# Approximation Schemes for Path Integration on Riemannian Manifolds

Juan Carlos Sampedro  Department of Mathematical Analysis and Applied
Mathematics
Faculty of Mathematical Science
Complutense University of Madrid
28040-Madrid
Spain<EMAIL_ADDRESS>
###### Abstract.

_Truth is much too complicated to allow anything but approximations._

John von Neumann

In this paper, we prove a finite dimensional approximation scheme for the
Wiener measure on closed Riemannian manifolds, establishing a generalization
for $L^{1}$-functionals, of the approach followed by Andersson and Driver on
[1]. We follow a new approach motived by categorical concept of colimit.

###### Key words and phrases:

Colimit, Finite Dimensional Approximations, Riemannian Manifolds, Stratonovich
Stochastic Integral, Wiener Measure

###### 2010 Mathematics Subject Classification:

28C20 (primary), 58D20, 46B25, 60H05 (secondary)

The author has been supported by the Research Grant PGC2018-097104-B-I00 of
the Spanish Ministry of Science, Technology and Universities, by the Institute
of Interdisciplinar Mathematics of Complutense University and by PhD Grant
PRE2019_1_0220 of the Basque Country Government.

##  1\. Introduction

In 1920, N. Wiener, based on Daniell’s recent interpretation of integral
[7]-[9], defined in [25] an integral for bounded and continuous functionals
$F:\mathcal{C}_{\mathbf{x}_{0}}[0,1]\to\mathbb{R}$, where the notation
$\mathcal{C}_{\mathbf{x}_{0}}[0,1]$ stands for the space of continuous
functions $\mathbf{u}:[0,1]\to\mathbb{R}$ with base point
$\mathbf{u}(0)=\mathbf{x}_{0}$. In later papers [25]-[30], he connected this
notion to that of Brownian motion and he defined the so-called Wiener process.
In posterior works, he generalized this results defining a probability measure
$\mu_{\mathbf{x}_{0}}$ on the measurable space
$(\mathcal{C}_{\mathbf{x}_{0}}[0,1],\mathcal{B}_{\mathbf{x}_{0}})$, where
$\mathcal{B}_{\mathbf{x}_{0}}$ stands for the Borel $\sigma$-algebra of
$\mathcal{C}_{\mathbf{x}_{0}}[0,1]$ endowed with the uniform convergence
topology. This measure is characterized by the following property: For each
finite partition $\mathcal{T}=\\{t_{1}<t_{2}<\cdots<t_{n}\\}\subset(0,1]$ and
each family $(B_{t})_{t\in\mathcal{T}}$ of Borel subsets of $\mathbb{R}$, the
identity

(1)
$\mu_{\mathbf{x}_{0}}\left(\pi^{-1}_{\mathcal{T}}(B_{t})_{t\in\mathcal{T}}\right)=\int_{B_{t_{1}}}\overset{(n)}{\cdots}\int_{B_{t_{n}}}\prod_{i=1}^{n}p_{t_{i}-t_{i-1}}(x_{i},x_{i-1})\prod_{i=1}^{n}dx_{i},\quad
t_{0}=0,\ x_{0}=\mathbf{x}_{0},$

holds, where $p_{t}(x,y)$ is the heat kernel of $\mathbb{R}$ and
$\pi_{\mathcal{T}}:\mathcal{C}_{\mathbf{x}_{0}}[0,1]\to\mathbb{R}^{\mathcal{T}}$
is the projector defined by

$\pi_{\mathcal{T}}:\mathcal{C}_{\mathbf{x}_{0}}[0,1]\longrightarrow\mathbb{R}^{\mathcal{T}},\quad\pi_{\mathcal{T}}(\mathbf{u}):=(\mathbf{u}(t))_{t\in\mathcal{T}}.$

At first, it seems there is no easy way to compute the integral of an
arbitrary measurable functional
$F:\mathcal{C}_{\mathbf{x}_{0}}[0,1]\to\mathbb{R}$. Nevertheless, Wiener
proved in [25] an analogue of Jessen’s formula [15, 23] for the measure
$\mu_{\mathbf{x}_{0}}$. More explicitly, he proved that given a bounded and
continuous functional on $\mathcal{C}_{\mathbf{x}_{0}}[0,1]$ and a partition
$\mathcal{T}=\left\\{\mathcal{T}_{n}\right\\}_{n\in\mathbb{N}}$,
$\mathcal{T}_{n}=\\{t^{n}_{i}\\}_{i=1}^{n}$ of $[0,1]$ with mesh zero,

$\lim_{n\to\infty}\max_{2\leq i\leq n}|t_{i}^{n}-t_{i-1}^{n}|=0,$

the integral of $F$ can be computed though finite dimensional integrals via

$\int_{\mathcal{C}_{\mathbf{x}_{0}}[0,1]}F(\mathbf{u})\
d\mu_{\mathbf{x}_{0}}(\mathbf{u})=\lim_{n\to\infty}\int_{\mathbb{R}^{n}}F_{n}(x_{1},x_{2},...,x_{n})\prod_{i=1}^{n}p_{t^{n}_{i}-t^{n}_{i-1}}(x_{i},x_{i-1})\prod_{i=1}^{n}dx_{i},$

where the functions $F_{n}:\mathbb{R}^{n}\to\mathbb{R}$ are defined by

$F_{n}(x_{1},x_{2},...,x_{n}):=F(\mathbf{u}_{(x_{1},x_{2},...)}),$

where $\mathbf{u}_{(x_{1},x_{2},...)}$ denotes the linear interpolation of the
points $x_{1},x_{2},...,x_{n}$, for each $n\in\mathbb{N}$. In [23], the author
generalized this formula to every $L^{1}$-functional proving that for each
$F\in L^{1}(\mathcal{C}_{\mathbf{x}_{0}}[0,1],\mu_{\mathbf{x}_{0}})$, there
exists a finite dimensional functional sequence
$(F_{n})_{n\in\mathbb{N}}\in\bigtimes_{n\in\mathbb{N}}L^{1}(\mathbb{R}^{n},\mu^{n}_{\mathbf{x}_{0}})$
such that

(2) $\int_{\mathcal{C}_{\mathbf{x}_{0}}[0,1]}F\
d\mu_{\mathbf{x}_{0}}=\lim_{n\to\infty}\int_{\mathbb{R}^{n}}F_{n}\
d\mu^{n}_{\mathbf{x}_{0}}\ \ \text{ where }\
d\mu^{n}_{\mathbf{x}_{0}}=\prod_{i=1}^{n}p_{t^{n}_{i}-t^{n}_{i-1}}(x_{i},x_{i-1})\prod_{i=1}^{n}dx_{i}.$

A similar discussion can be done for the category of Riemannian Manifolds.
Given a compact connected Riemannian manifold $(M,g)$ (closed Riemannian
manifold for short) of dimension $m$, we can construct, analogously to the
case of $[0,1]$, the measure space
$(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$. Here, the notation
$\mathcal{C}_{\mathbf{x}_{0}}(M)$ denotes the space of continuous curves
$\gamma:[0,1]\to M$ beginning at $\mathbf{x}_{0}$ and $\mu_{\mathbf{x}_{0}}$
the Wiener measure on $\mathcal{C}_{\mathbf{x}_{0}}(M)$, i.e., a measure
satisfying an analogue of equation (1) for this setting (see section 2 for
further details). Similar versions of Jessen type formula have been developed
for the category of Riemannian manifolds in [1]. In that article, Andersson
and Driver proved that given a bounded and continuous functional
$F:\mathcal{C}_{\mathbf{x}_{0}}(M)\to\mathbb{R}$, the identity

(3) $\int_{\mathcal{C}_{\mathbf{x}_{0}}(M)}F\
d\mu_{\mathbf{x}_{0}}=\lim_{n\to\infty}\int_{H_{\mathcal{T}_{n}}(M)}F(\sigma)\
d\nu_{\mathcal{T}_{n}}(\sigma)$

holds, where $(H_{\mathcal{T}_{n}}(M),\nu_{\mathcal{T}_{n}})$ is a finite
dimensional measure space based on the geometrical data of $(M,g)$ and
$\mathcal{T}=\left\\{\mathcal{T}_{n}\right\\}_{n\in\mathbb{N}},\mathcal{T}_{n}=\\{t^{n}_{i}\\}_{i=1}^{n}$,
is a partition of $[0,1]$ with mesh zero. Roughly, $H_{\mathcal{T}_{n}}(M)$ is
the space of piecewise geodesics paths in $M$, $\sigma:[0,1]\to M$, which
change directions only at the partition points
$\mathcal{T}_{n}=\\{t^{n}_{i}\\}_{i=1}^{n}$. The precise definition of the
pair $(H_{\mathcal{T}_{n}}(M),\nu_{\mathcal{T}_{n}})$ will be given in section
6.

After Andersson and Driver, several developments and generalizations have been
done by several authors, for instance this scheme has been generalized to heat
kernels on vector bundles in [2, 3] and further developed in [16, 17, 18, 19].
This finite dimensional approximations of path integrals have a great impact
in theoretical physics and in particular are extremely useful for the Feynman
path integral approach to quantum field theory [10, 11].

The aim of this article is to establish a generalization of equation (3) for
every integrable functional $F:\mathcal{C}_{\mathbf{x}_{0}}(M)\to\mathbb{R}$,
not necessarily bounded and continuous in the vein of the analogous result (2)
for the classical Wiener measure proved in [23]. To obtain this result, we use
a categorical point of view of this type of approximations. We prove that for
$1\leq p<\infty$, the space
$L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$ is the colimit
of certain diagram consisting on $L^{p}$-spaces of finite dimensional data
associated with $M$.

This generalization and structural result has many applications in stochastic
analysis. For instance, this scheme allows to embed the notion of the
Stratonovich stochastic integral

$\int_{0}^{1}f(X_{t})\circ dX_{t}\in
L^{2}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),$

in our approximation techniques, where $X$ denotes the Wiener process on
$(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$. Indeed, since the
Stratonovich stochastic integral is not, in general, a continuous and bounded
functional, we cannot apply the approximation (3) directly, but we can apply
the techniques of this article.

