-fk)<ϵ/8. Then (63)p1μUn∫Unfdμ-1μUn∫Unfkdμ<ϵ8,puk-u<ϵ4.Let N such that, for each n>N, (64)p1μUn∫UnfK+1dμ-uK+1<ϵ8.Then, by the same arguments, for each k>K, (65)p1μUn∫Unfkdμ-uk<3ϵ8.Gathering these inequalities, we get (66)p1μUn∫Unfdμ-u<ϵ8+3ϵ8+ϵ4<ϵ.


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

In this section,X=Rm with n∈N∗,λ is the Lebesgue measure, K=(Kn)n∈N is the renormalization procedure defined by (67)Kn=-n-1;n+1n.and L=(Ln)n∈N is the renormalization procedure defined by (68)Ln=x∈Rn;x≤n+1,where · is the Euclidian norm. We denote by ·∞ the sup norm and d∞ its associated distance. Let v∈Rn. We use the obvious notations K+v=(Kn+v)n∈N and L+v=(Ln+v)n∈N for the translated sequences. Let (A,B)∈P(X)2. We denote by AΔB=(A-B)∪(B-A) the symmetric difference of subsets.Proposition 29.
Letv∈Rm. Let f∈FλK (resp., f∈FλL) be a bounded function. Let U∈Renλ and v∈Rn. If (69)limn→+∞⁡λUnΔUn+vλUn=0,then (1)
f∈FλU+v (resp., f∈FλU+v) and WMVλUf=WMVλU+vf;(2)
forfv:x↦f(x-v) we have f∈FλU and WMVλUf=WMVλUfv,Proof.
We first notice that the second item is a reformulation of the first item: by change of variablesx↦x-v,WMVλK+v(f)=WMVλK(fv).
Let us now prove the first item. Letn∈N, and(70)1λUn∫Unfdλ-1λUn+v∫Un+vfdλ=1λUn∫Unfdλ-1λUn∫Un+vfdλ=1λUn∫Un-Un+vfdλ-∫Un+v-Unfdλ=1λKn∫UnΔUn+v1Un-Un+v-1Un+v-Unfdλ.Let M=supRm(f). Then (71)1λUn∫Unfdλ-1λUn+v∫Un+vfdλ≤MλUnΔUn+vλUn.Thus, we get the result.Lemma 30.
One has the following.(72)limn→+∞⁡λKnΔKn+vλKn=0,limn→+∞⁡λLnΔLn+vλLn=0.Proof.
We prove it for the sequenceK, and the proof is the same for the sequence L. We have Kn=(n+1)K0; thus λ(Kn)=(n+1)mλ(K0) and (73)λKnΔKn+v=n+1mλK0ΔK0+1n+1v.Let (74)An=x∈Rm∣d∞x,∂K0<2v∞n+1.We have K0ΔK0+1/n+1v⊂An and limn→+∞λ(An)=0. Thus, (75)limn→+∞⁡λKnΔKn+vλKn=0.Proposition 31.
Letv∈Rm. Let f∈FλK (resp., f∈FλL) be a bounded function.(1)
Thenf∈FλK+v (resp., f∈FλL+v) and WMVλK(f)=WMVλK+v(f) (resp., WMVλL(f)=WMVλL+v(f)).(2)
Letfv:x↦f(x-v). Then f∈FλK (resp., f∈FλL) and WMVλK(f)=WMVλK(fv) (resp., WMVλL(f)=WMVλL(fv)).Proof.
The proof forK and L 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,X is a smooth, locally compact, paracompact, connected, oriented, and noncompact manifold of dimension n≥1 equipped with a measure μ induced by a volume form ω and a Morse function F:X→R such that (76)μF≤a<+∞,∀a∈R.For the theory of Morse functions we refer to [19]. Notice that there exists some value A∈R such that μ(F<A)>0. Notice that we can have μ(X)∈]0;+∞].Definition 32.
Letf:X→V be a smooth function into a sclctvs V. Let t∈[A;+∞[. We define (77)Iμff,t=1μF≤t∫F≤tfxdμxand, if the limit exists, (78)WMVμFf=limt→+∞⁡IμFf,t.Of course this definition is the “continuum” version of the “sequential” Definition17. If V is metrizable, for any increasing sequence (αn)n∈N∈[A;+∞[N such that limn→+∞μF≤αn≥μ(X), setting Un=F≤αn, (79)WMVμUf=WMVμFfand conversely WMFμF(f) exists if WMFμU(f) exists and does not depend on the choice of the sequence (αn)n∈N.Moreover, sinceF is a Morse function, it has isolated critical points and changing X into X-C, where C is the set of critical points of F, for each t∈[A;+∞[, (80)F=t=F-1tis an (n-1)-dimensional manifold (disconnected or not). The first examples that we can give are definite positive quadratic forms on a vector space in which X is embedded.


