Abstract
We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasiregular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasiregular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.
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1 Introduction
How much geometric information can be recovered from associated analytic or probabilistic objects is the underlying issue for a range of questions most prominently embodied by M. Kac’s Can One Hear the Shape of a Drum? [9]. Here we take up this issue and investigate how much of the geometric structure of a space is encoded by an associated Markov semigroup or, equivalently, by a Markov process. To be more precise, let us briefly introduce the setting. For \(i=1,2\), let \((X^i,\mathcal {X}^i,\mathfrak {T}^i,\mu ^i)\) be a locally compact Polish \(\sigma \)finite Radonmeasure space and \({(T^i_t)}_{t\ge 0}\) be a subMarkovian semigroup on \(L^2(\mu ^i)\). They are intertwined by \(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) if
In [13] Lenz, Schmidt, and the second named author investigated the properties of intertwined subMarkovian semigroups in the case when the intertwining operator U is additionally an order isomorphism, that is, an invertible linear operator satisfying \(Uf\ge 0\) if and only if \(f\ge 0\). Suppose that \({(T^i_t)}_{t\ge 0}\) is the semigroup of a quasiregular symmetric Dirichlet form \(\big ({E^i,\mathscr {D}(E^i)}\big )\) on \(L^2(\mu ^i)\), and thus it is associated with a symmetric Markov process \(\textbf{M}^i\) with state space \(X^i\). It is then the main result of [13] that \(\textbf{M}^i\) (quasi)determines the topology \(\mathfrak {T}^i\), in the following sense.
Theorem 1.1
If the irreducible semigroups \({(T^i_t)}_{t\ge 0}\) are intertwined by an order isomorphism, then the corresponding Dirichlet spaces \((X^i,\mathfrak {T}^i,\mu ^i, E^i)\) are quasihomeomorphic.
Here, by a quasihomeomorphism^{Footnote 1} \(j:(X^1,\mathfrak {T}^1,\mu ^1, E^1)\rightarrow (X^2,\mathfrak {T}^2,\mu ^2, E^2)\) we mean a map such that for every \(\varepsilon >0\) there exist closed sets \(F^i\subset X^i\) with \(\text {Cap}(X^i\setminus F^i)<\varepsilon \) and j restricts to a homeomorphism \(j\big _{F^1}:F^1\rightarrow F^2\) (cf. [7, Sect. A.4, p. 429]).
Importantly, this completely characterizes how a Markov process \(\textbf{M}\) identifies the topological properties of its state space \((X,\mathfrak {T},\mu )\), and constitutes a sharp affirmative result in the spirit of M. Kac’ isospectrality question. For example,—as previously established by Arendt, Biegert, and ter Elst in [1]—a Brownian motion on a complete Riemannian manifold (M, g) characterizes both the differential and the Riemannian structure of (M, g).
The arguments in [13] rely however on the additional technical assumption that the semigroups \({(T^i_t)}_{t\ge 0}\) (or the Markov processes \(\textbf{M}^i\)) be irreducible. In terms of the semigroups, this is to say that
Our main contribution in this work is that to extend the results in [13] by removing the irreducibility assumption, thus proving Theorem 1.1 in its full generality. To this end, we employ the machinery of ergodic decompositions of Dirichlet spaces developed by the first named author in [3]. An ergodic decomposition of a Dirichlet space \(\big ({E^i,\mathscr {D}(E^i)}\big )\) is a collection of irreducible Dirichlet spaces \(\big ({E_\zeta ,\mathscr {D}(E_\zeta )}\big )\) indexed by a measure space \((Z,\mathcal {Z},\nu )\) decomposing \(\big ({E,\mathscr {D}(E)}\big )\) as the direct integral
Extending the uniqueness results in [3], we show that ergodic decompositions are unique up to unique automorphisms of their indexing spaces Z (Theorem 4.4). Profiting this fact, we prove as our main result (Theorem 4.5) that two Dirichlet spaces \(\big ({E^i,\mathscr {D}(E^i)}\big )\) are intertwined by order isomorphisms U if and only if their ergodic decompositions \(\zeta \mapsto \big ({E^i_\zeta ,\mathscr {D}(E^i_\zeta )}\big )\) are intertwined by an order isomorphism \(U_\zeta \), in which case U is decomposable over the ergodic decompositions and represented as a direct integral of the operators \(U_\zeta \).
2 Direct integrals
In this section, we review some of the basic theory of direct integrals of Hilbert spaces and quadratic forms for later use. More detailed accounts of the material discussed here can be found in [3, 5]. We refer to [6] for all measuretheoretical statements.
2.1 Measure spaces
To formulate the ergodic decomposition of Dirichlet forms in full generality, some rather technical aspects of measure theory are required. For the reader’s convenience, let us recall the relevant definitions, starting with some classes of measurable spaces that we will use.
Definition 2.1
(Measurable spaces) A measurable space \((X,\mathcal {X})\) is

separable if \(\mathcal {X}\) contains all singletons in X;

countably separated if there exists a countable family of sets in \(\mathcal {X}\) separating points in X;

countably generated if there exists a countable family of sets in \(\mathcal {X}\) generating \(\mathcal {X}\) as a \(\sigma \)algebra;

a standard Borel space if there exists a Polish topology \(\mathfrak {T}\) on X so that \(\mathcal {X}\) coincides with the Borel \(\sigma \)algebra induced by \(\mathfrak {T}\).
A \(\sigma \)finite measure space \((X,\mathcal {X},\mu )\) is standard if there exists a \(\mu \)conegligible set \(X_0\in \mathcal {X}\) such that \(X_0\) is a standard Borel space when regarded as a measurable subspace of \((X,\mathcal {X})\). We denote by \((X,\mathcal {X}^\mu ,\hat{\mu })\) the (Carathéodory) completion of \((X,\mathcal {X},\mu )\). A Hausdorff topological measure space \((X,\mathfrak {T},\mathcal {X},\mu )\) is a Radonmeasure space if \(\mathcal {X}^\mu \) coincides with the \(\mu \)completion of the Borel \(\sigma \)algebra and \(\hat{\mu }\) is a Radon measure, i.e., it is locally finite and inner regular with respect to the compact sets.
A \([\infty ,\infty ]\)valued function is called \(\mu \)measurable if it is measurable w.r.t. \(\mathcal {X}^\mu \). For measures \(\mu _1\), \(\mu _2\) we write \(\mu _1\sim \mu _2\) to indicate that \(\mu _1\) and \(\mu _2\) are equivalent, i.e., mutually absolutely continuous.
Let us now recall the notion of a perfect measure space together with some of its properties. A measure space \((X,\mathcal {X},\mu )\) is perfect (e.g., [6, 342K]) if whenever \(f:X\rightarrow {\mathbb R}\) is measurable and \(A\in \mathcal {X}\) with \(\mu A>0\), then there exists a compact set \(K\subset f(A)\) with \(f_\sharp \mu (K)>0\). Here, and everywhere in the following, \(f_\sharp \mu \) is the pushforward measure of \(\mu \) via f.
The following properties of perfect measure spaces are easily verified (cf. [6, 342X]). For a proof of (v), see [6, 416W(a)].
Lemma 2.2
Let \((X,\mathcal {X},\mu )\) be any measure space. Then,

(i)
\((X,\mathcal {X},\mu )\) is perfect if and only if so is its completion;

(ii)
if \((X,\mathcal {X},\mu )\) is perfect and \(\mathcal {X}_0\) is a \(\sigma \)subalgebra of \(\mathcal {X}\), then the restricted measure space \((X,\mathcal {X}_0,\mu \!\!\downharpoonright _{\mathcal {X}_0})\) is perfect;

(iii)
if \((X,\mathcal {X},\mu )\) is perfect and \(\lambda \) has a density w.r.t. \(\mu \), then \((X,\mathcal {X},\lambda )\) is perfect;

(iv)
if \((Z,\mathcal {Z})\) is a measurable space, \(f:X\rightarrow Z\) is \(\mathcal {X}/\mathcal {Z}\)measurable and \((X,\mathcal {X},\mu )\) is perfect, then \((Z,\mathcal {Z},f_\sharp \mu )\) is perfect;

(v)
every Radonmeasure space is perfect.
Definition 2.3
Let \((X,\mathcal {X},\mu )\) be a \(\sigma \)finite measure space, and \(\mathcal {X}^*\subset \mathcal {X}\) be a countably generated \(\sigma \)subalgebra. We say that:

\(\mathcal {X}\) is \(\mu \)essentially countably generated by \(\mathcal {X}^*\) if for each \(A\in \mathcal {X}\) there is \(A^*\in \mathcal {X}^*\) with \(\mu (A\triangle A^*)=0\);