The paper is organized as follows. In section two, we present the abstract
results about colimits in category theory that will be used throughout this
article and we recall the definition and construction of the Wiener measure on
closed Riemannan manifolds. In section three, we prove that for $1\leq
p<\infty$, the space
$L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$ is the colimit
of a diagram consisting on $L^{p}$-spaces of Cartesian products of a finite
number of $M$-factors. In section four, we derive a finite dimensional
approximation formula of type (2) for integrable functionals $F\in
L^{1}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, via a particular
realization of the colimit of the diagram defined in the preceding section. In
section five, we apply our approximation scheme to Stratonovich stochastic
integrals. The end of this article, section six, is dedicated to recall
Andersson and Driver’s scheme and to adapt the results developed in sections
three and four to their framework. In particular, we obtain a generalization
of (3) for every integrable functional.

It is convenient to remark that all the results obtained in this article can
be easily adapted to the category of Riemannian manifolds with boundary and to
the case of continuous paths with fixed initial and end point, the so-called
Pinned Wiener spaces. See for instance [4] for the definition and construction
of the Wiener measure on this spaces.

##  2\. Preliminaries

In this short section, we will recall some basic facts about colimits and the
Wiener measure on Riemannian manifolds that will be used trough this article.
It will be also useful to fix notation.

###  2.1. Basic notions about colimits

Let $\mathfrak{C}$ be a fixed category and $I$ a directed set. A diagram is a
functor $F:I\to\mathfrak{C}$. It can be represented as $(X_{i},\varphi_{ij})$
for a family of objects indexed by $I$, $\\{X_{i}:i\in I\\}$, and for each
$i\leq j$ a morphism $\varphi_{ij}:X_{i}\to X_{j}$ such that $\varphi_{ii}$ is
the identity on $X_{i}$ and $\varphi_{ik}=\varphi_{jk}\circ\varphi_{ij}$ for
each $i\leq j\leq k$. In general, we can consider diagrams as functors
$F:\mathfrak{J}\to\mathfrak{C}$ indexed by a general category $\mathfrak{J}$,
however, diagrams indexed by directed sets are enough for our purposes. A
cocone is a pair $(X,\phi_{i})$ where $X$ is an object of $\mathfrak{C}$ and
$\phi_{i}:X_{i}\to X$ is a morphism of $\mathfrak{C}$ such that
$\phi_{i}=\phi_{j}\circ\varphi_{ij}$ for every $i\leq j$. The class of cocones
of a given diagram forms itself a category. A colimit of the diagram
$(X_{i},\varphi_{ij})$ is defined to be a cocone $(L,\psi_{i})$ characterized
by the following universal property: For any cocone $(X,\phi_{i})$, there
exists a unique morphism $\varphi_{X}:L\to X$ making the diagram of Figure 1
commutative.

$\textstyle{X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{ij}}$$\scriptstyle{\psi_{i}}$$\scriptstyle{\phi_{i}}$$\textstyle{X_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{j}}$$\scriptstyle{\phi_{j}}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{X}}$$\textstyle{X}$
Figure 1. Diagram of the Universal Property

The colimits of a given diagram can be also characterized as initial objects
in the category of cocones. The colimit of a given diagram does not
necessarily exist, however if the colimit exists, it is unique up to a unique
isomorphism in the category of cocones. Any given isomorphism class of the
colimit is called a realization. The colimit of a given diagram
$(X_{i},\varphi_{ij})$ is commonly denoted as $L\equiv\underrightarrow{\lim}\
X_{i}$.

In this article, we will consider the category $\mathfrak{Ban}$ whose objects
are Banach spaces and whose morphisms are linear isometries. The proof of the
existence of the colimit for every diagram on $\mathfrak{Ban}$ can be found in
[24, App. L] or in [5] and references therein.

Though this article, we will make use of the following simple Lemma.

######  Lemma 2.1.

Let $(X_{i},\varphi_{ij})$ be a diagram on $\mathfrak{Ban}$. Then a cocone
$(X,\phi_{i})$ is a realization of the colimit of $(X_{i},\varphi_{ij})$ if
and only if $\bigcup_{i\in I}\phi_{i}(X_{i})$ is dense in $X$.

###### Proof.

Suppose that $(X,\phi_{i})$ defines a realization of the colimit of
$(X_{i},\varphi_{ij})$. Consider the cocone $(Y,\rho_{i})$ where

$Y:=\overline{\bigcup_{i\in I}\phi_{i}(X_{i})}\subset X,$

and where $\rho_{i}:X_{i}\to Y$ are the canonical inclusions. By the universal
property there exists a unique isometry $\varphi_{Y}:X\to Y$ making the
diagram of Figure 1 commutative. It is straightforward to prove that the
morphism $\varphi_{Y}$ is in fact an isomorphism, hence $\bigcup_{i\in
I}\phi_{i}(X_{i})$ is dense in $X$. On the other hand, take a cocone
$(X,\phi_{i})$ with $\bigcup_{i\in I}\phi_{i}(X_{i})$ dense in $X$. For any
other cocone $(Y,\rho_{i})$, define the morphism $\varphi_{Y}:X\to Y$ by

$\varphi_{Y}\circ\phi_{i}:=\rho_{i},\quad i\in I.$

Then it is easy to see that the morphism $\varphi_{Y}$ is the unique morphism
making the diagram of Figure 1 commutative. Hence $(X,\phi_{i})$ is a colimit.
This concludes the proof. ∎

###  2.2. Wiener measure on Riemannian manifolds

Along this article, $(M,g)$ denotes a compact connected Riemannian manifold
(closed Riemannian manifold for short) with a fixed base point
$\mathbf{x}_{0}\in M$. The notation $\mathcal{C}_{\mathbf{x}_{0}}(M)$ stands
for the space of continuous paths $\gamma\in\mathcal{C}([0,1],M)$ satisfying
$\gamma(0)=\mathbf{x}_{0}$, $\mathcal{B}_{\mathbf{x}_{0}}$ for the Borel
$\sigma$-algebra of $\mathcal{C}_{\mathbf{x}_{0}}(M)$ with respect to the
uniform convergence topology given by the induced metric of $M$ and
$\mu_{\mathbf{x}_{0}}$ for the Wiener measure on $M$ with base point
$\mathbf{x}_{0}$.

We recall the definition of the measure space
$(\mathcal{C}_{\mathbf{x}_{0}}(M),\mathcal{B}_{\mathbf{x}_{0}},\mu_{\mathbf{x}_{0}})$.
Consider in $(M,g)$ the measure $\mu:\mathcal{B}_{M}\to[0,+\infty]$ induced by
the metric $g$, where $\mathcal{B}_{M}$ denotes the Borel $\sigma$-algebra of
$M$. This measure is locally given by the expression

$d\mu=\sqrt{\det(g_{ij})_{ij}}\ dx_{1}\wedge\cdots\wedge dx_{m}$

where $m$ is the dimension of $M$ and $(g_{ij})_{ij}$ is the matrix of $g$ in
a local chart. For each closed Riemannian manifold $(M,g)$, there exists a
heat kernel $p_{t}(x,y)$, for $t>0$, $x,y\in M$, i.e., the Schwartz kernel of
the selfadjoint operator $e^{t\Delta}$ on $L^{2}(M,\mu)$, where $\Delta$
denotes the Laplace-Beltrami operator on $(M,g)$. The proof of the existence
of this map can be found in [4, 13]. A well-known consequence of the
Kolmogorov extension Theorem [31, Th. 6.1], is the existence of a probability
measure

$\mu_{\mathbf{x}_{0}}:\mathcal{B}\longrightarrow[0,+\infty]$

on $(M^{[0,1]},\mathcal{B})$, where $\mathcal{B}$ denotes the Borel
$\sigma$-algebra of $M^{[0,1]}$ with respect to the product topology,
satisfying the identity

(4)
$\mu_{\mathbf{x}_{0}}\left(\pi^{-1}_{\mathcal{T}}(B_{t})_{t\in\mathcal{T}\backslash\\{0\\}}\right)=\int_{B_{t_{1}}}\overset{(n)}{\cdots}\int_{B_{t_{n}}}\prod_{j=1}^{n}p_{t_{j}-t_{j-1}}(x_{j},x_{j-1})\prod_{j=1}^{n}d\mu(x_{j}),\quad
x_{0}=\mathbf{x}_{0},$

for each finite partition
$\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}\\}\subset[0,1]$ and each family of
Borel subsets
$(B_{t})_{t\in\mathcal{T}\backslash\\{0\\}}\subset\mathcal{B}_{M}$. Here and
in the sequel, the notation $\pi_{\mathcal{T}}$ stands for the projector
defined by

(5) $\pi_{\mathcal{T}}:M^{[0,1]}\longrightarrow
M^{\mathcal{T}\backslash\\{0\\}},\quad\pi_{\mathcal{T}}(\gamma_{t})_{t\in[0,1]}:=(\gamma_{t})_{t\in\mathcal{T}\backslash\\{0\\}}.$

Since $(M,g)$ is compact, it is, in particular, stochastically complete (see
for instance [13]), and therefore

$\int_{M}p_{t}(x,y)d\mu(y)=1$

for each $t>0$ and $x\in M$. This fact implies that the measure
$\mu_{\mathbf{x}_{0}}$ is of probability, that is,
$\mu_{\mathbf{x}_{0}}(M^{[0,1]})=1$.