## 3.7. Application: Homology as a Mean Value

LetM be a finite dimensional manifold equipped with a Riemannian metric g and the corresponding Laplace-Beltrami operator Δ and with finite dimensional de Rham cohomology space H∗(M,R). One of the standard results of Hodge theory is the onto and one-to-one map between H∗(M,R) and the space of L2-harmonic forms H made by integration over simplexes: (81)I:H⟶H∗M,R,α⟼Iα,where (82)Iα:ssimplex⟼Iαs=∫sα.We have assumed here that the simplex has the order of the harmonic form. This is mathematically coherent stating ∫sα=0 if s and α do not have the same order. Let λ be the Lebesgue measure on H with respect to the scalar product induced by the L2-scalar product. Let U=(Un)n∈N be the sequence of Euclidian balls centered at 0 such that, for each n∈N, the ball Un is of radius n.Proposition 33.
Assume thatH∗(M,R) is finite dimensional. Let s be a simplex. Let (83)φs=I·s1+I·s.The cohomology class of s is null if and only if (84)WMVλUφs=0.Proof.
(i) If the cohomology class ofs is null then, ∀α∈H,∫sα=0; thus φs(α)=0. Finally, (85)WMVλUφs=0.
(ii) If the cohomology class ofs is not null, let αs be the corresponding element in H. We have ∫sαs=1. Let πs be the projection onto the 1-dimensional vector space spanned by αs. Let n∈N∗. Let (86)Vn=α∈Unsuch  that∫sα>12.Then, (87)Vn=Un∩πs-1-1;1·αs.Moreover, (88)infα∈Vnφsα=12,λVn>λUn-1.Then (89)∫Unφsdλ≥∫Vnφsdλ≥λVn2.Thus (90)WMVλUφs=limn→+∞⁡1λUn∫Unφsdλ≥limn→+∞⁡λVn2λUn≥limn→+∞⁡λUn-12λUn=12≠0.Remark 34.
A very easy application of Theorem26 also shows that the map s↦WMVλU(φs) is a 0;1-valued map.