\(\mathcal {X}\) is \(\mu \)essentially countably generated if it is so by some \(\mathcal {X}^*\) as above.
Definition 2.4
(Almost isomorphisms) For \(i\in \{1,2\}\) let \((X^i,\mathcal {X}^i)\) be a measurable space, and \(\mathcal {N}^i\) be a \(\sigma \)ideal of \(\mathcal {X}^i\cup \mathcal {N}^i{{\,\mathrm{:=}\,}}\left\{ A\cup N : A\in \mathcal {X}^i, N\in \mathcal {N}^i\right\} \). We say that the triples \((X^i,\mathcal {X}^i,\mathcal {N}^i)\) are almost isomorphic if there exist sets \(N^i\in \mathcal {N}^i\) so that the spaces \(X^i\setminus N^i\) are strictly isomorphic when endowed with the restriction to \(X^i\setminus N^i\) of the \(\sigma \)algebra \(\mathcal {X}^i\cup \mathcal {N}^i\) and of the \(\sigma \)ideal \(\mathcal {N}^i\).
For \(i\in \{1,2\}\) let \((X^i,\mathcal {X}^i,\mu ^i)\) be measure spaces. We say that they are almost isomorphic if there exist \(\mu ^i\)negligible sets \(N^i\subset X^i\) so that the spaces \(X^i\setminus N^i\) are strictly isomorphic when endowed with the restriction to \(X^i\setminus N^i\) of the completed \(\sigma \)algebra \((\mathcal {X}^i)^{\mu ^i}\), of the completed measure \(\hat{\mu }^i\), and of the \(\sigma \)ideal of \(\mu ^i\)negligible sets \(\mathcal {N}^{\mu ^i}\).
2.2 Quadratic forms
Every Hilbert space is assumed to be separable and a real vector space.
Let \((H,\left\Vert {\,\cdot \,}\right\Vert )\) be a Hilbert space. For a bounded operator \(B:H\rightarrow H\) we write \(\left\Vert B\right\Vert _{\textrm{op}, H}\) for the operator norm of B; the subscript H is omitted whenever apparent from context. A quadratic form Q on H is a map \(Q:H\rightarrow [0,\infty ]\) such that
We always assume that the domain \(\mathscr {D}(Q)=\{u\in H:Q(u)<\infty \}\) is dense in H and write \((Q,\mathscr {D}(Q))\) for Q when we want to make the domain explicit.
For every quadratic form Q there exists a unique symmetric bilinear form whose diagonal coincides with Q. This bilinear form will also be denoted by Q, and we will use the term quadratic form for both objects interchangeably.
Additionally, for every \(\alpha >0\) we set
For \(\alpha >0\), we let \(\mathscr {D}(Q)_\alpha \) be the completion of \(\mathscr {D}(Q)\), endowed with the Hilbert norm \(Q_\alpha ^{1/2}\).
To every closed quadratic form \((Q,\mathscr {D}(Q))\) one can associate a unique nonnegative selfadjoint operator \(L\) such that \(\mathscr {D}(\sqrt{L})=\mathscr {D}(Q)\) and for all \(u,v\in \mathscr {D}(L)\). We denote the associated strongly continuous contraction semigroup by \(T_t{{\,\mathrm{:=}\,}}e^{tL}\), \(t>0\).
2.3 Direct integrals of Hilbert spaces
We recall the main definitions concerning direct integrals of separable Hilbert spaces, referring to [5, Sect. II.1, II.2] for a systematic treatment.
Definition 2.5
(Measurable fields, [5, Sect. II.1.3, Dfn. 1, p. 164]) Let \((Z,\mathcal {Z},\nu )\) be a \(\sigma \)finite measure space, \(\left( H_\zeta \right) _{\zeta \in Z}\) be a family of separable Hilbert spaces, and F be the linear space \(F{{\,\mathrm{:=}\,}}\prod _{\zeta \in Z} H_\zeta \). We say that \(\zeta \mapsto H_\zeta \) is a \(\nu \)measurable field of Hilbert spaces (with underlying space S) if there exists a linear subspace S of F with
 (a):

for every \(u\in S\), the function \(\zeta \mapsto \left\Vert u_\zeta \right\Vert _{\zeta }\) is \(\nu \)measurable;
 (b):

if \(v\in F\) is such that is \(\nu \)measurable for every \(u\in S\), then \(v\in S\);
 (c):

there exists a sequence \(\left( u_n\right) _n\subset S\) such that \(\left( u_{n,\zeta }\right) _n\) is a total sequence^{Footnote 2} in \(H_\zeta \) for every \(\zeta \in Z\).
Any such S is called a space of \(\nu \)measurable vector fields. Any sequence in S possessing property (c) is called a fundamental sequence.
Proposition 2.6
([5, Sect. II.1.4, Prop. 4, p. 167]) Let \(\mathcal {S}\) be a family of functions satisfying Definition 2.5 (a) and (c). Then, there exists exactly one space of \(\nu \)measurable vector fields S so that \(\mathcal {S}\subset S\).
Definition 2.7
(Direct integrals, [5, Sect. II.1.5, Prop. 5, p. 169]) A \(\nu \)measurable vector field u is called (\(\nu \))squareintegrable if
Two squareintegrable vector fields u, v are called (\(\nu \))equivalent if \(\left\Vert uv\right\Vert =0\). The space H of equivalence classes of squareintegrable vector fields, endowed with the nonrelabeled quotient norm \(\left\Vert {\,\cdot \,}\right\Vert \), is a Hilbert space [5, Sect. II.1.5, Prop. 5(i), p. 169], called the direct integral of \(\zeta \mapsto H_\zeta \) (with underlying space S) and denoted by
The superscript ‘S’ is omitted whenever S is clear from context.
In the following, it will occasionally be necessary to distinguish an element u of H from one of its representatives modulo \(\nu \)equivalence, say \(\hat{u}\) in S. In this case, we shall write \(u=\left[ \hat{u}\right] _{H}\).
Lemma 2.8
([3, Lem. 2.6]) If \((Z,\mathcal {Z})\) is \(\sigma \)finite countably generated, then H in (2.2) is separable.
We now turn to measurable fields of bounded operators.
Definition 2.9
(Measurable fields of bounded operators, decomposable operators) Let H be as in (2.2). A field of bounded operators \(\zeta \mapsto B_\zeta \in \mathcal {B}(H_\zeta )\) is called \(\nu \)measurable (with underlying space S) if \(\zeta \mapsto B_\zeta u_\zeta \in H_\zeta \) is a \(\nu \)measurable vector field for every \(\nu \)measurable vector field u. Set \(\left\Vert B_\zeta \right\Vert _{\textrm{op}, \zeta }{{\,\mathrm{:=}\,}}\left\Vert B_\zeta \right\Vert _{\textrm{op}, H_\zeta }\). A \(\nu \)measurable vector field of bounded operators is called \(\nu \)essentially bounded if \(\nu \). A bounded operator \(B\in \mathcal {B}(H)\) is called decomposable if Bu is represented by a \(\nu \)essentially bounded \(\nu \)measurable field of bounded operators \(\zeta \mapsto B_\zeta \), in which case we write
2.4 Direct integrals of quadratic forms
We briefly recall all the relevant notions concerning direct integrals of quadratic forms according to [3].
Definition 2.10
(Direct integral of quadratic forms) Let \((Z,\mathcal {Z},\nu )\) be a \(\sigma \)finite countably generated measure space. For \(\zeta \in Z\) let \((Q_\zeta ,D_\zeta )\) be a closable (densely defined) quadratic form on a Hilbert space \(H_\zeta \). We say that \(\zeta \mapsto (Q_\zeta ,D_\zeta )\) is a \(\nu \)measurable field of quadratic forms on Z if
 (a):

\(\zeta \mapsto H_\zeta \) is a \(\nu \)measurable field of Hilbert spaces on Z with underlying space \(S_H\);
 (b):

\(\zeta \mapsto \mathscr {D}(Q_\zeta )_1\) is a \(\nu \)measurable field of Hilbert spaces on Z with underlying space \(S_Q\);
 (c):

\(S_Q\) is a linear subspace of \(S_H\) under the canonical identification of \(\mathscr {D}(Q_\zeta )\) with a subspace of \(H_\zeta \).
We further denote by
the direct integral of \(\zeta \mapsto (Q_\zeta ,\mathscr {D}(Q_\zeta ))\), i.e., the quadratic form defined on H as in (2.2) given by
Remark 2.11
(Separability) It is implicit in our definition of \(\nu \)measurable field of Hilbert spaces that \(H_\zeta \) is separable for every \(\zeta \in Z\). As a consequence, \(\mathscr {D}(Q_\zeta )_1\) is \((Q_\zeta )^{1/2}_1\)separable.
Proposition 2.12
([3, Prop. 2.13]) Let \((Q,\mathscr {D}(Q))\) be a direct integral of quadratic forms. Then,
 (i):

\(\big ({Q,\mathscr {D}(Q)}\big )\) is a densely defined closed quadratic form on H;
 (ii):