On the other hand, by [4, Cr. 2.19], the measure $\mu_{\mathbf{x}_{0}}$
satisfies the identity
$\mu_{\mathbf{x}_{0}}(\mathcal{C}^{\alpha}_{\mathbf{x}_{0}}(M))=1\text{ for
each }\alpha\in(0,1/2)$ where $\mathcal{C}^{\alpha}_{\mathbf{x}_{0}}(M)$
stands for the subset of $M^{[0,1]}$ consisting of Hölder continuous paths
$\gamma:[0,1]\to M$ of exponent $\alpha\in(0,1)$ satisfying
$\gamma(0)=\mathbf{x}_{0}$. Therefore, since $\mathcal{C}_{\mathbf{x}_{0}}(M)$
is a Borel subset of $M^{[0,1]}$ (see [12, Th. 10.28]) containing
$\mathcal{C}^{\alpha}_{\mathbf{x}_{0}}(M)$ and since

$\mathcal{B}\cap\mathcal{C}_{\mathbf{x}_{0}}(M):=\\{B\cap\mathcal{C}_{\mathbf{x}_{0}}(M):B\in\mathcal{B}\\},$

coincides with the Borel $\sigma$-algebra $\mathcal{B}_{\mathbf{x}_{0}}$ (see
[20, Prop. 2.2]), we can consider the restricted probability space
$(\mathcal{C}_{\mathbf{x}_{0}}(M),\mathcal{B}_{\mathbf{x}_{0}},\mu_{\mathbf{x}_{0}})$.
The restricted measure $\mu_{\mathbf{x}_{0}}$ is called the Wiener measure of
$M$ with base point $\mathbf{x}_{0}$. The proof of these facts can be found
for instance in [4, 13] and references therein.

##  3\. Approximation Scheme for the Wiener Measure

In this section, we will prove that the spaces
$L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, $1\leq
p<\infty$, are realizations of the colimit of the diagram
$(L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}}),\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})$.
Let us start by defining the diagram
$(L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}}),\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})$.
Consider the directed set $\mathcal{P}$ consisting on partitions of $[0,1]$

$\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}\\},$

partially ordered by inclusion. For notational simplicity, we will use the
notation $M^{\mathcal{T}}$ to denote

$M^{\mathcal{T}}:=\bigtimes_{t\in\mathcal{T}\backslash\\{0\\}}M.$

For a partition $\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}\\}$ in
$\mathcal{P}$, consider the probability space
$(M^{\mathcal{T}},\bigotimes_{i=1}^{n}\mathcal{B}_{M},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})$
where $\mu_{\mathbf{x}_{0}}^{\mathcal{T}}$ is the measure defined by the
density

$d\mu_{\mathbf{x}_{0}}^{\mathcal{T}}=\prod_{i=1}^{n}p_{t_{i}-t_{i-1}}(x_{t_{i}},x_{t_{i-1}})\prod_{i=1}^{n}d\mu(x_{t_{i}}),\quad
x_{0}=\mathbf{x}_{0},$

and $\bigotimes_{i=1}^{n}\mathcal{B}_{M}$ denotes the product
$\sigma$-algebra. For each pair of partitions
$\mathcal{T},\mathcal{T}^{\prime}$ with
$\mathcal{T}\subset\mathcal{T}^{\prime}$, we obtain a measurable projection
map

$\pi_{\mathcal{T}\mathcal{T}^{\prime}}:M^{\mathcal{T}^{\prime}}\longrightarrow
M^{\mathcal{T}},\quad\pi_{\mathcal{T}\mathcal{T}^{\prime}}(x_{t})_{t\in\mathcal{T}^{\prime}\backslash\\{0\\}}:=(x_{t})_{t\in\mathcal{T}\backslash\\{0\\}}.$

We therefore obtain a diagram in the category of measure spaces, indexed by
the directed set $\mathcal{P}$. For each pair of partitions
$\mathcal{T},\mathcal{T}^{\prime}$ with
$\mathcal{T}\subset\mathcal{T}^{\prime}$, we can construct the isometry

$\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}}:L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})\longrightarrow
L^{p}(M^{\mathcal{T}^{\prime}},\mu^{\mathcal{T}^{\prime}}_{\mathbf{x}_{0}}),\quad\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}}(f):=f\circ\pi_{\mathcal{T}\mathcal{T}^{\prime}}.$

Applying this construction to the entire diagram, we obtain a diagram in
$\mathfrak{Ban}$ indexed by the directed set $\mathcal{P}$, where all arrow
directions are now swapped, as the $L^{p}$-construction is contravariant. This
diagram forms a diagram in $\mathfrak{Ban}$ that will be subsequently denoted
by
$(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}),\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})$.

It will be proved that the cocone
$(L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\phi_{\mathcal{T}})$,
where $\phi_{\mathcal{T}}$ are the morphisms defined by

$\phi_{\mathcal{T}}:L^{p}(M^{\mathcal{T}},\nu^{\mathcal{T}}_{\mathbf{x}_{0}})\longrightarrow
L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\quad\phi_{\mathcal{T}}(f):=f\circ\pi_{\mathcal{T}}|_{\mathcal{C}_{\mathbf{x}_{0}}(M)},$

defines a realization of the colimit of
$(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}),\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})$.
The following Lemma will be useful for this purpose.

######  Lemma 3.1.

Let $1\leq p<\infty$, then the subspace
$\bigcup_{\mathcal{T}\in\mathcal{P}}\phi_{\mathcal{T}}(L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}}))$
is dense in $L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$.

###### Proof.

As we have recalled in section 2.2, the following equality of
$\sigma$-algebras

(6)
$\mathcal{B}_{\mathbf{x}_{0}}=\mathcal{B}\cap\mathcal{C}_{\mathbf{x}_{0}}(M),$

holds. By definition, the product $\sigma$-algebra $\mathcal{B}$ is generated
by the cylinder sets and therefore by (6), it follows that

$\mathcal{B}_{\mathbf{x}_{0}}=\sigma(\mathcal{R}),\quad\mathcal{R}:=\\{\pi^{-1}_{\mathcal{T}}(B_{t})_{t\in\mathcal{T}}:(B_{t})_{t\in\mathcal{T}}\subset\mathcal{B}_{M},\mathcal{T}\subset(0,1]\text{
finite}\\}\cap\mathcal{C}_{\mathbf{x}_{0}}(M).$

Since $(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$ has finite
measure, by [6, Lem. 3.4.6], the subspace
$\text{Span}\left\\{\chi_{R}:R\in\mathcal{R}\right\\}$, where $\chi_{R}$
denoted the characteristic function of $R$, is dense in
$L^{p}\left(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}\right)$. From
the inclusion

$\left\\{\chi_{R}:R\in\mathcal{R}\right\\}\subset\bigcup_{\mathcal{T}\in\mathcal{P}}\phi_{\mathcal{T}}(L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})),$

it follows that
$\bigcup_{\mathcal{T}\in\mathcal{P}}\phi_{\mathcal{T}}(L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}}))$
is dense in
$L^{p}\left(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}\right)$. This
concludes the proof. ∎

Finally, we prove the main result of this section.

######  Theorem 3.2.

Let $1\leq p<\infty$, then the cocone
$(L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\phi_{\mathcal{T}})$
defines a realization of the colimit of
$(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}),\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})$.

###### Proof.

By Lemma 3.1, the subspace

$\bigcup_{\mathcal{T}\in\mathcal{P}}\phi_{\mathcal{T}}(L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}}))\subset
L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$

is dense in $L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$.
Therefore, we can apply Lemma 2.1 to
$(L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\phi_{\mathcal{T}})$
obtaining the result. ∎

##  4\. Derivation of the Limit Formula

In this section, we will prove that the integral of a given integrable
functional can be expressed as the limit of finite dimensional integrals. To
relate the colimit structure obtained in the last section with the integration
procedure, we have to provide a particular realization of the colimit of the
directed system
$(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}),\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})$.
To define this realization, we need the following definition.

An element
$(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\in\bigtimes_{\mathcal{T}\in\mathcal{P}}L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})$
is said to be co-Cauchy if for each $\varepsilon>0$, there exists
$\mathcal{R}\in\mathcal{P}$ such that

$\|\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}}(f_{\mathcal{T}})-f_{\mathcal{T^{\prime}}}\|_{L^{p}_{\mathcal{T}}}<\varepsilon,\text{
for all }\mathcal{T},\mathcal{T}^{\prime}\in\mathcal{P}\text{ with
}\mathcal{R}\subset\mathcal{T}\subset\mathcal{T}^{\prime}.$

For notational simplicity we denote $L^{p}\equiv
L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$ and
$L^{p}_{\mathcal{T}}\equiv
L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})$.

We define the space
$\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))$ by

(7)
$\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})):=\Big{\\{}(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\in\bigtimes_{\mathcal{T}\in\mathcal{P}}L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}):(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\text{
is co-Cauchy}\Big{\\}}\Big{/}\sim,$

where we relate
$(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\sim(g_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}$
if

$\lim_{\mathcal{T}}\|f_{\mathcal{T}}-g_{\mathcal{T}}\|_{L^{p}_{\mathcal{T}}}=0.$

We define in
$\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))$ the
norm

$\|(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\|_{\mathfrak{L}}:=\lim_{\mathcal{T}}\|f_{\mathcal{T}}\|_{L^{p}_{\mathcal{T}}}.$

All the limits involved are considered as limits of nets. For further
considerations, it is imperative to obtain the completeness of the space
$\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))$,
which is not obvious from the definition. For this reason, we identify it
directly with $L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$
via the following result.