## 4. The Mean Value on Infinite Products

### 4.1. Mean Value on an Infinite Product of Measured Spaces

LetΛ be an infinite (countable, continuous, or other) set of indexes. Let (Xλ,μλ)λ∈Λ or for short (Xλ)Λ be a family of measured spaces as before. We assume that, on each space Xλ, we havefixed a sequence Uλ∈Renμλ. Let Xc=∏λ∈ΛXλ be the Cartesian product of the sequence (Xλ)Λ.Definition 35.
Letf∈C0(X) for the product topology. f is calledcylindrical if and only if there exists Λ~, a finite subset of Λ, and a map f~∈C0(∏λ∈Λ~Xλ) such that (91)fxλΛ=f~xλΛ~,∀xλΛ∈X.Then, we set, if f~∈F⊗λ∈Λ~μλ∏λ∈Λ~Uλ, (92)WMVf=WMV⊗λ∈Λ~μλ∏λ∈Λ~Uλf~.In this case we use the notation: f∈F (here, subsidiary notations are omitted since the sequence of measures and the sequences of renormalization are fixed in this section).Notice that if we haveΛ~⊂Λ0~ with the notations used in the definition, and since f is constant with respect to the variables xλ indexed by λ∈Λ0~-Λ~, the definition of WMV(f) does not depend on the choice of Λ~, which makes it coherent.Theorem 36.
Letf be a cylindrical function associated with the finite set of indexes Λ~=λ1,…,λn and with the function f~∈C0(∏λ∈Λ~Xλ).(1)
Letλ∈Λ~. Let us fix Uλ∈Renμλ. Then ∏λ∈Λ~Uλ=∏λ∈Λ~(Uλ)nn∈N∈Ren⊗λ∈Λ~μλ.(2)
If both sides are defined, for each scalar-valued mapf=fλ1⊗⋯⊗fλn∈Fμλ1Uλ1⊗⋯⊗Fμλ1Uλn, we have (93)WMV⊗λ∈Λ~μλ∏λ∈Λ~Uλf~=∏λ∈Λ~WMVμλUλfλ.For convenience of notations, we shall writeWMV(f~) instead of WMV⊗λ∈Λ~μλ∏λ∈Λ~Uλ(f~). Let us now consider an arbitrary map f:X→V which is not cylindrical (V is a sclctvs). Theorem 28 gives us a way to extend the notion of mean value by uniform convergence of sequences of cylindrical maps. But we shall do this not only for X, but also for classes of functions defined on a class of subsets of X. These classes are the following ones.Definition 37.
LetD⊂X. The domain D is calledadmissible if and only if (94)⨂λ∈Λ~μλ∏λ∈Λ~Uλ,n-DΛ~,n,x=0,∀x∈D,∀Λ~finite  subset  ofΛ,∀n∈N,where (95)DΛ~,n,x=u∈∏λ∈Λ~Uλ,n∣∃x′∈D,∀λ∈Λ~,xλ′=uλ∧∀λ∈Λ-Λ~,xλ′=xλ.Definition 38.
LetD be an admissible domain. A function f:D→V iscylindrical if its value depends only on a finite number of coordinates indexed by a fixed finite subset of Λ.The mean value of a cylindrical functionf is immediately computed, since its trace is defined on ∏λ∈Λ~Uλ,n up to a subset of measure 0.Theorem 39.
LetV be a sclctvs. Let f:D→V be the uniform limit of a sequence (fn)n∈N of cylindrical functions on D with a mean value on D. Then,(1)
the sequence(WMV(fn))n∈N has a limit;(2)
this limit does not depend on the sequence(fn)n∈N but only on f.Proof.
Letun=WMVμU(fn).
(i) Let us prove that(un) has a limit u∈V. Let p be a norm on V. Let ϵ∈R+∗. There exists N∈N such that, for each (n,m)∈N2, (96)supx∈X⁡p⁡fn-fm<ϵ.Thus, by Lemma 27 with f1=0 and f2=fn-fm, (97)pun-um=pWMVμUfn-fm≤ϵ.Thus, the sequence (un) is a Cauchy sequence. Since V is complete, the sequence (un) has a limit u∈V.
(ii) Now, let us consider another sequence(fn′) of cylindrical functions which converges uniformly to f. In order to finish the proof of the theorem, let us prove that u=limn→+∞⁡WMVμU(fn′).
Letn∈N. We define (98)fn′′=fn/2′ifnis  evenfn-1/2ifnis  odd.This sequence again converges uniformly to f and is thus a Cauchy sequence. The sequence (WMVμU(fn′′)n∈N has a limit u′∈V. Extracting the subsequences (fn)n∈N=(f2n+1′′)n∈N and (fn′)n∈N=(f2n+1′′)n∈N we get (99)u′=limn→+∞⁡WMVμUfn′′=limn→+∞⁡WMVμUfn=u,limn→+∞⁡WMVμUfn′=limn→+∞⁡WMVμUfn′′=u.By the way, the following definition is justified.Definition 40.
LetV be a sclctvs. Let f:D→V be the uniform limit of a sequence (fn)n∈N of cylindrical functions on D with a mean value on D. Then, (100)WMVμUf=limn→+∞⁡WMVμUfn.Trivially, the mapWMVμU is linear as well as in the context of Proposition 18.