\(\zeta \mapsto T_{\zeta ,t}\) is \(\nu \)measurable fields of bounded operators for every \(t>0\);
 (iii):

the semigroup associated with Q is given by
(2.4)
Under the assumptions of Proposition 2.12, assertion (i) of the same Proposition implies that the space of \(\nu \)measurable vector fields \(S_H\) is uniquely determined by \(S_Q\) as a consequence of Proposition 2.6. Thus, everywhere in the following when referring to a direct integral of quadratic forms we shall—with abuse of notation—write S in place of both \(S_H\) and \(S_Q\).
2.5 Directintegral representation of \(L^2\)spaces
In order to introduce directintegral representations of Dirichlet forms, we need to construct direct integrals of concrete Hilbert spaces in such a way to additionally preserve the Riesz structure of Lebesgue spaces implicitly used to phrase the subMarkovianity property (2.11). To this end, we shall need the concept of a disintegration of measures.
Disintegrations Let \((X,\mathcal {X},\mu )\) and \((Z,\mathcal {Z},\nu )\) be (nontrivial) measure spaces. A map \(s:(X,\mathcal {X})\rightarrow (Z,\mathcal {Z})\) is inversemeasurepreserving if \(s_\sharp \mu {{\,\mathrm{:=}\,}}\mu \circ s^{1}=\nu \).
Definition 2.13
(Disintegrations [6, 452E]) A pseudodisintegration of \(\mu \) over \(\nu \) is any family of nonzero measures \(\left( \mu _\zeta \right) _{\zeta \in Z}\) on \((X,\mathcal {X})\) so that \(\zeta \mapsto \mu _\zeta A\) is \(\nu \)measurable and
A pseudodisintegration is

separated if there exists a family of pairwise disjoint sets \(\left\{ A_\zeta \right\} _{\zeta \in Z}\subset \mathcal {X}^\mu \) so that \(A_\zeta \) is \(\mu _\zeta \)conegligible for \(\nu \)a.e. \(\zeta \in Z\), henceforth called a separating family for \(\left( \mu _\zeta \right) _{\zeta \in Z}\);

sseparated if it is separated and there exists a \(\mathcal {X}/\mathcal {Z}\)measurable map \(s:X\rightarrow Z\) so that \(\left\{ s^{1}(\zeta )\right\} _{\zeta \in Z}\) is a separating family;

consistent with s if
$$\begin{aligned} \mu \big ({A\cap s^{1}(B)}\big )=\int _B\mu _\zeta A \mathop {}\!\textrm{d}\nu (\zeta ),\,\qquad A\in \mathcal {X},\,B\in \mathcal {Z}\end{aligned}$$(2.5) 
strongly consistent with s if it is consistent with s and \(s^{1}(\zeta )\) is \(\mu _\zeta \)conegligible for \(\nu \)a.e. \(\zeta \in Z\).
A disintegration of \(\mu \) over \(\nu \) is a pseudodisintegration additionally so that \(\mu _\zeta \) is a subprobability measure for every \(\zeta \in Z\). A disintegration is

\(\nu \)essentially unique if the measures \(\mu _\zeta \) are uniquely determined for \(\nu \)a.e. \(\zeta \in Z\).
If \(\left( \mu _\zeta \right) _{\zeta \in Z}\) is a pseudodisintegration of \(\mu \) over \(\nu \), then
whenever the lefthand side makes sense, [6, 452F]. We note that a disintegration \(\left( \mu _\zeta \right) _{\zeta \in Z}\) of \(\mu \) over \(\nu \) strongly consistent with a map s is automatically separated, with separating family \(\{s^{1}(\zeta )\}_{\zeta \in Z}\).
Direct integrals and disintegrations Let \((X,\mathcal {X},\mu )\) be \(\sigma \)finite standard, \((Z,\mathcal {Z},\nu )\) be \(\sigma \)finite countably generated, and \(\left( \mu _\zeta \right) _{\zeta \in Z}\) be a pseudodisintegration of \(\mu \) over \(\nu \). Denote by

\(\mathcal {L}^0(\mu )\) the space of \(\mu \)measurable realvalued functions (not: \(\mu \)classes) on X;

\(\mathcal {L}^\infty (\mu )\) the space of uniformly bounded (not: \(\mu \)essentially uniformly bounded) functions in \(\mathcal {L}^0(\mu )\);

\(\mathcal {L}^p(\mu )\) the space of pintegrable functions in \(\mathcal {L}^0(\mu )\).
For a family \(\mathcal {A}\subset \mathcal {L}^0(\mu )\), let \(\left[ \mathcal {A}\right] _{\mu }\) denote the family of the corresponding \(\mu \)classes.
Let now \(F{{\,\mathrm{:=}\,}}\prod _{\zeta \in Z} L^2(\mu _\zeta )\). The diagonal embedding of \(\mathcal {L}^2(\mu )\) into F, regarded up to \(\mu _\zeta \)classes, is the map \(\delta :f\mapsto {(\zeta \mapsto \delta (f)_\zeta )}\), where
Since \(f\in \mathcal {L}^2(\mu )\), we have \(\delta (f)_\zeta =\left[ f\right] _{\mu _\zeta }\) for \(\nu \)a.e. \(\zeta \in Z\) by (2.6), and \(\delta \) therefore is well defined as linear morphism mapping \(\mu \)classes to Hclasses (see Proposition 2.15). Now, assume that
Since \(\left[ \mathcal {A}\right] _{\mu }\) is dense in \(L^2(\mu )\) and the latter is separable, then there exists a countable family \(\mathcal {U}\subset \mathcal {A}\) so that \(\left[ \mathcal {U}\right] _{\mu _\zeta }\) is total in \(L^2(\mu _\zeta )\) for \(\nu \)a.e. \(\zeta \in Z\). Thus for every \(\mathcal {A}\) as in (2.8) there exists a unique space of \(\nu \)measurable vector fields \(S=S_\mathcal {A}\) containing \(\delta (\mathcal {A})\), generated by \(\delta (\mathcal {A})\) in the sense of Proposition 2.6. We denote by H the corresponding direct integral of Hilbert spaces
Remark 2.14
(cf. [8, Sect. 7.2, p. 84]) The direct integral H constructed in (2.9) is a Banach lattice (e.g., [6, 354A(b)]) for the order
Proposition 2.15
([3, Prop. 2.25]) Let \((X,\mathcal {X},\mu )\) be \(\sigma \)finite standard, \((Z,\mathcal {Z},\nu )\) be \(\sigma \)finite countably generated, and \(\left( \mu _\zeta \right) _{\zeta \in Z}\) be a pseudodisintegration of \(\mu \) over \(\nu \). Then, the morphism
 (i):

is well defined, linear, and an isometry of Hilbert spaces, additionally unitary if \(\left( \mu _\zeta \right) _{\zeta \in Z}\) is separated;
 (ii):

is a Riesz homomorphism (e.g., [6, 351H]). In particular,

for each \(f\in \mathcal {L}^2(\mu )\), it holds that \((\iota \left[ f\right] _{\mu })_\zeta \ge {\textbf{0}}_{L^2(\mu _\zeta )}\) for \(\nu \)a.e. \(\zeta \in Z\) if and only if \(f\ge 0\) \(\mu \)a.e.;

for each \(f,g\in \mathcal {L}^2(\mu )\), it holds that \((\iota \left[ f \wedge g\right] _{\mu })_\zeta = (\iota \left[ f\right] _{\mu })_\zeta \wedge (\iota \left[ g\right] _{\mu })_\zeta \) for \(\nu \)a.e. \(\zeta \in Z\).
2.6 Dirichlet forms
We recall a standard setting for the theory of Dirichlet forms, following [15].
Assumption 2.16
The quadruple \((X,\mathfrak {T},\mathcal {X},\mu )\) is so that \((X,\mathfrak {T})\) is a metrizable Luzin space with Borel \(\sigma \)algebra \(\mathcal {X}\) and \(\hat{\mu }\) is a Radon measure on \((X,\mathfrak {T},\mathcal {X}^\mu )\) with full support.
By [6, 415D(iii), 424G] any space \((X,\mathcal {X},\mu )\) satisfying Assumption 2.16 is \(\sigma \)finite standard. The support of a (\(\mu \))measurable function \(f:X\rightarrow {\mathbb R}\) (possibly defined only on a \(\mu \)conegligible set) is defined as the support of the measure \(\leftf\right\cdot \mu \). Every such f has a support, independent of the \(\mu \)representative of f, cf. [15, p. 148].
A closed positive semidefinite quadratic form \((Q,\mathscr {D}(Q))\) on \(L^2(\mu )\) is a (symmetric) Dirichlet form if
We shall denote Dirichlet forms by \((E,\mathscr {D}(E))\). A Dirichlet form \((E,\mathscr {D}(E))\) is regular if \((X,\mathfrak {T})\) is (additionally) locally compact and \(\mathscr {D}(E)\cap \mathcal {C}_0(X)\) is both \(E^{1/2}_1\)dense in \(\mathscr {D}(E)\) and uniformly dense in \(\mathcal {C}_0(X)\).
Finally, we are interested in the notion of invariant sets of a Dirichlet form \((E,\mathscr {D}(E))\) on \(L^2(\mu )\). We say that \(A\subset X\) is Einvariant if it is \(\mu \)measurable and any of the following equivalent conditions holds.