######  Theorem 4.1.

The following spaces are isometrically isomorphic for $1\leq p<\infty$

$\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))\simeq
L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}).$

###### Proof.

Let us prove that the map

$\mathfrak{I}_{p}:\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))\longrightarrow
L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\quad(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\mapsto\lim_{\mathcal{T}}\phi_{\mathcal{T}}(f_{\mathcal{T}}).$

defines an isometric isomorphism. The map $\mathfrak{I}_{p}$ is well defined
since the net
$(\phi_{\mathcal{T}}(f_{\mathcal{T}}))_{\mathcal{T}\in\mathcal{P}}$ converges
in $L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, a fact that
follows from the identity

$\displaystyle\|\phi_{\mathcal{T}}(f_{\mathcal{T}})-\phi_{\mathcal{T}^{\prime}}(f_{\mathcal{T}^{\prime}})\|_{L^{p}}$
$\displaystyle=\|(\phi_{\mathcal{T}^{\prime}}\circ\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})(f_{\mathcal{T}})-\phi_{\mathcal{T}^{\prime}}(f_{\mathcal{T}^{\prime}})\|_{L^{p}}$
$\displaystyle=\|\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}}(f_{\mathcal{T}})-f_{\mathcal{T}^{\prime}}\|_{L^{p}_{\mathcal{T}^{\prime}}},\quad\mathcal{T}\subset\mathcal{T}^{\prime}$

and the co-Cauchy property. It is easily seen that $\mathfrak{I}_{p}$ is an
isometry since

$\|\mathfrak{I}_{p}(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\|_{L^{p}}=\|\lim_{\mathcal{T}}\phi_{\mathcal{T}}(f_{\mathcal{T}})\|_{L^{p}}=\lim_{\mathcal{T}}\|f_{\mathcal{T}}\|_{L^{p}_{\mathcal{T}}}=\|(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\|_{\mathfrak{L}}.$

Finally, we prove that $\mathfrak{I}_{p}$ is onto. Let $f\in
L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$ and take
$\mathcal{D}=\\{t_{i}\\}_{i\in\mathbb{N}\cup\\{0\\}}$, $t_{0}=0$, a dense
countable subset of $[0,1]$. Define the sequence of partitions
$\mathcal{P}=\\{\mathcal{P}_{n}\\}_{n\in\mathbb{N}}$ by

$\mathcal{P}_{n}:=\\{t_{i}\\}_{i=0}^{n},\quad n\in\mathbb{N}.$

Clearly $\mathcal{P}_{n}\subset\mathcal{P}_{n+1}$, $n\in\mathbb{N}$ and
$\bigcup_{n\in\mathbb{N}}\mathcal{P}_{n}=\mathcal{D}$. It is easily seen using
the same techniques of the proof of Lemma 3.1, that
$\bigcup_{n\in\mathbb{N}}\phi_{\mathcal{P}_{n}}(L^{p}(M^{\mathcal{P}_{n}},\mu_{\mathbf{x}_{0}}^{\mathcal{P}_{n}}))$
is dense in $L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$.
Hence there exists a sequence

$(f_{n})_{n\in\mathbb{N}},\quad f_{n}\in
L^{p}(M^{\mathcal{P}_{N_{n}}},\mu_{\mathbf{x}_{0}}^{\mathcal{P}_{N_{n}}})\text{
for some }N_{n}\in\mathbb{N},$

such that

$\lim_{n\to\infty}\phi_{\mathcal{P}_{N_{n}}}(f_{n})=f\text{ in
}L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}).$

Take an strictly increasing sequence
$(M_{n})_{n\in\mathbb{N}}\subset\mathbb{N}$ such that $N_{n}\leq
M_{n},n\in\mathbb{N}$. Then, in particular,
$\mathcal{P}_{N_{n}}\subset\mathcal{P}_{M_{n}}$ and

$g_{n}:=\pi^{\ast}_{\mathcal{P}_{N_{n}}\mathcal{P}_{M_{n}}}(f_{n})\in
L^{p}(M^{\mathcal{P}_{M_{n}}},\mu_{\mathbf{x}_{0}}^{\mathcal{P}_{M_{n}}}),\quad
n\in\mathbb{N}.$

Define the sequence

$(F_{n})_{n\in\mathbb{N}},\quad
F_{n}:=\left\\{\begin{array}[]{ll}\pi^{\ast}_{\mathcal{P}_{M_{m}}\mathcal{P}_{n}}(g_{n})&\text{
if }M_{m}\leq n<M_{m+1}\\\ 0&\text{ if }n<M_{1}\end{array}\right.$

then

$(F_{n})_{n\in\mathbb{N}}\in\bigtimes_{n\in\mathbb{N}}L^{p}(M^{\mathcal{P}_{n}},\mu_{\mathbf{x}_{0}}^{\mathcal{P}_{n}})\text{
and }\lim_{n\to\infty}\phi_{\mathcal{P}_{n}}(F_{n})=f\text{ in
}L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}).$

Define the associated net $(F_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}$ by

$F_{\mathcal{T}}:=\left\\{\begin{array}[]{ll}\pi^{\ast}_{\mathcal{P}_{n}\mathcal{T}}(F_{n})&\text{
if }\mathcal{P}_{n}\subset\mathcal{T}\text{ but
}\mathcal{P}_{n+1}\not\subset\mathcal{T}\\\ 0&\text{ else }\end{array}\right.$

Finally, we have that
$(F_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\in\bigtimes_{\mathcal{T}\in\mathcal{P}}L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})$
and $\lim_{\mathcal{T}}\phi_{\mathcal{T}}(F_{\mathcal{T}})=f$, which implies
that $\mathfrak{I}_{p}(F_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}=f$. This
concludes the proof. ∎

It is appropriate to note the following in accordance with the proof of
Theorem 4.1. One is tempted to think that in order to prove the surjectivity
of $\mathfrak{I}_{p}$, for each $f\in
L^{p}(\mathcal{C}_{\mathbf{x}_{0}},\mu_{\mathbf{x}_{0}})$, it is sufficient to
use Lemma 2.1, choosing a sequence
$(f_{\mathcal{T}})_{\mathcal{T}}\in\bigcup_{\mathcal{T}\in\mathcal{P}}\phi_{\mathcal{T}}(L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}}))$
such that $\lim_{\mathcal{T}}\phi_{\mathcal{T}}(f_{\mathcal{T}})=f$.
Nonetheless, it must be observed that the sequence
$(f_{\mathcal{T}})_{\mathcal{T}}$ is not necessarily contained in the
Cartesian product
$\bigtimes_{\mathcal{T}\in\mathcal{P}}L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})$
which makes things a bit more involved.

We define the morphisms
$\psi_{\mathcal{R}}:L^{p}(M^{\mathcal{R}},\mu^{\mathcal{R}}_{\mathbf{x}_{0}})\to\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))$,
$\mathcal{R}\in\mathcal{P}$, through

(8)
$\psi_{\mathcal{R}}(f_{\mathcal{R}})=(g_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}},\quad
g_{\mathcal{T}}:=\left\\{\begin{array}[]{ll}\pi^{\ast}_{\mathcal{R}\mathcal{T}}(f_{\mathcal{R}})&\text{
if }\mathcal{R}\subset\mathcal{T}\\\ 0&\text{ else }\end{array}\right.$

By Theorem 4.1, the space
$\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))$ is a
Banach space and therefore the pair
$(\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})),\psi_{\mathcal{R}})$
defines a cocone. The colimit of
$(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}),\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})$
will be identify with
$(\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})),\psi_{\mathcal{R}})$.

######  Theorem 4.2.

The cocone
$(\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})),\psi_{\mathcal{T}})$
defines another relization of the colimit of
$(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}),\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})$.

###### Proof.

Since the colimit is unique up to a unique isomorphism on the category of
cocones, it is enough to prove that
$(\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})),\psi_{\mathcal{T}})$
is isomorphic to
$(L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\phi_{\mathcal{T}})$
in the category of cocones. By the proof of Theorem 4.1, the map

$\mathfrak{I}_{p}:\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))\longrightarrow
L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\quad(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\mapsto\lim_{\mathcal{T}}\phi_{\mathcal{T}}(f_{\mathcal{T}}).$

is an isometric isomorphism. To prove that it defines an isomorphism of
cocones, we need to verify the commutativity of the diagram of Figure 2 for
each $\mathcal{T}\in\mathcal{P}$.

$\textstyle{L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}}}$$\scriptstyle{\psi_{\mathcal{T}}}$$\scriptstyle{\phi_{\mathcal{T}}}$$\textstyle{L^{p}(M^{\mathcal{T}^{\prime}},\mu^{\mathcal{T}^{\prime}}_{\mathbf{x}_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{\mathcal{T}^{\prime}}}$$\scriptstyle{\phi_{\mathcal{T}^{\prime}}}$$\textstyle{\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{I}_{p}}$$\textstyle{L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})}$
Figure 2. Diagram of the Universal Property II

Take $f_{\mathcal{T}}\in
L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})$, then by a simple
computation we obtain

$(\mathfrak{I}_{p}\circ\psi_{\mathcal{T}})(f_{\mathcal{T}})=\lim_{\mathcal{Q}}h_{\mathcal{Q}},\quad
h_{\mathcal{Q}}:=\left\\{\begin{array}[]{ll}\phi_{\mathcal{T}}(f_{\mathcal{T}})&\text{
if }\mathcal{T}\subset\mathcal{Q}\\\ 0&\text{ else }\end{array}\right.$

and thus
$(\mathfrak{I}_{p}\circ\psi_{\mathcal{T}})(f_{\mathcal{T}})=\phi_{\mathcal{T}}(f_{\mathcal{T}})$.
Hence the diagram of Figure 2 commutes and the proof is concluded. ∎

Here in after, as a direct consequence of Theorem 4.2, we can and we shall
denote
$\mathfrak{L}(L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}))\equiv\underrightarrow{\lim}\
L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})$.

As a rather direct application of the isometric property of
$\mathfrak{I}_{p}$, we obtain our integral limit approximation. Indeed, if
$F\in L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, there
exists an element
$(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\in\underrightarrow{\lim}\
L^{p}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})$ such that
$\|F\|_{L^{p}}=\|(f_{\mathcal{T}})_{\mathcal{T}}\|_{\mathfrak{L}}$. Therefore,
we have

$\int_{\mathcal{C}_{\mathbf{x}_{0}}(M)}|F|^{p}\
d\mu_{\mathbf{x}_{0}}=\lim_{\mathcal{T}}\int_{M^{\mathcal{T}}}|f_{\mathcal{T}}|^{p}\
d\mu_{\mathbf{x}_{0}}^{\mathcal{T}}.$

Furthermore, if we take into account that given $F\in
L^{1}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, we can write it
as $F=F^{+}-F^{-}$ with $F^{+},F^{-}$ positive and $F^{+},F^{-}\in
L^{1}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, then we get the
following result.