### 4.2. Application: The Mean Value on Marked Infinite Configurations

LetX be a locally compact and paracompact manifold, orientable, and let μ be a measure on X induced by a volume form. In the following, we have either of the following:(i)
ifX is compact, setting x0∈X, (101)Γ=unn∈N∈XN∣lim⁡un=x0,∀n,m∈N2,n≠m⟹un≠um;(ii)
ifX is not compact, setting (Kn)n∈N an exhaustive sequence of compact subspaces of X, (102)OΓ=unn∈N∈XN∣∀p∈N,un;n∈N∩Kp<+∞,∀n,m∈N2,n≠m⟹un≠um.The first setting was first defined by Ismaginov, Vershik, Gel’fand, and Graev; see, for example, [20] for a recent reference, and the second one has been extensively studied by Albeverio, Daletskii, Kondratiev, and Lytvynov; see, for example, [21]. Alternatively, Γ can be seen as a set of countable sums of Dirac measures equipped with the topology of vague convergence.For the following, we also need the set of ordered finitek-configurations: (103)OΓk=u1,…,uk∈Xk∣∀n,m∈N2,1≤n<m≤k⟹un≠um.Assume now that X is equipped with a Radon measure μ. Let usfixU∈Renμ. Notice first that, for each (n,k)∈N∗×N, (104)μ⊗nUkn=μ⊗nx1,…,xn∈Ukn∣∀i,j,0≤i<j≤n⟹xi≠xj.In other words, the set of n-uples for which there exists two coordinates that are equal is of measure 0. This shows that OΓ is an admissible domain in XN and enables us to write, for a bounded cylindrical function f, (105)WMVμUf=WMVμ⊗OrdfUOrdff~,since f~ is defined up to a subset of measure 0 on each UkOrd(f), for k∈N. By the way, Theorem 39 applies in this setting. Notice also that the normalization sequence U on OΓ is induced from the normalization sequence on XN. This implies heuristically that cylindrical functions with a weak mean value with respect to U are in a sense small perturbations of functions on XN. This is why we can modify the sequence U on OΓ in the following way: let φ:R+→R+∗ be a function such that limx→+∞φ=0. Then, if f is a cylindrical function on OΓ, we set (106)Uφn=Un-xi1≤i≤n∣∃i,jsuch  thati<j∧dxi,xj<φn.