\(T_t({{\,\mathrm{\textbf{1}}\,}}_A f)={{\,\mathrm{\textbf{1}}\,}}_A T_t f\) \(\mu \)a.e. for any \(f\in L^2(\mu )\) and \(t>0\);

\(T_t({{\,\mathrm{\textbf{1}}\,}}_A f)=0\) \(\mu \)a.e. on \(A^\complement \) for any \(f\in L^2(\mu )\) and \(t>0\);

\({{\,\mathrm{\textbf{1}}\,}}_A f\in \mathscr {D}(E)\) for any \(f\in \mathscr {D}(E)\) and
(2.12)
The form \((E,\mathscr {D}(E))\) is irreducible if every invariant set is either negligible or conegligible.
2.7 Direct integrals of Dirichlet forms
Let \((X,\mathcal {X},\mu )\) be \(\sigma \)finite standard, \((Z,\mathcal {Z},\nu )\) be \(\sigma \)finite countably generated, and \(\left( \mu _\zeta \right) _{\zeta \in Z}\) be a pseudodisintegration of \(\mu \) over \(\nu \). Further let \(\zeta \mapsto (Q_\zeta , \mathscr {D}(Q_\zeta ))\) be a \(\nu \)measurable field of quadratic forms, each densely defined in \(L^2(\mu _\zeta )\) with separable domain, and denote by \((Q,\mathscr {D}(Q))\) their direct integral in the sense of Definition 2.10.
Definition 2.17
([3, Dfn. 2.26]) We say that \((Q,\mathscr {D}(Q))\) is compatible with the pseudodisintegration \(\left( \mu _\zeta \right) _{\zeta \in Z}\) if the space \(S_Q\) underlying \(\zeta \mapsto \mathscr {D}(Q_\zeta )_1\) is of the form \(S_\mathcal {A}\) for some \(\mathcal {A}\) as in (2.8) and additionally satisfying \(\mathcal {A}\subset \mathscr {D}(Q)\).
Proposition 2.18
([3, Prop. 2.29]) Let \((X,\mathcal {X},\mu )\) be \(\sigma \)finite standard, \((Z,\mathcal {Z},\nu )\) be \(\sigma \)finite countably generated, and \(\left( \mu _\zeta \right) _{\zeta \in Z}\) be a separated pseudodisintegration of \(\mu \) over \(\nu \). Further let \((E,\mathscr {D}(E))\) be a direct integral of quadratic forms \(\zeta \mapsto (E_\zeta , \mathscr {D}(E_\zeta ))\) compatible with \(\left( \mu _\zeta \right) _{\zeta \in Z}\). Then, \((E,\mathscr {D}(E))\) is a Dirichlet form on \(L^2(\mu )\) if and only if \((E_\zeta , \mathscr {D}(E_\zeta ))\) is so on \(L^2(\mu _\zeta )\) for \(\nu \)a.e. \(\zeta \in Z\).
Proposition 2.18 motivates the following definition.
Definition 2.19
A quadratic form \((E,\mathscr {D}(E))\) on \(L^2(\mu )\) is a direct integral of Dirichlet forms \(\zeta \mapsto (E_\zeta ,\mathscr {D}(E_\zeta ))\) if it is a direct integral of quadratic forms \(\zeta \mapsto (E_\zeta ,\mathscr {D}(E_\zeta ))\), for each \(\zeta \) the form \((E_\zeta ,\mathscr {D}(E_\zeta ))\) is a Dirichlet form on \(L^2(\mu _\zeta )\), and the direct integral is additionally compatible with the separated pseudodisintegration \(\left( \mu _\zeta \right) _\zeta \) in the sense of Definition 2.17.
We refer to [3] for further comments on direct integrals of Dirichlet forms and related notions.
3 Order isomorphisms and intertwining operators
In this section, we first recall the notion of order isomorphisms between \(L^2\)spaces together with their main representation theorem, which shows that they are weighted composition operators. In the second part, we study intertwining operators and give a first result connecting intertwining operators and direct integral decompositions (Proposition 3.7).
Definition 3.1
(Orderpreserving operators, order isomorphisms) For \(i=1,2\) let \((X^i,\mathcal {X}^i,\mu ^i)\) be \(\sigma \)finite countably generated and countably separated. A linear operator \(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) is orderpreserving (also: positivitypreserving) if \(f\ge 0\) implies \(Uf\ge 0\) for each \(f\in L^2(\mu ^1)\). An order isomorphism is an invertible orderpreserving linear operator with orderpreserving inverse.
The structure of order isomorphisms between \(L^2\)spaces is characterized by the following Banach–Lampertitype theorem.
Proposition 3.2
(Order isomorphism as weighted composition operator) If \(U:L^2(X^1,\mu ^1)\rightarrow L^2(X^2,\mu ^2)\) is an order isomorphism, then there exist a measurable map \(h:X^2 \rightarrow (0,\infty )\) and an \(\mathcal {X}^2/\mathcal {X}^1\)measurable almost isomorphism \(\tau :X^2\rightarrow X^1\) such that
The maps h and \(\tau \) are unique up to equality almost everywhere.
Proof
See [16, Thm. 5.1] for existence in the case of finite measures \(\mu ^i\). For the general case argue as follows. Since \((X^i,\mathcal {X}^i,\mu ^i)\) is \(\sigma \)finite standard, there exists a function \(f_i\) satisfying \(f_i>0\) \(\mu ^i\)a.e., and \(\left\Vert f_i\right\Vert _{L^2(\mu ^i)}=1\). Thus, the multiplication operator \(M_i{{\,\mathrm{:=}\,}}M_{f_i^{1/2}}:L^2(\mu ^i)\rightarrow L^2(f_i\mu ^i)\) is an order isomorphism with (order) inverse \(M_i^{1}=M_{f_i^{1/2}}\), and therefore the operator \(U'{{\,\mathrm{:=}\,}}M_2\circ U\circ M_1^{1}:L^2(f_1\mu ^1)\rightarrow L^2(f_2\mu ^2)\) is an order isomorphism as well. Applying the assertion for \(U'\) yields maps \(h'\) and \(\tau '\). Finally, letting \(h{{\,\mathrm{:=}\,}}f_2^{1/2}\circ \tau \cdot f_1^{1/2} \cdot h'\) and \(\tau {{\,\mathrm{:=}\,}}\tau '\) yields the assertion in the \(\sigma \)finite case.
For uniqueness, one can similarly reduce to the case of finite measure. Then \(h=U1\) is determined uniquely up to equality a.e. Moreover, if \(\tau _1,\tau _2:X^2\rightarrow X^1\) are \(\mathcal X^2\)/\(\mathcal X^1\)measurable maps such that \(h\cdot (f\circ \tau _1)=h\cdot (f\circ \tau _2)\) for all \(f\in L^2(X^1,\mu ^1)\), then \(\mu ^1(\tau _1^{1}(B)\cap \tau _2^{1}(B))=0\) for all \(B\in \mathcal X^2\). From [6, 343F] it follows that \(\tau _1=\tau _2\) \(\mu ^2\)a.e. \(\square \)
The maps h and \(\tau \) associated with an order isomorphism U according to the previous proposition are the main players in the present article. We call h the scaling and \(\tau \) the transformation associated with U.
Lemma 3.3
(Adjoint of an order isomorphism, [13, Lem. 1.2]) Let \(U:L^2(X^1,\mu ^1)\rightarrow L^2(X^2,\mu ^2)\) be an order isomorphism with associated scaling h and transformation \(\tau \). Then \(\tau _\sharp \mu ^2\) and \(\mu ^1\) are mutually absolutely continuous. Moreover, the adjoint of U is given by
Definition 3.4
(Intertwining operators) For \(i=1,2\) let \(V^{\scriptscriptstyle {(i)}}\) be a Banach space and \((A^{\scriptscriptstyle {(i)}},\mathscr {D}(A^{\scriptscriptstyle {(i)}}))\) be a (possibly unbounded) densely defined operator on \(V^{\scriptscriptstyle {(i)}}\). An invertible bounded linear operator \(U:V^{\scriptscriptstyle {(1)}}\rightarrow V^{\scriptscriptstyle {(2)}}\) is said to intertwine \(A^{\scriptscriptstyle {(1)}}\) and \(A^{\scriptscriptstyle {(2)}}\) if
Two families of operators \(\left( A^{\scriptscriptstyle {(i)}}_t\right) _{t\in T}\), \(i=1,2\), are intertwined by U if U intertwines \(A^{\scriptscriptstyle {(1)}}_t\), \(A^{\scriptscriptstyle {(2)}}_t\) for all \(t\in T\).
The following is not difficult to show.
Lemma 3.5
([11, Prop. A.1]) Two strongly continuous contraction semigroups on Banach spaces are intertwined by U if and only if so are their generators.
The next result is a variation of [13, Cor. 2.5], replacing the irreducibility assumption on the semigroups with the unitariness of the intertwining operator U. A proof is analogous, and therefore it is omitted.
Proposition 3.6
For \(i=1,2\) let \((X^i,\mathcal {X}^i,\mu ^i)\) be \(\sigma \)finite standard, \(\left( T^{\scriptscriptstyle {(i)}}_t\right) _{t\ge 0}\) be a subMarkovian semigroup on \(L^2(\mu ^i)\) with corresponding Dirichlet form \((E^i,\mathscr {D}(E^i))\), and \(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) be a unitary order isomorphism intertwining \(\left( T^{\scriptscriptstyle {(1)}}_t\right) _{t\ge 0}\) and \(\left( T^{\scriptscriptstyle {(2)}}_t\right) _{t\ge 0}\). Then,
Proposition 3.7
For \(i=1,2\) let \((X^i,\mathcal {X}^i,\mu ^i)\) be \(\sigma \)finite standard, \((Z,\mathcal {Z},\nu )\) be \(\sigma \)finite countably generated, and \({\big (\mu ^i_\zeta \big )}_{\zeta \in Z}\) be a separated pseudodisintegration of \(\mu ^i\) over \(\nu \). Further let \(\zeta \mapsto U_\zeta \) be a \(\nu \)measurable field of bounded operators \(U_\zeta :L^2(\mu ^1_\zeta )\rightarrow L^2(\mu ^2_\zeta )\) in the sense of Definition 2.9, and set
Then, U is a bounded operator \(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) and
 (i):