######  Theorem 4.3.

Let $F\in L^{1}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, then
there exists
$(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\in\bigtimes_{\mathcal{T}\in\mathcal{P}}L^{1}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})$
such that

$\int_{\mathcal{C}_{\mathbf{x}_{0}}(M)}F\
d\mu_{\mathbf{x}_{0}}=\lim_{\mathcal{T}}\int_{M^{\mathcal{T}}}f_{\mathcal{T}}\
d\mu_{\mathbf{x}_{0}}^{\mathcal{T}}.$

##  5\. Stratonovich Stochastic Integral

In this section, we will apply the developed theory to a particular example,
the Stratonovich stochastic integral. Our approach can be used to get a finite
dimensional approximation scheme for this type of integrals, in contrast to
the Andersson and Driver’s framework [1] that only can be used if the involved
functional is bounded and continuous.

Let us firstly recall briefly some basic facts about stochastic integration.
Let $(X,Y)=(\\{X_{t}\\}_{t\in[0,1]},\\{Y_{t}\\}_{t\in[0,1]})$ be a pair of
bounded $\mathbb{R}$-valued semimartingales defined in the probability space
$(\Omega,\mathcal{F},\mu)$. Then, the Stratonovich integral of $X$ with
respect to $Y$ is defined by the relation

$\int_{0}^{1}X_{t}\circ
dY_{t}:=\lim_{L^{2}(\mu)}\sum_{i=1}^{n}\frac{X_{t^{n}_{i}}+X_{t^{n}_{i-1}}}{2}(Y_{t^{n}_{i}}-Y_{t^{n}_{i-1}})\in
L^{2}(\Omega,\mu)$

where $\mathcal{T}=\\{\mathcal{T}_{n}\\}_{n\in\mathbb{N}}$,
$\mathcal{T}_{n}:=\\{t^{n}_{i}\\}_{i=0}^{n}$, is a fixed partition with mesh
zero of $[0,1]$. It is related to the Itô stochastic integral by the relation

$\int_{0}^{1}X_{t}\circ dY_{t}=\int_{0}^{1}X_{t}\ dY_{t}+[X,Y]_{t}$

where $[X,Y]_{t}$ denotes the covariation of the processes $(X,Y)$ and
$dY_{t}$ denotes the Itô differential. In the case in which
$(X,Y)=(\\{X_{t}\\}_{t\in[0,1]},\\{Y_{t}\\}_{t\in[0,1]})$ are bounded
$\mathbb{R}^{N}$-valued semimartingales, we define

$\int_{0}^{1}X_{t}\circ dY_{t}:=\sum_{i=1}^{N}\int_{0}^{1}X^{i}_{t}\circ
dY^{i}_{t}.$

It is worth to mention that the usual definition of the Stratonovich integral
is under convergence in probability [21, Th. 26, Ch. V]. Since we will deal
with semimartingales defined on compact manifolds, we only need the definition
for bounded ones, in which case, the convergence in probability implies the
$L^{2}$-convergence as a consequence of Vitali’s convergence Theorem.

It must be taken into account that if $(M,g)$ is a closed Riemannian manifold
embedded in the Euclidean space $\mathbb{R}^{N}$, by [14, Pr. 3.2.1], the
$M$-valued stochastic process $X=\\{X_{t}\\}_{t\in[0,1]}$ defined by the
coordinate functionals of
$(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, defines a
$\mathbb{R}^{N}$-valued bounded semimartingale. This implies by [14, Pr.
1.2.7, (i)] that $\\{f(X_{t})\\}_{t\in[0,1]}$ is a real valued semimartingale
for each $f\in\mathcal{C}^{\infty}(M)$. Therefore the Stratonovich stochastic
integral of $f(X)$ with respect to $X$, where
$f\in\mathcal{C}^{\infty}(M,\mathbb{R}^{N})$, $f=(f_{1},f_{2},...,f_{N})$, is
well defined.

The main result of this section is the following. It establishes which is
exactly the preimage of the functional $\int_{0}^{1}f(X_{t})\circ dX_{t}\in
L^{2}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$ via the
identification $\mathfrak{I}_{2}:\underrightarrow{\lim}\
L^{2}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})\to
L^{2}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$.

######  Theorem 5.1.

Let $(M,g)$ be a closed Riemannian manifold embedded in the Euclidean space
$\mathbb{R}^{N}$, $f\in\mathcal{C}^{\infty}(M,\mathbb{R}^{N})$,
$f=(f_{1},f_{2},...,f_{N})$, and $\\{X_{t}\\}_{t\in[0,1]}$ the $M$-valued
semimartingale defined by the coordinate functionals of
$(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$. Then

$\mathfrak{I}_{2}\left(\sum_{j=1}^{N}\sum_{i=1}^{n}\frac{f_{j}(\mathbf{x}_{t_{i}})+f_{j}(\mathbf{x}_{t_{i-1}})}{2}(x^{j}_{t_{i}}-x^{j}_{t_{i-1}})\right)_{\mathcal{T}\in\mathcal{P}}=\int_{0}^{1}f(X_{t})\circ
dX_{t},$

where $\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}\\}$ and
$\mathbf{x}_{t_{i}}=(x^{1}_{t_{i}},\cdots,x^{N}_{t_{i}})\in
M\subset\mathbb{R}^{N}$.

###### Proof.

Since $M$ is compact and embedded in $\mathbb{R}^{N}$, the process
$X=\\{X_{t}\\}_{t\in[0,1]}$ is, in particular, a bounded
$\mathbb{R}^{N}$-valued semimartingale and its Stratonovich integral is
defined in the usual manner by

$\displaystyle\int_{0}^{1}f(X_{t})\circ dX_{t}$
$\displaystyle:=\sum_{j=1}^{N}\int_{0}^{1}f_{j}(X_{t})\circ dX^{j}_{t}$
$\displaystyle=\lim_{L^{2}(\mu_{\mathbf{x}_{0}})}\sum_{j=1}^{N}\sum_{i=1}^{n}\frac{f_{j}(X_{t^{n}_{i}})+f_{j}(X_{t^{n}_{i-1}})}{2}(X^{j}_{t^{n}_{i}}-X^{j}_{t^{n}_{i-1}}),$

for every partition $\mathcal{Q}=\\{\mathcal{Q}_{n}\\}_{n\in\mathbb{N}}$,
$\mathcal{Q}_{n}:=\\{t^{n}_{i}\\}_{i=0}^{n}$ with mesh zero of $[0,1]$
satisfying $t^{n}_{0}=0$ for each $n\in\mathbb{N}$. This limit can be
expressed as a convergence of nets via

$\int_{0}^{1}f(X_{t})\circ
dX_{t}=\lim_{\mathcal{T}}\sum_{j=1}^{N}\sum_{i=1}^{n}\frac{f_{j}(X_{t_{i}})+f_{j}(X_{t_{i-1}})}{2}(X^{j}_{t_{i}}-X^{j}_{t_{i-1}}),$

where $\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}\\}$. The limit is understood
in $L^{2}$-convergence. On the other hand, we define the maps
$F^{j}_{\mathcal{T}}:M^{\mathcal{T}}\to\mathbb{R}$, $j\in\\{1,2,...,N\\}$, by

$F^{j}_{\mathcal{T}}(\mathbf{x}_{t_{i}})_{i=1}^{n}:=\sum_{i=1}^{n}\frac{f_{j}(\mathbf{x}_{t_{i}})+f_{j}(\mathbf{x}_{t_{i-1}})}{2}(x^{j}_{t_{i}}-x^{j}_{t_{i-1}}),$

where $\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}\\}$ and
$\mathbf{x}_{t_{i}}=(x^{1}_{t_{i}},x^{2}_{t_{i}},\cdots,x^{n}_{t_{i}})$. Then,
by the boundedness of each $F^{j}_{\mathcal{T}}$, it follows that
$F^{j}_{\mathcal{T}}\in
L^{2}(M^{\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}})$ and by the
definition of the morphisms
$\phi_{\mathcal{T}}:L^{2}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})\to
L^{2}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, we obtain

$\phi_{\mathcal{T}}(F^{j}_{\mathcal{T}})=\sum_{i=1}^{n}\frac{f_{j}(X_{t_{i}})+f_{j}(X_{t_{i-1}})}{2}(X^{j}_{t_{i}}-X^{j}_{t_{i-1}}).$

Therefore, if we prove that
$(F^{j}_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\in\underrightarrow{\lim}\
L^{2}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})$ for each
$j\in\\{1,2,...,N\\}$, by the definition of $\mathfrak{I}_{2}$ and the
linearity of $\phi_{\mathcal{T}}$, we will deduce

$\displaystyle\mathfrak{I}_{2}\left(\sum_{j=1}^{N}F^{j}_{\mathcal{T}}\right)_{\mathcal{T}\in\mathcal{P}}$
$\displaystyle=\lim_{\mathcal{T}}\sum_{j=1}^{N}\phi_{\mathcal{T}}(F^{j}_{\mathcal{T}})$
$\displaystyle=\lim_{\mathcal{T}}\sum_{j=1}^{N}\sum_{i=1}^{n}\frac{f_{j}(X_{t_{i}})+f_{j}(X_{t_{i-1}})}{2}(X^{j}_{t_{i}}-X^{j}_{t_{i-1}})$
$\displaystyle=\int_{0}^{1}f(X_{t})\circ dX_{t},$

obtaining the required result. Finally, we prove that, indeed,
$(F^{j}_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\in\underrightarrow{\lim}\
L^{2}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})$ for each
$j\in\\{1,2,...,N\\}$. Since each $\phi_{\mathcal{T}}$ is an isometric
isomorphism, we have

$\displaystyle\|\pi_{\mathcal{R}\mathcal{T}}^{\ast}(F^{j}_{\mathcal{R}})-F^{j}_{\mathcal{T}}\|_{L^{2}_{\mathcal{T}}}$
$\displaystyle=\|(\phi_{\mathcal{T}}\circ\pi_{\mathcal{R}\mathcal{T}}^{\ast})(F^{j}_{\mathcal{R}})-\phi_{\mathcal{T}}(F^{j}_{\mathcal{T}})\|_{L^{2}}$
$\displaystyle=\|\phi_{\mathcal{R}}(F^{j}_{\mathcal{R}})-\phi_{\mathcal{T}}(F^{j}_{\mathcal{T}})\|_{L^{2}},\quad\mathcal{R}\subset\mathcal{T}.$

The convergence of
$\\{\phi_{\mathcal{T}}(F^{j}_{\mathcal{T}})\\}_{\mathcal{T}\in\mathcal{P}}$
(that is justified by the existence of the integral $\int_{0}^{1}f(X_{t})\circ
dX_{t}$) together with the last identity, implies the co-Cauchy property for
$(F^{j}_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}$. Hence
$(F^{j}_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\in\underrightarrow{\lim}\
L^{2}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})$ and the proof is
concluded. ∎

As a direct consequence of the preceding result and the isometric property of
$\mathfrak{I}_{2}$, we deduce the following finite dimensional approximation
under the same hypothesis of Theorem 5.1,

$\int_{\mathcal{C}_{\mathbf{x}_{0}}(M)}\left|\int_{0}^{1}f(X_{t})\circ
dX_{t}\right|^{2}d\mu_{\mathbf{x}_{0}}=\lim_{\mathcal{T}}\int_{M^{\mathcal{T}}}\left|\sum_{j=1}^{N}\sum_{i=1}^{n}\frac{f_{j}(\mathbf{x}_{t_{i}})+f_{j}(\mathbf{x}_{t_{i-1}})}{2}(x^{j}_{t_{i}}-x^{j}_{t_{i-1}})\right|^{2}d\mu^{\mathcal{T}}_{\mathbf{x}_{0}},$

where $\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}\\}$. Finally, it is
convenient to remark that all the considerations made in this section can be
adapted easily to the cover the Itô stochastic integration.