## 4.1. Mean Value on an Infinite Product of Measured Spaces

LetΛ be an infinite (countable, continuous, or other) set of indexes. Let (Xλ,μλ)λ∈Λ or for short (Xλ)Λ be a family of measured spaces as before. We assume that, on each space Xλ, we havefixed a sequence Uλ∈Renμλ. Let Xc=∏λ∈ΛXλ be the Cartesian product of the sequence (Xλ)Λ.Definition 35.
Letf∈C0(X) for the product topology. f is calledcylindrical if and only if there exists Λ~, a finite subset of Λ, and a map f~∈C0(∏λ∈Λ~Xλ) such that (91)fxλΛ=f~xλΛ~,∀xλΛ∈X.Then, we set, if f~∈F⊗λ∈Λ~μλ∏λ∈Λ~Uλ, (92)WMVf=WMV⊗λ∈Λ~μλ∏λ∈Λ~Uλf~.In this case we use the notation: f∈F (here, subsidiary notations are omitted since the sequence of measures and the sequences of renormalization are fixed in this section).Notice that if we haveΛ~⊂Λ0~ with the notations used in the definition, and since f is constant with respect to the variables xλ indexed by λ∈Λ0~-Λ~, the definition of WMV(f) does not depend on the choice of Λ~, which makes it coherent.Theorem 36.
Letf be a cylindrical function associated with the finite set of indexes Λ~=λ1,…,λn and with the function f~∈C0(∏λ∈Λ~Xλ).(1)
Letλ∈Λ~. Let us fix Uλ∈Renμλ. Then ∏λ∈Λ~Uλ=∏λ∈Λ~(Uλ)nn∈N∈Ren⊗λ∈Λ~μλ.(2)
If both sides are defined, for each scalar-valued mapf=fλ1⊗⋯⊗fλn∈Fμλ1Uλ1⊗⋯⊗Fμλ1Uλn, we have (93)WMV⊗λ∈Λ~μλ∏λ∈Λ~Uλf~=∏λ∈Λ~WMVμλUλfλ.For convenience of notations, we shall writeWMV(f~) instead of WMV⊗λ∈Λ~μλ∏λ∈Λ~Uλ(f~). Let us now consider an arbitrary map f:X→V which is not cylindrical (V is a sclctvs). Theorem 28 gives us a way to extend the notion of mean value by uniform convergence of sequences of cylindrical maps. But we shall do this not only for X, but also for classes of functions defined on a class of subsets of X. These classes are the following ones.Definition 37.
LetD⊂X. The domain D is calledadmissible if and only if (94)⨂λ∈Λ~μλ∏λ∈Λ~Uλ,n-DΛ~,n,x=0,∀x∈D,∀Λ~finite  subset  ofΛ,∀n∈N,where (95)DΛ~,n,x=u∈∏λ∈Λ~Uλ,n∣∃x′∈D,∀λ∈Λ~,xλ′=uλ∧∀λ∈Λ-Λ~,xλ′=xλ.Definition 38.
LetD be an admissible domain. A function f:D→V iscylindrical if its value depends only on a finite number of coordinates indexed by a fixed finite subset of Λ.The mean value of a cylindrical functionf is immediately computed, since its trace is defined on ∏λ∈Λ~Uλ,n up to a subset of measure 0.Theorem 39.
LetV be a sclctvs. Let f:D→V be the uniform limit of a sequence (fn)n∈N of cylindrical functions on D with a mean value on D. Then,(1)
the sequence(WMV(fn))n∈N has a limit;(2)
this limit does not depend on the sequence(fn)n∈N but only on f.Proof.
Letun=WMVμU(fn).
(i) Let us prove that(un) has a limit u∈V. Let p be a norm on V. Let ϵ∈R+∗. There exists N∈N such that, for each (n,m)∈N2, (96)supx∈X⁡p⁡fn-fm<ϵ.Thus, by Lemma 27 with f1=0 and f2=fn-fm, (97)pun-um=pWMVμUfn-fm≤ϵ.Thus, the sequence (un) is a Cauchy sequence. Since V is complete, the sequence (un) has a limit u∈V.
(ii) Now, let us consider another sequence(fn′) of cylindrical functions which converges uniformly to f. In order to finish the proof of the theorem, let us prove that u=limn→+∞⁡WMVμU(fn′).
Letn∈N. We define (98)fn′′=fn/2′ifnis  evenfn-1/2ifnis  odd.This sequence again converges uniformly to f and is thus a Cauchy sequence. The sequence (WMVμU(fn′′)n∈N has a limit u′∈V. Extracting the subsequences (fn)n∈N=(f2n+1′′)n∈N and (fn′)n∈N=(f2n+1′′)n∈N we get (99)u′=limn→+∞⁡WMVμUfn′′=limn→+∞⁡WMVμUfn=u,limn→+∞⁡WMVμUfn′=limn→+∞⁡WMVμUfn′′=u.By the way, the following definition is justified.Definition 40.
LetV be a sclctvs. Let f:D→V be the uniform limit of a sequence (fn)n∈N of cylindrical functions on D with a mean value on D. Then, (100)WMVμUf=limn→+∞⁡WMVμUfn.Trivially, the mapWMVμU is linear as well as in the context of Proposition 18.