\(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) is unitary if and only if \(U_\zeta \) is so for \(\nu \)a.e. \(\zeta \in Z\), in which case
 (ii):

\(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) is orderpreserving if and only if \(U_\zeta \) is so for \(\nu \)a.e. \(\zeta \in Z\).
For \(i=1,2\) further let \(\zeta \mapsto Q_\zeta ^i\) be a \(\nu \)measurable field of closed quadratic forms \(\big ({Q_\zeta ^i,\mathscr {D}(Q^i_\zeta )}\big )\) on \(L^2(\mu ^i_\zeta )\) in the sense of Definition 2.10, each with semigroup \({\big (T^{\scriptscriptstyle {(i)}}_{\zeta ,t}\big )}_{t\ge 0}\), \(\zeta \in Z\), and set
Then, for each fixed \(t>0\), the operator \(T^{\scriptscriptstyle {(i)}}_t\) satisfies \(T^{\scriptscriptstyle {(i)}}_t\in \mathcal {B}\big ({L^2(\mu ^i)}\big )\) and
 (iii):

\(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) intertwines \(T^{\scriptscriptstyle {(1)}}_t\) and \(T^{\scriptscriptstyle {(2)}}_t\) if and only if \(U_\zeta \) intertwines \(T^{\scriptscriptstyle {(1)}}_{\zeta ,t}\) and \(T^{\scriptscriptstyle {(2)}}_{\zeta ,t}\) for \(\nu \)a.e. \(\zeta \in Z\).
Proof
By definition of U, we have
Let \(i=1,2\). Since the disintegration \({(\mu ^i_\zeta )}_{\zeta \in Z}\) is separated, by Proposition 2.15 (i) there exists a unitary order isomorphism \(\iota ^{\scriptscriptstyle {(i)}}\) satisfying (2.10). Since \(\iota ^{\scriptscriptstyle {(i)}}\) is as well an order isomorphism by Proposition 2.15 (ii), in the rest of the proof we may identify U with \(\iota ^{\scriptscriptstyle {(2)}}\circ U\circ (\iota ^{\scriptscriptstyle {(1)}})^{1}:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\).
(i) is [5, Sect. II.2.3, Example, p. 182].
(ii) Both implications are a consequence of the definition of the order structure on the direct integral of \(\zeta \mapsto L^2(\mu _\zeta )\), together with Proposition 2.15 (i).
(iii) Let \(t>0\) be fixed. Assume first that \(U_\zeta \) intertwines \(T^{\scriptscriptstyle {(1)}}_{\zeta , t}\), \(T^{\scriptscriptstyle {(2)}}_{\zeta ,t}\). By [5, §II.2.3, Prop. 3, p. 182],
Vice versa assume that U intertwines \(T^{\scriptscriptstyle {(1)}}_t\) and \(T^{\scriptscriptstyle {(2)}}_t\). By Proposition 2.12 (iii) and [5, §II.2.3, Prop. 3, p. 182],
hence, by a further application of [5, §II.2.3, Prop. 3, p. 182], and by [5, §II.2.3, Prop. 2, p. 181],
which concludes the proof. \(\square \)
4 Intertwining of ergodic decompositions
In this section, we describe the structure of order isomorphisms intertwining Dirichlet forms that are not necessarily irreducible. Informally, one might expect that one can simply decompose the Dirichlet forms into their “connected components” and apply the known result for irreducible Dirichlet forms. However, it is technically nontrivial to make this notion of “connected components” rigorous for abstract Dirichlet forms.
We rely here on the notion of ergodic decompositions introduced in [3]. We will first show that these ergodic decompositions are essentially unique (Theorem 4.4) and that intertwining order isomorphisms carry ergodic decompositions to ergodic decompositions (Theorem 4.5). As a consequence, whenever the Dirichlet forms in question admit ergodic decompositions, intertwining order isomorphisms act componentwise as anticipated by the informal discussion above (Corollary 4.6).
4.1 Ergodic decompositions
Let us first introduce a rigorous definition for the ergodic decomposition of a regular Dirichlet form.
Definition 4.1
Let \((X,\mathfrak {T},\mathcal {X},\mu )\) be a locally compact Polish Radonmeasure space, and \((E,\mathscr {D}(E))\) be a regular Dirichlet form on \(L^2(\mu )\). We say that \((E,\mathscr {D}(E))\) admits an ergodic decomposition \(\zeta \mapsto (E_\zeta ,\mathscr {D}(E_\zeta ))\) indexed by \((Z,\mathcal {Z},\nu )\) if there exist
 (i):

a separable countably generated probability space \((Z,\mathcal {Z},\nu )\) and a measurable map \(s:(X,\mathcal {X})\rightarrow (Z,\mathcal {Z})\);
 (ii):

a pseudodisintegration \(\left( \mu _\zeta \right) _{\zeta \in Z}\) of \(\mu \) over \(\nu \), strongly consistent with s such that if \(s^{1}(\zeta )\) is endowed with the subspace topology and the trace \(\sigma \)algebra inherited by \((X,\mathfrak {T},\mathcal {X}^\mu )\), then \((s^{1}(\zeta ),\hat{\mu }_\zeta )\) is a Radonmeasure space for \(\nu \)a.e. \(\zeta \in Z\);
 (iii):

a \(\nu \)measurable field \(\zeta \mapsto (E_\zeta ,\mathscr {D}(E_\zeta ))\) of regular irreducible Dirichlet forms \((E_\zeta ,\mathscr {D}(E_\zeta ))\) on \(L^2(\mu _\zeta )\);
such that
as a direct integral of Dirichlet forms in the sense of Definition 2.19.
Remark 4.2
Since the pseudodisintegration \(\left( \mu _\zeta \right) _{\zeta \in Z}\) in Definition 4.1(ii) is sseparated, the map s is implicitly always assumed to be surjective.
Remark 4.3
The existence of ergodic decompositions has been shown:
 (a):

in [3] under the additional assumptions that either \(\mu \) be a finite measure, or that the form \((E,\mathscr {D}(E))\) be strongly local and admitting a carré du champ operator;
 (b):

by K. Kuwae in [12] under the additional assumption that the transition probabilities of the Markov process properly associated to the Dirichlet form be absolutely continuous w.r.t. the reference measure \(\mu \).
It is one further result of [12] that, under the same assumption as in (b) above, the set Z indexing the ergodic decomposition is in fact at most countable, which greatly simplifies the discussion. This type of ergodic decomposition however rules out some interesting examples, in particular from infinitedimensional analysis (see [3, §3.4] and references therein).
Let us also point out that, again in the case of (b) above, our main result (i.e., the extension of the results in [13] to the nonirreducible case) has also been obtained by L. Li and H. Lin in [14, Thm. 4.6].
Ergodic decompositions of Dirichlet forms are essentially projectively unique, in the sense of the next theorem, expanding the scope of all the projective uniqueness results in [3].
Theorem 4.4
Let \((X,\mathcal {X},\mu )\) be a locally compact Polish Radonmeasure space, and \((E,\mathscr {D}(E))\) be a regular Dirichlet form on \(L^2(\mu )\). Further assume that \((E,\mathscr {D}(E))\) admits

an ergodic decomposition \(\zeta \mapsto (E^1_\zeta ,\mathscr {D}(E^1_\zeta ))\) indexed by \((Z^1,\mathcal {Z}^1,\nu ^1)\);