##  6\. Approximation Scheme for the Geometric Framework

In this final section, we will give an approximation scheme based in the
geometric measure introduced by Andersson and Driver in [1]. Up to here, we
have proved that the spaces
$L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, $1\leq
p<\infty$, define a realization of the colimit of the diagram
$(L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}}),\pi^{\ast}_{\mathcal{T}\mathcal{T}^{\prime}})$,
where the measures $\mu_{\mathbf{x}_{0}}^{\mathcal{T}}$ are given by

$d\mu_{\mathbf{x}_{0}}^{\mathcal{T}}=\prod_{i=1}^{n}p_{t_{i}-t_{i-1}}(x_{t_{i}},x_{t_{i-1}})\prod_{i=1}^{n}d\mu(x_{t_{i}}),\quad\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}\\}.$

Moreover, we have provided a particular realization, $\underrightarrow{\lim}\
L^{p}(M^{\mathcal{T}},\mu_{\mathbf{x}_{0}}^{\mathcal{T}})$, of the colimit
that allows to stablish a finite dimensional approximation result for
integrable functionals.

However, observe that this approximation scheme does not follows the
philosophy of Andersson and Driver approach materialized in equation (3). For
this reason, in this section, we adapt our scheme to cover the measure
considered by Andersson and Driven in [1]. We start recalling some facts and
definitions of [1]. Through this section, the notation
$\mathcal{C}_{0}(\mathbb{R}^{m})$ stands for the space of continuous paths
$\gamma:[0,1]\to\mathbb{R}^{m}$ such that $\gamma(0)=0$.

###  6.1. Piecewise Linear Path Space

Consider $H(\mathbb{R}^{m})\subset\mathcal{C}_{0}(\mathbb{R}^{m})$ the
subspace consisting on finite energy paths

$H(\mathbb{R}^{m}):=\\{\gamma\in\mathcal{C}_{0}(\mathbb{R}^{m}):\gamma\text{
is absolutely continuous and }E_{\mathbb{R}^{m}}(\gamma)<\infty\\},$

where the energy functional is defined by

$E_{\mathbb{R}^{m}}(\gamma):=\int_{0}^{1}\langle\gamma^{\prime}(s),\gamma^{\prime}(s)\rangle\
ds.$

For each partition $\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}=1\\}$, the space
of piecewise linear paths on $\mathbb{R}^{m}$ with respect to $\mathcal{T}$,
subsequently denoted by $H_{\mathcal{T}}(\mathbb{R}^{m})$, is defined through

$H_{\mathcal{T}}(\mathbb{R}^{m}):=\\{\gamma\in\mathcal{C}_{0}(\mathbb{R}^{m}):\gamma\text{
is linear for }t\notin\mathcal{T}\\}.$

Clearly $H_{\mathcal{T}}(\mathbb{R}^{m})$ is linearly isomorphic to
$\mathbb{R}^{m\times\mathcal{T}}$, where

$\mathbb{R}^{m\times\mathcal{T}}:=\bigtimes_{t\in\mathcal{T}\backslash\\{0\\}}\mathbb{R}^{m},$

via the linear isomorphism

$\Pi_{\mathcal{T}}:H_{\mathcal{T}}(\mathbb{R}^{m})\longrightarrow\mathbb{R}^{m\times\mathcal{T}},\quad\Pi_{\mathcal{T}}(\gamma)=(\gamma(t_{1}),\gamma(t_{2}),...,\gamma(t_{n})).$

Since $H_{\mathcal{T}}(\mathbb{R}^{m})$ is linear, it follows that
$T_{\gamma}H_{\mathcal{T}}(\mathbb{R}^{m})\simeq
H_{\mathcal{T}}(\mathbb{R}^{m})$ for each $\gamma\in
H_{\mathcal{T}}(\mathbb{R}^{m})$. We introduce the Riemmanian metric on
$H_{\mathcal{T}}(\mathbb{R}^{m})$,
$h_{\mathcal{T}}\in\Gamma(T^{*}H_{\mathcal{T}}(\mathbb{R}^{m})\otimes
T^{*}H_{\mathcal{T}}(\mathbb{R}^{m}))$, defined by

$h_{\mathcal{T}}(u,v):=\int_{0}^{1}\langle u^{\prime}(s),v^{\prime}(s)\rangle\
ds,\quad u,v\in T_{\gamma}H_{\mathcal{T}}(\mathbb{R}^{m}),\ \gamma\in
H_{\mathcal{T}}(\mathbb{R}^{m}),$

and its corresponding volume form
$\operatorname{Vol}_{h_{\mathcal{T}}}\in\Gamma(\wedge^{m\times
n}TH_{\mathcal{T}}(\mathbb{R}^{m}))$ determinated by

$\operatorname{Vol}_{h_{\mathcal{T}}}(u_{1},u_{2},...,u_{m\times
n}):=\sqrt{\det(h_{\mathcal{T}}(u_{i},u_{j})_{ij})},$

where $\\{u_{1},u_{2},...,u_{m\times n}\\}\subset
T_{\gamma}H_{\mathcal{T}}(\mathbb{R}^{m})$ is an oriented basis and $\gamma\in
H_{\mathcal{T}}(\mathbb{R}^{m})$. Finally, we introduce a Borel measure
$\mu_{\mathcal{T}}$ on $H_{\mathcal{T}}(\mathbb{R}^{m})$. As usual, when we
deal with a measure associated with a Riemannian structure, we choose the
Borel $\sigma$-algebra associated with the topology induced by the
corresponding Riemannian metric. Along this section, this will be done several
times without specifying it again.

######  Definition 6.1.

For each partition $\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}=1\\}$ of
$[0,1]$, we denote by $\mu_{\mathcal{T}}$ the Borel measure on
$H_{\mathcal{T}}(\mathbb{R}^{m})$ defined by the density

$d\mu_{\mathcal{T}}=\frac{1}{(\sqrt{2\pi})^{mn}}\exp\left\\{-\frac{1}{2}E_{\mathbb{R}^{m}}\right\\}\operatorname{Vol}_{h_{\mathcal{T}}}.$

###  6.2. Piecewise Geodesic Path Space

Now, we define the curved analogue of the measure space
$(H_{\mathcal{T}}(\mathbb{R}^{m}),\mu_{\mathcal{T}})$. Let $(M,g)$ be a closed
Riemannian manifold of dimension $m$. Consider
$H(M)\subset\mathcal{C}_{\mathbf{x}_{0}}(M)$ to be the Hilbert manifold of
finite energy paths, defined by

$H(M):=\\{\gamma\in\mathcal{C}_{\mathbf{x}_{0}}(M):\gamma\text{ is absolutely
continuous and }E(\gamma)<\infty\\}$

where the energy functional $E$ is given through

$E(\gamma):=\int_{0}^{1}g(\gamma^{\prime}(s),\gamma^{\prime}(s))\ ds.$

Recall that $\gamma\in\mathcal{C}_{\mathbf{x}_{0}}(M)$ is said to be
absolutely continuous if $f\circ\gamma$ is absolutely continuous for all
$f\in\mathcal{C}^{\infty}(M)$. The tangent space $T_{\gamma}H(M)$ to $H(M)$ at
$\gamma$ can be identified with the space of absolutely continuous vector
fields $X:[0,1]\to TM$ along $\gamma$ such that $X(0)=0$ and
$G^{1}(X,X)<\infty$ where

$G^{1}(X,X):=\int_{0}^{1}g\left(\frac{\nabla X(t)}{dt},\frac{\nabla
X(t)}{dt}\right)\ dt.$

As usual, we denote

$\frac{\nabla
X(t)}{dt}:=//_{t}(\gamma)\frac{d}{dt}\\{//_{t}(\gamma)^{-1}X(t)\\},$

where $//_{t}(\gamma):T_{\mathbf{x}_{0}}M\to T_{\gamma(\mathbf{x}_{0})}M$
denotes the parallel translation along $\gamma$ relative to the Levy-Civita
covariant derivative $\nabla$.