## 4.2. Application: The Mean Value on Marked Infinite Configurations

LetX be a locally compact and paracompact manifold, orientable, and let μ be a measure on X induced by a volume form. In the following, we have either of the following:(i)
ifX is compact, setting x0∈X, (101)Γ=unn∈N∈XN∣lim⁡un=x0,∀n,m∈N2,n≠m⟹un≠um;(ii)
ifX is not compact, setting (Kn)n∈N an exhaustive sequence of compact subspaces of X, (102)OΓ=unn∈N∈XN∣∀p∈N,un;n∈N∩Kp<+∞,∀n,m∈N2,n≠m⟹un≠um.The first setting was first defined by Ismaginov, Vershik, Gel’fand, and Graev; see, for example, [20] for a recent reference, and the second one has been extensively studied by Albeverio, Daletskii, Kondratiev, and Lytvynov; see, for example, [21]. Alternatively, Γ can be seen as a set of countable sums of Dirac measures equipped with the topology of vague convergence.For the following, we also need the set of ordered finitek-configurations: (103)OΓk=u1,…,uk∈Xk∣∀n,m∈N2,1≤n<m≤k⟹un≠um.Assume now that X is equipped with a Radon measure μ. Let usfixU∈Renμ. Notice first that, for each (n,k)∈N∗×N, (104)μ⊗nUkn=μ⊗nx1,…,xn∈Ukn∣∀i,j,0≤i<j≤n⟹xi≠xj.In other words, the set of n-uples for which there exists two coordinates that are equal is of measure 0. This shows that OΓ is an admissible domain in XN and enables us to write, for a bounded cylindrical function f, (105)WMVμUf=WMVμ⊗OrdfUOrdff~,since f~ is defined up to a subset of measure 0 on each UkOrd(f), for k∈N. By the way, Theorem 39 applies in this setting. Notice also that the normalization sequence U on OΓ is induced from the normalization sequence on XN. This implies heuristically that cylindrical functions with a weak mean value with respect to U are in a sense small perturbations of functions on XN. This is why we can modify the sequence U on OΓ in the following way: let φ:R+→R+∗ be a function such that limx→+∞φ=0. Then, if f is a cylindrical function on OΓ, we set (106)Uφn=Un-xi1≤i≤n∣∃i,jsuch  thati<j∧dxi,xj<φn.


## 5. Mean Value for Heuristic Lebesgue Measures

Definition 41.
Anormalized Fréchet space is a pair (F,H), where(1)
F is a Fréchet space,(2)
H is a Hilbert space,(3)
F⊂H,(4)
F is dense in H.Another way to understand this definition is the following: we choose a pre-Hilbert norm on the Fréchet spaceF. Then, H is the completion of F.Definition 42.
LetV be a sclctvs. A function f:F→V iscylindrical if there exists Ff, a finite dimensional affine subspace of F, for which, if π is the orthogonal projection, π:F→Ff such that (107)fx=f∘πx,∀x∈F.Remark 43.
This construction is canonical in many examples. For instance, the mapφs defined in the example of Section 3.7 extends to a cylindrical function on L2-forms, and hence the construction described above applies to this extension map.Proposition 44.
Let(fn)n∈N be a sequence of cylindrical functions. There exists a unique sequence (Ffn)n∈N increasing for ⊂, for which, ∀m∈N, Ffm is the minimal affine space for which (108)fn∘πm=fn,∀n≤m.Proof.
We construct the sequence by induction:(i)
Ff0 is the minimal affine subspace of F for which Definition 42 applies to f0.(ii)
Letn∈N. Assume that we have constructed Ffn. Let F~ be the minimal affine subspace of F for which Definition 42 applies to fn+1. We set (109)Ffn+1=Ffn+F~.
(this is the minimal affine subspace ofF which contains both Ffn and F~.) If π~ and πn+1 are the orthogonal projections into F~ and Ffn+1, then (110)fn+1=fn+1∘π~=fn+1∘πn+1.This ends the proof.We now develop renormalization procedures onF inspired from Section 4.1, using orthogonal projections to 1-dimensional vector subspaces. In these approaches, a finite dimensional Euclidian space is equipped with its Lebesgue measure noted by λ in any dimension. The Euclidian norms are induced by the pre-Hilbert norm on F for any finite dimensional vector subspace of F.