an ergodic decomposition \(\eta \mapsto (E^2_\eta ,\mathscr {D}(E^2_\eta ))\) indexed by \((Z^2,\mathcal {Z}^2,\nu ^2)\).
Then, there exists an almost isomorphism \(\varrho :(Z^1,\mathcal {Z}^1,\mathcal {N}^{\nu ^1})\rightarrow (Z^2,\mathcal {Z}^2,\mathcal {N}^{\nu ^2})\) and a \(\mathcal {Z}^2\)measurable function \(h:Z^2\rightarrow (0,\infty )\) so that \(\varrho _\sharp \nu ^1\sim \nu ^2\) and \(E^1_{\zeta } = h\big ({\varrho (\zeta )}\big ) E^2_{\varrho (\zeta )}\) for \(\nu ^1\)a.e. \(\zeta \in Z^1\).
Proof
Let \(\mathcal {X}_0\) be the family of \(\mu \)measurable Einvariant subsets of X, and note that \(\mathcal {X}_0\) is a \(\sigma \)subalgebra of \(\mathcal {X}^\mu \), e.g., [7, Lem. 1.6.1, p. 53]. Let \(\mu _0\) be the restriction of \(\hat{\mu }\) to \((X,\mathcal {X}_0)\). By our Assumption 2.16, \(\mathcal {X}\) is countably generated, thus \(\mathcal {X}_0\) is \(\mu _0\)essentially countably generated by \(\mathcal {X}^*{{\,\mathrm{:=}\,}}\mathcal {X}\cap \mathcal {X}_0\).
Further let \(\mathcal {C}\) be a special standard core for \((E,\mathscr {D}(E))\) witnessing the compatibility of the direct integral representation, i.e., so that \(\mathcal {C}=\mathcal {A}\) as in Definition 2.17. Note that for every \(u\in \mathcal {C}\) we may choose \(u_\zeta \equiv u\) as a \(\mu _\zeta \)representative of the diagonal embedding \(\delta (u)_\zeta \).
Step 1: Invariant sets For simplicity of notation, in this step let Z denote either \(Z^1\) or \(Z^2\), and analogously for all the other symbols.
Claim I: For every \(B\in \mathcal {Z}\) the set \(A{{\,\mathrm{:=}\,}}s^{1}(B)\) is Einvariant Since \(\left( \mu _\zeta \right) _{\zeta \in Z}\) is sseparated, \(\left( X_\zeta \right) _{\zeta \in Z}\), with \(X_\zeta {{\,\mathrm{:=}\,}}s^{1}(\zeta )\), is a separating family in the sense of Definition 2.13. The multiplication operator \(M_{X_\zeta }{{\,\mathrm{:=}\,}}M_{{{\,\mathrm{\textbf{1}}\,}}_{X_\zeta }}:L^2(\mu _\zeta )\rightarrow L^2(\mu _\zeta )\) satisfies \(M_{X_\zeta }=\textrm{id}_{L^2(\mu _\zeta )}\), hence \(T_{\zeta ,t} M_{X_\zeta }=M_{X_\zeta } T_{\zeta ,t}\). As a consequence,
that is, \(A{{\,\mathrm{:=}\,}}s^{1}(B)\) is Einvariant.
Claim II: for every Einvariant \(A\in \mathcal {X}_0\) there exists \(B\in \mathcal {Z}\) with \(\mu _0(A\triangle s^{1}(B))=0\). It suffices to show the statement for \(A\in \mathcal {X}^*\). Since \((Z,\mathcal {Z})\) is separable countably generated, it is countably separated, hence by [10, Prop. 12.1(iii)] there exists a separable metrizable topology \(\mathfrak {T}_Z\) on Z so that \(\mathcal {Z}\) is the Borel \(\sigma \)algebra generated by \(\mathfrak {T}_Z\). As a consequence, \(s:X\rightarrow Z\) is Borel, and therefore s(A) is analytic by [6, 423G(b)], thus \(\mathcal {Z}\)universally measurable by [6, 434D(c)], hence finally \(\nu \)measurable. In particular, there exist \(B_0\subset s(A)\subset B_1\) with \(B_0,B_1\in \mathcal {Z}\) and \(\nu (B_1\setminus B_0)=0\). By the previous claim, we have that \(A_0{{\,\mathrm{:=}\,}}s^{1}(B_0)\), \(A_1{{\,\mathrm{:=}\,}}s^{1}(B_1)\in \mathcal {X}_0\). The conclusion follows since \(A_0\subset A\subset A_1\) by definition, and since \(\mu _0(A_1\setminus A_0)=0\) combining (2.5) with \(\nu (B_1\setminus B_0)=0\).
Claim III: \(s^*\mathcal {Z}{{\,\mathrm{:=}\,}}\left\{ s^{1}(B): B\in \mathcal {Z}\right\} \) is \(\mu _0\)essentially countably generating \(\mathcal {X}_0\). It suffices to combine the previous two claims with the fact that \(\mathcal {Z}\) is countably generated by assumption.
Claim IV: The measure space \((Z,\mathcal {Z},\nu )\) is perfect. We shall use Lemma 2.2 in numerous instances, without explicit mention. As a Radon space \((X,\mathfrak {T},\mathcal {X}^\mu ,\mu )\) is perfect. Since it is \(\sigma \)finite, there exists a probability measure \(\lambda '\) equivalent to \(\hat{\mu }\) on \(\mathcal {X}^\mu \), and the space \((X,\mathcal {X}^{\lambda '},\lambda ')\) is perfect as well. Let \(\lambda {{\,\mathrm{:=}\,}}\lambda '\!\!\downharpoonright _{s^*\mathcal {Z}}\) and \(\sigma {{\,\mathrm{:=}\,}}s_\sharp \lambda \), and note that the restricted measure space \((X, s^*\mathcal {Z},\lambda )\) is perfect. Thus \((Z,\mathcal {Z},\sigma )\) is perfect, too.
Now, let \(B\in \mathcal {Z}\) with \(\nu B=0\). Then, \(s^{1}(B)\) is \(\mu \)negligible by (2.5). Since \(\lambda \sim \mu \) on \(s^*\mathcal {Z}\), the set \(s^{1}(B)\) is as well \(\lambda \)negligible, hence B is as well \(\sigma \)negligible, which shows that \(\sigma \ll \nu \). Vice versa let \(B\in \mathcal {Z}\) with \(\sigma B=0\). Then, \(s^{1}(B)\) is \(\lambda \)negligible, and therefore \(\mu \)negligible. Since \(\mu _\zeta \) is nonzero for every \(\zeta \in Z\) by definition of pseudodisintegration, we conclude again from (2.5) that B is as well \(\nu \)negligible, which shows \(\nu \ll \sigma \). Thus \(\nu \sim \sigma \).
Finally, since both \(\nu \) and \(\sigma \) are finite measures, \(\nu \) has a density w.r.t. \(\sigma \) by the Radon–Nikodym Theorem, and therefore \((Z,\mathcal {Z},\nu )\) is perfect since so is \((Z,\mathcal {Z},\sigma )\).
Step 2: Almost isomorphism Throughout this step, let \(i=1,2\). By construction, the measure algebra \((\mathfrak {Z}_i, \bar{\sigma }_i)\) of \((Z,\mathcal {Z}^i,\sigma _i)\) is isomorphic to the measure algebra of \((X, s_i^*\mathcal {Z}^i,\lambda ^i)\). Furthermore, by Step 1: Claim III the \(\sigma \)algebra \(\mathcal {Z}^i\) is \(\mu _0\)essentially countably generating the algebra \(\mathcal {X}_0\) of Einvariant sets. Thus, \(\mathcal {Z}^i\) is as well \(\lambda _0{{\,\mathrm{:=}\,}}\lambda '\!\!\downharpoonright _{\mathcal {X}^*}\)essentially countably generating \(\mathcal {X}_0\), and therefore
As a consequence, since the measure algebra of a measure space coincides with that of its completion, the measure algebras \((\mathfrak {Z}^1, \bar{\sigma }^1)\) and \((\mathfrak {Z}^2, \bar{\sigma }^2)\) are isomorphic. By all of the above, the spaces \((Z^i,(\mathcal {Z}^i)^{\sigma _i},\hat{\sigma }_i)\) are countably separated perfect complete probability spaces, with isomorphic measure algebras. Therefore, the corresponding measure spaces \((Z^i,\mathcal {Z}^i,\sigma ^i)\) are almost isomorphic by [6, 344C], via some almost isomorphism \(\varrho :Z^1\rightarrow Z^2\) satisfying \(\varrho _\sharp \sigma ^1=\sigma ^2\). Since \(\sigma ^i\sim \nu ^i\) by the proof of Step 1 Claim IV, we conclude that \(\varrho _\sharp \nu ^1\sim \nu ^2\).
Step 3: Forms identifications Since \(\varrho _\sharp \nu ^1\sim \nu ^2\), there exists a measurable \(h:(Z^2,\mathcal {Z}^2)\rightarrow (0,\infty )\) satisfying \(h \cdot \varrho _\sharp \nu ^1=\nu ^2\). Now
which shows that \({\big (h(\varrho (\zeta ))^{1} \cdot \mu ^2_{\varrho (\zeta )}\big )}_{\zeta \in Z^1}\) is a pseudodisintegration of \(\mu \) over \(\nu ^1\), strongly consistent with \(\varrho ^{1}\circ s^2\).
Furthermore, exchanging the roles of \(\nu ^1\) and \(\nu ^2\) in Step 2 we have that \(\varrho _\sharp \nu ^1\sim \nu ^2\) and \(\varrho ^{1}_\sharp \nu ^2\sim \nu ^1\). Thus, \(f:Z^2\rightarrow {\mathbb R}\) is \(\nu ^2\)measurable if and only if \(f\circ \varrho :Z^1\rightarrow {\mathbb R}\) is \(\nu ^1\)measurable. As a consequence, the function \(h\circ \varrho :Z^1\rightarrow (0,\infty )\) is \(\nu ^1\)measurable and, since \(\eta \mapsto (E^2_\eta ,\mathscr {D}(E^2_\eta ))\) is a \(\nu ^2\)measurable field of quadratic forms, for every \(u,v\in \mathcal {C}\) the map
is \(\nu ^1\)measurable, which shows that \(\zeta \mapsto \big ({h(\rho (\zeta )) E^2_{\varrho (\zeta )},\mathscr {D}(E^2_{\varrho (\zeta )})}\big )\) is a \(\nu ^1\)measurable field of quadratic forms on \(L^2(\mu ^2_{\varrho (\zeta )})\). The corresponding direct integral form \(\big ({\widetilde{E},\mathscr {D}(\widetilde{E})}\big )\) is compatible with the disintegration in (4.1) in the sense of Definition 2.17 with underlying space \(S_\mathcal {C}\), since \(\mathcal {C}\) is a core for
for \(\nu ^1\)a.e. \(\zeta \in Z^1\). Finally, let us show that the form \(\big ({\widetilde{E},\mathscr {D}(\widetilde{E})}\big )\) coincides with \((E,\mathscr {D}(E))\). It suffices to note that, for all \(u,v\in \mathcal {C}\) we have
\(\square \)
4.2 Intertwining
We are now ready to state our main result.
Theorem 4.5
For \(i=1,2\), let \((X^i,\mathfrak {T}^i,\mathcal {X}^i,\mu ^i)\) be a locally compact Polish Radonmeasure space, and \((E^i,\mathscr {D}(E^i))\) be a regular Dirichlet form on \(L^2(\mu ^i)\) with semigroup \(\left( T^{\scriptscriptstyle {(i)}}_t\right) _{t\ge 0}\). Suppose further that
 (a):