Let $\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}=1\\}$ be a partition of
$[0,1]$. We define the subspace of $H(M)$,

$H_{\mathcal{T}}(M):=\left\\{\gamma\in
H(M)\cap\mathcal{C}^{2}([0,1]\backslash\mathcal{T},M):\nabla\gamma^{\prime}(t)/dt=0\text{
for }t\notin\mathcal{T}\right\\},$

consisting on piecewise geodesic paths in $H(M)$ which change directions only
at the partition points. The space $H_{\mathcal{T}}(M)$ is a finite
dimensional submanifold of $H(M)$ of dimension $n\times m$. For $\gamma\in
H_{\mathcal{T}}(M)$, the tangent space $T_{\gamma}H_{\mathcal{T}}(M)$ can be
identified with elements $X\in T_{\gamma}H(M)$ satisfying the Jacobi equations
on $[0,1]\backslash\mathcal{T}$. In other words, $X\in T_{\gamma}H(M)$ is in
$T_{\gamma}H_{\mathcal{T}}(M)$ if and only if

$\frac{\nabla^{2}}{dt^{2}}X(t)=R(\gamma^{\prime}(t),X(t))\gamma^{\prime}(t),$

where $R$ is the curvature tensor of $\nabla$. We can give to
$H_{\mathcal{T}}(M)$ a Riemannian structure introducing the
$\mathcal{T}$-metric. The $\mathcal{T}$-metric
$g_{\mathcal{T}}\in\Gamma(T^{*}H_{\mathcal{T}}(M)\otimes
T^{*}H_{\mathcal{T}}(M))$ is defined by

$g_{\mathcal{T}}(X,Y):=\sum_{i=1}^{n}g\left(\frac{\nabla
X(t_{i-1}+)}{dt},\frac{\nabla Y(t_{i-1}+)}{dt}\right)\Delta_{i}t,\quad X,Y\in
T_{\gamma}H_{\mathcal{T}}(M),$

for each $\gamma\in H_{\mathcal{T}}(M)$, where the notation $\nabla
X(t_{i-1}+)/dt$ is a shorthand of $\lim_{t\downarrow t_{i-1}}\nabla X(t)/dt$.
We denote by $\operatorname{Vol}_{g_{\mathcal{T}}}\in\Gamma(\wedge^{m\times
m}TH_{\mathcal{T}}(M))$ the volume form associated to $g_{\mathcal{T}}$. It is
determinated by

$\operatorname{Vol}_{g_{\mathcal{T}}}(X_{1},X_{2},...,X_{m\times
n}):=\sqrt{\det(g_{\mathcal{T}}(X_{i},X_{j})_{ij})},$

where $\\{X_{1},X_{2},...,X_{m\times n}\\}\subset
T_{\gamma}H_{\mathcal{T}}(M)$ is an oriented basis and $\gamma\in
H_{\mathcal{T}}(M)$.

######  Definition 6.2.

For each partition $\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}=1\\}$ of
$[0,1]$, we denote by $\nu_{\mathcal{T}}$ the Borel measure on
$H_{\mathcal{T}}(M)$ defined by the density

$d\nu_{\mathcal{T}}:=\frac{1}{(\sqrt{2\pi})^{mn}}\exp\left\\{-\frac{1}{2}E\right\\}\operatorname{Vol}_{g_{\mathcal{T}}}.$

This measure spaces
$(H_{\mathcal{T}}(M),\mathcal{B}_{\mathcal{T}},\nu_{\mathcal{T}})$, where
$\mathcal{B}_{\mathcal{T}}$ stands for the Borel $\sigma$-algebra of
$H_{\mathcal{T}}(M)$, will be the finite dimensional candidates for the
approximation scheme.

###  6.3. Cartan’s Development map

In general, it is not quite easy to deal with the manifold
$H_{\mathcal{T}}(M)$. Due to this fact, we will identify this space through
the well known space $H_{\mathcal{T}}(\mathbb{R}^{m})$. The cited
identification is done via the Cartan’s development map. The Cartan’s
development map $\Phi:H(\mathbb{R}^{m})\to H(M)$ is defined, for $\alpha\in
H(\mathbb{R}^{m})$, by $\Phi(\alpha):=\gamma$, where $\gamma\in H(M)$ is the
unique solution of the ordinary differential equation

(9)
$\gamma^{\prime}(t)=//_{t}(\gamma)\alpha^{\prime}(t),\quad\gamma(0)=\mathbf{x}_{0}.$

The anti-development map $\Phi^{-1}:H(M)\to H(\mathbb{R}^{m})$ is defined by
$\Phi^{-1}(\gamma):=\alpha$ where $\alpha\in H(\mathbb{R}^{m})$ is given by

$\alpha(t):=\int_{0}^{t}//_{r}^{-1}(\gamma)\gamma^{\prime}(r)\ dr.$

The map $\Phi:H(\mathbb{R}^{m})\to H(M)$ is bijective and smooth, hence it
defines a diffeomorphism of infinite dimensional Hilbert manifolds, but it is
not in general an isometry of Riemannian manifolds. The development map
$\Phi:H(\mathbb{R}^{m})\to H(M)$ has the property

$\Phi(H_{\mathcal{T}}(\mathbb{R}^{m}))=H_{\mathcal{T}}(M).$

We shall denote $\Phi|_{H_{\mathcal{T}}(\mathbb{R}^{m})}$ by
$\Phi_{\mathcal{T}}$.

A fundamental property of the Cartan’s development map is that it preserves
the $\mathcal{T}$-measure, in the sense that

(10) $\mu_{\mathcal{T}}(B)=\nu_{\mathcal{T}}(\Phi_{\mathcal{T}}(B)),$

for each Borel subset $B$ of $H_{\mathcal{T}}(\mathbb{R}^{m})$.

Moreover, it can be seen that the development map relates the measure
$\nu_{\mathcal{T}}$ of $H_{\mathcal{T}}(M)$ with the well known heat kernel
measure in the flat space $\mathbb{R}^{m\times\mathcal{T}}$. For a partition
$\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}=1\\}$, the heat kernel measure
$\lambda_{0}^{\mathcal{T}}$ is the Borel measure on
$\mathbb{R}^{m\times\mathcal{T}}$ defined by

$d\lambda_{0}^{\mathcal{T}}:=\prod_{i=1}^{n}p_{t_{i}-t_{i-1}}(x_{t_{i}},x_{t_{i-1}})\prod_{i=1}^{n}dx_{t_{i}},$

where $x_{0}=0$ and $p_{t}(x,y)$ is the heat kernel of $\mathbb{R}^{m}$. In
[1, Lem. 4.11], it is proved the identity

(11)
$\mu_{\mathcal{T}}(\Pi_{\mathcal{T}}^{-1}(B))=\lambda_{0}^{\mathcal{T}}(B)$

for each Borel subset $B$ of $\mathbb{R}^{m\times\mathcal{T}}$. Hence joining
equations (10) and (11) yields

$\nu_{\mathcal{T}}((\Phi_{\mathcal{T}}\circ\Pi_{\mathcal{T}}^{-1})(B))=\lambda_{0}^{\mathcal{T}}(B).$

As a consequence, we can reduce the structure of the space
$L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)$ to the more familiar
one $L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda^{\mathcal{T}}_{0})$. That
is, for $1\leq p<\infty$, the following spaces are identified

$L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\simeq
L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda^{\mathcal{T}}_{0}),$

via the isometric isomorphism

(12)
$\Lambda_{\mathcal{T}}:L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\longrightarrow
L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda^{\mathcal{T}}_{0}),\quad\Lambda_{\mathcal{T}}(f):=f\circ\Phi_{\mathcal{T}}\circ\Pi_{\mathcal{T}}^{-1}.$

It can be also introduced an almost everywhere defined stochastic extension of
the development map, called the stochastic development map
$\tilde{\Phi}:\mathcal{C}_{0}(\mathbb{R}^{m})\to\mathcal{C}_{\mathbf{x}_{0}}(M)$
and its corresponding anti-development map
$\tilde{\Phi}^{-1}:\mathcal{C}_{\mathbf{x}_{0}}(M)\to\mathcal{C}_{0}(\mathbb{R}^{m})$.
It can be proved that they preserve the Wiener measure in the sense that

$\lambda_{0}(B)=\mu_{\mathbf{x}_{0}}(\tilde{\Phi}(B)),$

for each Borel subset $B$ of $\mathcal{C}_{0}(\mathbb{R}^{m})$ where
$\lambda_{0}$ is the Wiener measure on the classical Wiener space
$\mathcal{C}_{0}(\mathbb{R}^{m})$ (see for instance [23]). Hence, thanks to
the stochastic development map, we can reduce the structure of the space
$L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$ to the more
familiar one $L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0})$ under the
philosophy of (12). For $1\leq p<\infty$, the following spaces are identified

$L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})\simeq
L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0}),$

via the map

(13)
$\Lambda:L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})\longrightarrow
L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0}),\quad\Lambda(f):=f\circ\tilde{\Phi}.$

###  6.4. Finite Dimensional Approximation Scheme

To prove an analogue of Theorem 3.2 for the geometric measure
$\nu_{\mathcal{T}}$, we will follow the following philosophy. Firstly we shall
prove the approximation scheme for the classical Wiener measure space
$(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0})$ and then we will make use the
identifications provided by equations (12) and (13), to translate the result
to the geometric framework.

Let $\mathcal{P}$ be the directed set consisting on partitions of $[0,1]$

$\mathcal{T}=\\{0=t_{0}<t_{1}<\cdots<t_{n}=1\\},$

partially ordered by inclusion and consider the projectors

$\pi_{\mathcal{T}\mathcal{T}^{\prime}}:\mathbb{R}^{m\times\mathcal{T}^{\prime}}\longrightarrow\mathbb{R}^{m\times\mathcal{T}},\quad\pi_{\mathcal{T}\mathcal{T}^{\prime}}(x_{t})_{t\in\mathcal{T}^{\prime}\backslash\\{0\\}}:=(x_{t})_{t\in\mathcal{T}\backslash\\{0\\}}$
$\pi_{\mathcal{T}}:\mathbb{R}^{m\times[0,1]}\longrightarrow\mathbb{R}^{m\times\mathcal{T}},\quad\pi_{\mathcal{T}}(x_{t})_{t\in[0,1]}:=(x_{t})_{t\in\mathcal{T}\backslash\\{0\\}}.$

In analogy with the preceding sections, we define the diagram
$(L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}}),\eta_{\mathcal{T}\mathcal{T}^{\prime}})$
where the morphisms $\eta_{\mathcal{T}\mathcal{T}^{\prime}}$,
$\mathcal{T}\subset\mathcal{T}^{\prime}$, are defined through

$\eta_{\mathcal{T}\mathcal{T}^{\prime}}:L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}})\longrightarrow
L^{p}(\mathbb{R}^{m\times\mathcal{T}^{\prime}},\lambda_{0}^{\mathcal{T}^{\prime}}),\quad\eta_{\mathcal{T}\mathcal{T}^{\prime}}(f):=f\circ\pi_{\mathcal{T}\mathcal{T}^{\prime}}.$

Using the same techniques of section 4, it is easily proved that for each
$1\leq p<\infty$, the cocone
$(L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0}),\varphi_{\mathcal{T}})$
defines a realization of the colimit of
$(L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}}),\eta_{\mathcal{T}\mathcal{T}^{\prime}})$,
where the morphisms $\varphi_{\mathcal{T}}$, $\mathcal{T}\in\mathcal{P}$, are
given by

$\varphi_{\mathcal{T}}:L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}})\longrightarrow
L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0}),\quad\varphi_{\mathcal{T}}(f):=f\circ\pi_{\mathcal{T}}|_{\mathcal{C}_{\mathbf{x}_{0}}(\mathbb{R}^{m})}.$

This establishes the approximation scheme for the classical Wiener space
$(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0})$. Now, we proceed to define the
diagram and cocone for the geometric measure $\nu_{\mathcal{T}}$.