### 5.1. Mean Value by Infinite Product

Letf be a bounded function which is the uniform limit of a sequence of cylindrical functions (fn)n∈N. Here, an orthonormal basis (ek)k∈N is obtained by induction, completing at each step an orthonormal basis of Ffn by an orthonormal basis of Ffn+1. Thus we can identify F with a subset D of RN which is invariant under change of a finite number of coordinates. This qualifies it as admissible since, with the notations used in Definition 37, (111)∏λ∈Λ~Uλ,n-DΛ~,n,x=∅,for any set of renormalization procedures in RN as defined in Section 4.1 so that Theorem 39 applies. We note by (112)WMVλfthis value. We remark that we already know by Theorem 39 that this mean value does not depend on the sequence (fn)n∈Nonly once the sequence(Ffn)n∈Nis fixed. In other words, two sequences (fn)n∈N and (fn′)n∈N which converge uniformly to f a priori lead to the same mean value if Ffn=Ffn′ (maybe up to reindexation). From heuristic calculations, it seems to come from the choice of the renormalization procedure, which is dependent on the basis chosen, more than from the sequence (Ffn)n∈N.


### 5.2. Invariance

We notice three types of invariance: scale invariance, translation invariance, and invariance under the orthogonal (or unitary) group.Proposition 45.
Letα∈N∗. Let f be a function on an infinite dimensional vector space F with mean value. Let fα:x∈F↦f(αx). Then fα has a mean value and (113)WMVλfα=WMVλf.Proof.
Let(fn)n∈N be a sequence which converges uniformly to f. Then, with the notations above, the sequence ((fn)α)n∈N converges uniformly to fα. Let m=dim⁡(Ffn).(114)WMVλfnα=WMVα-mλfn=WMVλfn.By Proposition 20 and remarking that for the fixed renormalization sequence above, this change of variables consists in extracting a subsequence of renormalization. Thus, taking the limit, we get (115)WMVλfα=WMVλf.Proposition 46.
Letv∈F. Let f be a function on F with mean value. Let fv:x∈F↦f(x+v). Then fv has a mean value and (116)WMVλfv=WMVλf.Proof.
Let(fn)n∈N be a sequence which converges uniformly to f. Let vn=πn(v)∈Ffn. We have (fn)v=(fn)vn. Then, (117)WMVλfv=limn→+∞⁡WMVλfnvn=limn→+∞⁡WMVλfnby  Proposition31=WMVλf.Proposition 47.
LetUF be the group of unitary operators of H which restricts to a bounded map F→F together with its inverse. Let u∈UF. Let f be a map with mean value. Then f∘u has a mean value and (118)WMVλf∘u=WMVλf.This last proposition becomes obvious after remarking that we transform the sequence(Ffn)n∈N into the orthogonal sequence (119)u-1Ffnn∈N=Ffn∘un∈N.This remark shows that we get the same mean value for f∘u as for f by changing the orthogonal sequence.