\((E^1,\mathscr {D}(E^1))\) admits ergodic decomposition \(\zeta \mapsto (E^1_\zeta ,\mathscr {D}(E^1_\zeta ))\) indexed by \((Z,\mathcal {Z},\nu )\);
 (b):

there exists a unitary order isomorphism \(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) intertwining \(\left( T^{\scriptscriptstyle {(1)}}_t\right) _{t\ge 0}\), \(\left( T^{\scriptscriptstyle {(2)}}_t\right) _{t\ge 0}\), represented by a scaling h and a transformation \(\tau \) in the sense of Proposition 3.2.
Finally, set
Then,
 (i):

\({\big (\mu ^2_\zeta \big )}_{\zeta \in Z}\) is a separated pseudodisintegration of \(\mu ^2\) over \((Z,\mathcal {Z},\nu )\);
 (ii):

\(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) is decomposable over \((Z,\mathcal {Z},\nu )\) and represented by a \(\nu \)measurable field of order isomorphisms \(U_\zeta :L^2(\mu ^1_\zeta )\rightarrow L^2(\mu ^2_\zeta )\).
For \(\zeta \in Z\) and \(t>0\) further define an operator \(T^{\scriptscriptstyle {(2)}}_{\zeta ,t}:L^2(\mu ^2_\zeta )\rightarrow L^2(\mu ^2_\zeta )\) by
Then,
 (iii):

\({\big (T^{\scriptscriptstyle {(2)}}_{\zeta ,t}\big )}_{t\ge 0}\) is a subMarkovian strongly continuous contraction semigroup on \(L^2(\mu ^2_\zeta )\) for \(\nu \)a.e. \(\zeta \in Z\);
 (iv):

\((E^2,\mathscr {D}(E^2))\) admits an ergodic decomposition \(\zeta \mapsto (E^2_\zeta , \mathscr {D}(E^2_\zeta ))\) indexed by \((Z,\mathcal {Z},\nu )\), where the Dirichlet form \((E^2_\zeta ,\mathscr {D}(E^2_\zeta ))\) is the image form of \((E^1_\zeta ,\mathscr {D}(E^1_\zeta ))\) via \(U_\zeta \) for \(\nu \)a.e. \(\zeta \in Z\) and the associated semigroup is \({\big (T^{\scriptscriptstyle {(2)}}_{\zeta ,t}\big )}_{t\ge 0}\).
Proof
Up to (re)defining \(\tau \), resp. \(\tau ^{1}\), on a \(\mu ^1\), resp. \(\mu ^2\), negligible set, we can assume without loss of generality that \(\tau \) is defined everywhere. Note however that \(\tau \) is neither surjective nor injective.
Step 1: Disintegrations By Lemma 3.3, the definition of \({\big (\mu ^2_\zeta \big )}_{\zeta \in Z}\) in (4.2) is wellposed. By definition of pseudodisintegration, \(\zeta \mapsto \mu ^1_\zeta f\) is \(\nu \)measurable for every \(\mathcal {X}^1\)measurable realvalued f on \(X^1\). As a consequence, the map
is as well \(\nu \)measurable, by \(\mathcal {X}^2/\mathcal {X}^1\)measurability of \(\tau \). Furthermore, by (2.5) for \({\big (\mu ^1_\zeta \big )}_{\zeta \in Z}\),
that is, the family \({\big (\mu ^2_\zeta \big )}_{\zeta }\) is a pseudodisintegration of \(\mu ^2\) over \((Z,\mathcal {Z},\nu )\). Since \(\tau ^{1}\) is \(\mu ^1\)a.e. injective, it is as well \(\mu ^1_\zeta \)a.e. injective for \(\nu \)a.e \(\zeta \in Z\). As a consequence, the pseudodisintegration \({\big (\mu ^2_\zeta \big )}_{\zeta }\) is as well separated, which concludes a proof of (i).
Step 2: Directintegral representations of \(\mathrm {L^2}\)spaces. Since \({\big (\mu ^1_\zeta \big )}_{\zeta \in Z}\) is separated by assumption and since \({\big (\mu ^2_\zeta \big )}_{\zeta \in Z}\) is separated by (i), Proposition 2.15 yields the directintegral representations
for every space of \(\nu \)measurable vector fields \(S^i=S_{\mathcal {A}^i}\) induced by \(\mathcal {A}^i\) as in (2.8). In the following, we shall always choose \(\mathcal {A}^1{{\,\mathrm{:=}\,}}\mathcal {C}\) to be a special standard core for the (regular) form \((E^1,\mathscr {D}(E^1))\), and set \(\mathcal {A}^2{{\,\mathrm{:=}\,}}\left\{ h\cdot f\circ \tau : f \in \mathcal {A}^1\right\} \). In light of Proposition 3.6 we have \(\mathcal {A}^2\subset \mathscr {D}(E^2)\). Having fixed the spaces \(S^i\) throughout the proof, we omit them from the notation.
As a consequence, if \(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) is decomposable, and represented by a \(\nu \)measurable field of operators \(\zeta \mapsto U_\zeta \), then \(U_\zeta :L^2(\mu ^1_\zeta )\rightarrow L^2(\mu ^2_\zeta )\) for \(\nu \)a.e. \(\zeta \in Z\).
Step 3: Decomposability of U. For every \(A\in (\mathcal {X}^1)^{\mu ^1}\), and every \(f\in L^2(\mu ^1)\),
Now, let A be \(E^1\)invariant. Combining (4.4) and the definition of invariant set yields
whence, precomposing with \(U^{1}\) on both sides yields that \(\tau ^{1}(A)\) is \(E^2\)invariant.
Arguing as in Step 3 in the proof of [3, Thm. 3.4], it follows that U commutes with all the operators in \(\mathcal {B}(L^2(\mu ^1))\) diagonalizable over Z in the sense of [5, §II.2.4, Dfn. 3, p. 185]. By the characterization of decomposable operators via diagonalizable operators [5, §II.2.5, Thm. 1, p. 187], the operator U is decomposable, and represented by a \(\nu \)measurable field of bounded operators \(\zeta \mapsto U_\zeta :L^2(\mu ^1_\zeta )\rightarrow L^2(\mu ^2_\zeta )\). For \(\nu \)a.e. \(\zeta \in Z\), the operator \(U_\zeta \) is additionally an order isomorphism, by the forward implication in Proposition 3.7. This concludes the proof of (ii).
Step 4: Image forms Since \(U_\zeta \) is unitary and \({\big (T^{\scriptscriptstyle {(1)}}_{\zeta ,t}\big )}_{t\ge 0}\) is a strongly continuous contraction semigroup on \(L^2(\mu ^1_\zeta )\) for \(\nu \)a.e. \(\zeta \in Z\), the same holds for \({\big (T^{\scriptscriptstyle {(2)}}_{\zeta ,t}\big )}_{t\ge 0}\). For \(t>0\), let
By Step 2 above, \(\tilde{T}^{\scriptscriptstyle {(2)}}_t\) is a bounded operator in \(\mathcal {B}(L^2(\mu ^2))\). By the decomposability of U shown in Step 4 and by definition (4.3) of \(T^{\scriptscriptstyle {(2)}}_{\zeta ,t}\), we may apply the reverse implication in Proposition 3.7(iii) to obtain
and conclude that \(\tilde{T}^{\scriptscriptstyle {(2)}}_t=T^{\scriptscriptstyle {(2)}}_t\) for each \(t>0\), since U is unitary.
For \(\nu \)a.e. \(\zeta \in Z\), let \(\big ({E^2_\zeta ,\mathscr {D}(E^2_\zeta )}\big )\) be the quadratic form on \(L^2(\mu ^2_\zeta )\) corresponding to \({\big (T^{\scriptscriptstyle {(2)}}_{\zeta ,t}\big )}_{t\ge 0}\).
Step 5: Directintegral representation The direct integral representation of \((E^2,\mathscr {D}(E^2))\) as a direct integral of Dirichlet forms follows from that of \(\left( T^{\scriptscriptstyle {(2)}}_t\right) _{t\ge 0}\) by Proposition 2.12, provided we show
 (a):