Let us consider the diagram
$(L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right),\varrho_{\mathcal{T}\mathcal{T}^{\prime}})$
where for each $\mathcal{T}\subset\mathcal{T}^{\prime}$,
$\varrho_{\mathcal{T}\mathcal{T}^{\prime}}$ are the morphisms defined by

$\varrho_{\mathcal{T}\mathcal{T}^{\prime}}:L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\longrightarrow
L^{p}\left(H_{\mathcal{T}^{\prime}}(M),\nu_{\mathcal{T}^{\prime}}\right),\quad\varrho_{\mathcal{T}\mathcal{T}^{\prime}}:=\Lambda_{\mathcal{T}^{\prime}}^{-1}\circ\eta_{\mathcal{T}\mathcal{T}^{\prime}}\circ\Lambda_{\mathcal{T}},$

in other words, the unique morphisms making the diagram of Figure 3
commutative.

$\textstyle{L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varrho_{\mathcal{T}\mathcal{T}^{\prime}}}$$\scriptstyle{\Lambda_{\mathcal{T}}}$$\textstyle{L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda^{\mathcal{T}}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{\mathcal{T}\mathcal{T}^{\prime}}}$$\textstyle{L^{p}\left(H_{\mathcal{T}^{\prime}}(M),\nu_{\mathcal{T}^{\prime}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Lambda_{\mathcal{T}^{\prime}}}$$\textstyle{L^{p}(\mathbb{R}^{m\times\mathcal{T}^{\prime}},\lambda^{\mathcal{T}^{\prime}}_{0})}$
Figure 3. Definition of the morphisms
$\varrho_{\mathcal{T}\mathcal{T}^{\prime}}$

The main result of this subsection is to prove that the cocone
$(L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\theta_{\mathcal{T}})$
is a realization of the colimit of the diagram
$(L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right),\varrho_{\mathcal{T}\mathcal{T}^{\prime}})$,
where the morphisms $\theta_{\mathcal{T}}$, $\mathcal{T}\in\mathcal{P}$, are
defined by

$\theta_{\mathcal{T}}:L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\longrightarrow
L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\quad\theta_{\mathcal{T}}:=\Lambda^{-1}\circ\varphi_{\mathcal{T}}\circ\Lambda_{\mathcal{T}}.$

In other words, the morphisms $\theta_{\mathcal{T}}$ are the unique morphisms
making commutative the diagram of Figure 4. This identification establishes an
analogue of Theorem 3.2 for the geometrical framework. The definition of the
involved morphisms
$\varrho_{\mathcal{T}\mathcal{T}^{\prime}},\theta_{\mathcal{T}}$ are given
through the commutativity of the diagrams of Figure 3 and 4, respectively, in
order to make things natural, in the categorical meaning of the word. This
will become clear in the proof of the main Theorem of this subsection, Theorem
6.3.

$\textstyle{L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Lambda_{\mathcal{T}}}$$\scriptstyle{\theta_{\mathcal{T}}}$$\textstyle{L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Lambda}$$\textstyle{L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\mathcal{T}}}$$\textstyle{L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0})}$
Figure 4. Definition of the morphisms $\theta_{\mathcal{T}}$

######  Theorem 6.3.

The cocone
$(L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\theta_{\mathcal{T}})$
defines a realization of the colimit of
$(L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right),\varrho_{\mathcal{T}\mathcal{T}^{\prime}})$.

###### Proof.

By (12), we have a family of isometric isomorphisms indexed by $\mathcal{T}$,

$\Lambda_{\mathcal{T}}:L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\to
L^{p}(\mathbb{R}^{m\times\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}).$

By the commutativity of the diagram of Figure 3, this family of isomorphisms
$(\Lambda_{\mathcal{T}})_{\mathcal{T}}$ establishes a natural isomorphism
between the diagrams
$(L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right),\varrho_{\mathcal{T}\mathcal{T}^{\prime}})$
and
$(L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}}),\eta_{\mathcal{T}\mathcal{T}^{\prime}})$.
Hence the colimits of this two diagrams are isomorphic via a naturally defined
isomorphism, see for instance the reference [22, Cor. 3.6.3] for a proof of
this fact. As the colimit of
$(L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}}),\eta_{\mathcal{T}\mathcal{T}^{\prime}})$
is
$(L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0}),\varphi_{\mathcal{T}})$,
to prove the Theorem it is enough to prove that the cocone
$(L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}),\theta_{\mathcal{T}})$
is isomorphic to
$(L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0}),\varphi_{\mathcal{T}})$
via a naturally defined isomorphism. The commutativity of the diagram of
Figure 4, establishes that $\Lambda$ is our required isomorphism. This
concludes the proof. ∎

###  6.5. Derivation of the Limit Formula

In this final subsection, we prove an analogue of Theorem 4.3 for the
geometric measure $\nu_{\mathcal{T}}$. Let us denote by
$\underrightarrow{\lim}\
L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}})$ and
$\underrightarrow{\lim}\
L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)$, the analogues of the
space defined by (7), under the natural changes to adapt it to
$(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}})$ and
$\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)$, respectively. We omit the
explicit description of this spaces for notational convenience. Rephrasing the
arguments of section 4, it can be proved that the operator

(14) $\mathfrak{I}_{p}:\underrightarrow{\lim}\
L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}})\longrightarrow
L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0}),\quad(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\mapsto\lim_{\mathcal{T}}\varphi_{\mathcal{T}}(f_{\mathcal{T}})$

is an isometric isomorphism. Thanks to this isomorphism and the
identifications provided by (12) and (13), we prove the final result of this
article, Theorem 6.4. As a direct consequence of Theorem 6.3 and the following
Theorem 6.4, we obtain that the cocone

$(\underrightarrow{\lim}\
L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right),\psi_{\mathcal{R}}),$

where the morphisms $\psi_{\mathcal{R}}$ are the corresponding analogues of
(8) for $(H_{\mathcal{T}}(M),\nu_{\mathcal{T}})$, defines a realization of the
colimit of
$(L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right),\varrho_{\mathcal{T}\mathcal{T}^{\prime}})$.

######  Theorem 6.4.

Let $1\leq p<\infty$, then the following spaces are isometrically isomorphic

(15) $\underrightarrow{\lim}\
L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\simeq
L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}}).$

In consequence, for every $F\in
L^{1}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})$, there exists an
element
$(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}\in\bigtimes_{\mathcal{T}\in\mathcal{P}}L^{1}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)$
such that

(16) $\int_{\mathcal{C}_{\mathbf{x}_{0}}(M)}F\
d\mu_{\mathbf{x}_{0}}=\lim_{\mathcal{T}}\int_{H_{\mathcal{T}}(M)}f_{\mathcal{T}}\
d\nu^{\mathcal{T}}.$

###### Proof.

By (12), we have a family of isometric isomorphisms indexed by $\mathcal{T}$,

$\Lambda_{\mathcal{T}}:L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\to
L^{p}(\mathbb{R}^{m\times\mathcal{T}},\mu^{\mathcal{T}}_{\mathbf{x}_{0}}).$

It is easily seen using the definition of the morphisms
$\varrho_{\mathcal{T}\mathcal{T}^{\prime}}$, Figure 3, that the operator

$\Sigma_{p}:\underrightarrow{\lim}\
L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\longrightarrow\underrightarrow{\lim}\
L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}}),\quad\Sigma_{p}(f_{\mathcal{T}})_{\mathcal{T}\in\mathcal{P}}:=(\Lambda_{\mathcal{T}}(f_{\mathcal{T}}))_{\mathcal{T}\in\mathcal{P}}$

defines an isometric isomorphism. Finally, the composition operator

$\textstyle{\underrightarrow{\lim}\
L^{p}\left(H_{\mathcal{T}}(M),\nu_{\mathcal{T}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma_{p}}$$\textstyle{L^{p}(\mathcal{C}_{\mathbf{x}_{0}}(M),\mu_{\mathbf{x}_{0}})}$$\textstyle{\underrightarrow{\lim}\
L^{p}(\mathbb{R}^{m\times\mathcal{T}},\lambda_{0}^{\mathcal{T}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{I}_{p}}$$\textstyle{L^{p}(\mathcal{C}_{0}(\mathbb{R}^{m}),\lambda_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Lambda^{-1}}$

where the morphisms $\mathfrak{I}_{p}$ and $\Lambda$ are given by (14) and
(13), respectively, defines an isometric isomorphism. This concludes the proof
of (15). Formula (16) follows from the isometric property of the induced
operator. ∎

##  7\. Acknowledgements

The author is very grateful to the animous referee for his/her extremely
useful advises that without any doubt has improved the presentation and
correctness of the article and has improved deeply the results of it.
Moreover, the author is also very fortunate to have been advised by Professors
Christian Bär and Matthias Ludewig, two experts in this field.

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