### 5.3. Invariance by Restriction

With the same notations as above, letG be a vector subspace of F such that (120)⨁n∈NFfn⊂G.As a consequence, if g is the restriction of f to G, the sequence (fn)n∈N of cylindrical functions on F restricts to a sequence (gn)n∈N of cylindrical functions on G. Then, for uniform convergence, (121)limn→+∞⁡gn=gand for fixed n∈N we get through restriction to Ffn, (122)WMVλgn=WMVλfn.Taking the limit, we get (123)WMVλg=WMVλf.This shows the restriction property announced in the Introduction.


## 5.1. Mean Value by Infinite Product

Letf be a bounded function which is the uniform limit of a sequence of cylindrical functions (fn)n∈N. Here, an orthonormal basis (ek)k∈N is obtained by induction, completing at each step an orthonormal basis of Ffn by an orthonormal basis of Ffn+1. Thus we can identify F with a subset D of RN which is invariant under change of a finite number of coordinates. This qualifies it as admissible since, with the notations used in Definition 37, (111)∏λ∈Λ~Uλ,n-DΛ~,n,x=∅,for any set of renormalization procedures in RN as defined in Section 4.1 so that Theorem 39 applies. We note by (112)WMVλfthis value. We remark that we already know by Theorem 39 that this mean value does not depend on the sequence (fn)n∈Nonly once the sequence(Ffn)n∈Nis fixed. In other words, two sequences (fn)n∈N and (fn′)n∈N which converge uniformly to f a priori lead to the same mean value if Ffn=Ffn′ (maybe up to reindexation). From heuristic calculations, it seems to come from the choice of the renormalization procedure, which is dependent on the basis chosen, more than from the sequence (Ffn)n∈N.


## 5.2. Invariance

We notice three types of invariance: scale invariance, translation invariance, and invariance under the orthogonal (or unitary) group.Proposition 45.
Letα∈N∗. Let f be a function on an infinite dimensional vector space F with mean value. Let fα:x∈F↦f(αx). Then fα has a mean value and (113)WMVλfα=WMVλf.Proof.
Let(fn)n∈N be a sequence which converges uniformly to f. Then, with the notations above, the sequence ((fn)α)n∈N converges uniformly to fα. Let m=dim⁡(Ffn).(114)WMVλfnα=WMVα-mλfn=WMVλfn.By Proposition 20 and remarking that for the fixed renormalization sequence above, this change of variables consists in extracting a subsequence of renormalization. Thus, taking the limit, we get (115)WMVλfα=WMVλf.Proposition 46.
Letv∈F. Let f be a function on F with mean value. Let fv:x∈F↦f(x+v). Then fv has a mean value and (116)WMVλfv=WMVλf.Proof.
Let(fn)n∈N be a sequence which converges uniformly to f. Let vn=πn(v)∈Ffn. We have (fn)v=(fn)vn. Then, (117)WMVλfv=limn→+∞⁡WMVλfnvn=limn→+∞⁡WMVλfnby  Proposition31=WMVλf.Proposition 47.
LetUF be the group of unitary operators of H which restricts to a bounded map F→F together with its inverse. Let u∈UF. Let f be a map with mean value. Then f∘u has a mean value and (118)WMVλf∘u=WMVλf.This last proposition becomes obvious after remarking that we transform the sequence(Ffn)n∈N into the orthogonal sequence (119)u-1Ffnn∈N=Ffn∘un∈N.This remark shows that we get the same mean value for f∘u as for f by changing the orthogonal sequence.


## 5.3. Invariance by Restriction

With the same notations as above, letG be a vector subspace of F such that (120)⨁n∈NFfn⊂G.As a consequence, if g is the restriction of f to G, the sequence (fn)n∈N of cylindrical functions on F restricts to a sequence (gn)n∈N of cylindrical functions on G. Then, for uniform convergence, (121)limn→+∞⁡gn=gand for fixed n∈N we get through restriction to Ffn, (122)WMVλgn=WMVλfn.Taking the limit, we get (123)WMVλg=WMVλf.This shows the restriction property announced in the Introduction.



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*Source: 9853672-2017-01-30.xml*