the compatibility of \((E^2,\mathscr {D}(E^2))\) with the representation by \(\zeta \mapsto E^2_\zeta \) in the sense of Definition 2.17; and that
 (b):

\(\big ({E^2_\zeta ,\mathscr {D}(E^2_\zeta )}\big )\) is a Dirichlet form for \(\nu \)a.e. \(\zeta \in Z\).
In light of Step 2, in order to show (a) it suffices to show that \(\mathcal {A}^2\) is contained in \(\mathscr {D}(E^2)\), which follows from Proposition 3.6. In order to show (b), note that, since \(\left( T^{\scriptscriptstyle {(2)}}_t\right) _{t\ge 0}\) is subMarkovian, it follows by Proposition 2.18 that \({\big (T^{\scriptscriptstyle {(2)}}_{\zeta ,t}\big )}_{t\ge 0}\) is subMarkovian for \(\nu \)a.e. \(\zeta \in Z\). \(\square \)
Corollary 4.6
For \(i=1,2\) let \((X^i,\mathfrak {T}^i,\mathcal {X}^i,\mu ^i)\) be a locally compact Polish Radonmeasure space, and \((E^i,\mathscr {D}(E^i))\) be a regular Dirichlet form on \(L^2(\mu ^i)\) with semigroup \(\left( T^{\scriptscriptstyle {(i)}}_t\right) _{t\ge 0}\), each admitting ergodic decomposition \(\zeta \mapsto \big ({E^i_\zeta ,\mathscr {D}(E^i_\zeta )}\big )\) indexed by \((Z^i,\mathcal {Z}^i,\nu ^i)\). Suppose further that there exists a unitary order isomorphism \(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) intertwining \(\left( T^{\scriptscriptstyle {(1)}}_t\right) _{t\ge 0}\), \(\left( T^{\scriptscriptstyle {(2)}}_t\right) _{t\ge 0}\), represented by a scaling h and a transformation \(\tau \) in the sense of Proposition 3.2. Then, there exists an almost isomorphism \(\varrho :(Z^1,\mathcal {Z}^1,\nu ^1)\rightarrow (Z^2,\mathcal {Z}^2,\nu ^2)\) such that
 (i):

\(U:L^2(\mu ^1)\rightarrow L^2(\mu ^2)\) is decomposable over \((Z^1,\mathcal {Z}^1,\nu ^1)\) and represented by a \(\nu ^1\)measurable field of order isomorphisms \(U_\zeta :L^2(\mu ^1_\zeta )\rightarrow L^2(\mu ^2_{\varrho (\zeta )})\);
 (ii):

\(U^{1}:L^2(\mu ^2)\rightarrow L^2(\mu ^1)\) is decomposable over \((Z^2,\mathcal {Z}^2,\nu ^2)\). Furthermore, the field of order isomorphisms \((U_\zeta )^{1}:L^2(\mu ^2_{\varrho (\zeta )})\rightarrow L^2(\mu ^2_\zeta )\) is \(\nu ^2\)measurable, and it represents \(U^{1}\);
 (iii):

for \(\nu ^1\)a.e. \(\zeta \in Z^1\), the order isomorphism \(U_\zeta \) intertwines the semigroup \({\big (T^{\scriptscriptstyle {(1)}}_{\zeta ,t}\big )}_{t\ge 0}\) associated to \(\big ({E^1_\zeta ,\mathscr {D}(E^1_\zeta )}\big )\) with the semigroup \({\big (T^{\scriptscriptstyle {(2)}}_{\varrho (\zeta ),t}\big )}_{t\ge 0}\) associated to \(\big ({E^2_{\varrho (\zeta )},\mathscr {D}(E^2_{\varrho (\zeta )})}\big )\).
Example 4.7
Theorem 4.5 and Corollary 4.6 provide an alternative proof for the construction of the ergodic decomposition of the Dirichlet–Ferguson diffusion [4] on the space of probability measures over a closed Riemannian manifold, obtained by ad hoc techniques in [4, Thm. 6.15(ii)]. The ergodic decomposition of the associated form consists of uncountably many components, and therefore the result does not follow from the aforementioned work [14] (cf. Remark 4.3 above).
Theorem 4.5 has the following natural converse, a proof of which is a straightforward application of Proposition 3.7
Proposition 4.8
Let \((X,\mathfrak {T},\mathcal {X},\mu )\) be a locally compact Polish Radonmeasure space, \((Z,\mathcal {Z},\nu )\) be a separable countably generated probability space, and \(s:(X,\mathcal {X}^\mu )\rightarrow (Z,\mathcal {Z})\) be a measurable map. For \(i=1,2\) further let \({(\mu ^i_\zeta )}_{\zeta \in Z}\) be an sseparated pseudodisintegration of \(\mu \) over \((Z,\mathcal {Z},\nu )\), and \(\zeta \mapsto (E^i_\zeta ,\mathscr {D}(E^i_\zeta ))\) be a \(\nu \)measurable field of Dirichlet forms compatible with the disintegration. Further assume that there exists a \(\nu \)measurable field \(\zeta \mapsto U_\zeta \) of unitary order isomorphisms \(U_\zeta :L^2(\mu _\zeta )\rightarrow L^2(\mu _\zeta )\) intertwining the semigroups of \(\big ({E^i_\zeta ,\mathscr {D}(E^i_\zeta )}\big )\) as in (4.3).
Then, the direct integral forms are intertwined by the unitary order isomorphism .
By the transfer method for quasiregular Dirichlet spaces, e.g., [2], it is possible to extend all the previous results to the quasiregular case. The Definition 4.1 of ergodic decomposition is readily adapted by letting \((X,\mathfrak {T},\mathcal {X},\mu )\) be satisfying Assumption 2.16 (as opposed to: locally compact Polish). We only spell out the adaptation of Theorem 4.5. The easy adaptation of Proposition 4.4 and Corollary 4.6 is left to the reader.
Corollary 4.9
For \(i=1,2\), let \((X^i,\mathfrak {T}^i,\mathcal {X}^i,\mu ^i)\) be satisfying Assumption 2.16, and \((E^i,\mathscr {D}(E)^i)\) be a quasiregular Dirichlet form on \(L^2(\mu ^i)\) with semigroup \(\left( T^{\scriptscriptstyle {(i)}}_t\right) _{t\ge 0}\). Let all other assumptions of Theorem 4.5 hold. Then, all the conclusions of that Theorem hold as well.
Remark 4.10
(On quasihomeomorphisms) It is shown in [13, Thm. 3.11] that the transformation \(\tau \) associated to any order isomorphism U intertwining two irreducible regular Dirichlet forms has a version \(\tilde{\tau }\) which is additionally a quasihomeomorphism. This result has been extended by Li and Lin in [14] to the case of uptocountable ergodic decompositions as in Remark 4.3(b). Whereas there is a reasonable expectation for this result to hold—at least under the additional assumption that U be unitary, as in Theorem 4.5—a proof seems currently beyond reach.
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20 April 2023
In the original publication, “Lorenzo Dello” is the first name of the author and “Schiavo” is the surname. In this version, the surname has been changed to "Dello Schiavo
Notes
In the rest of the paper, by a quasihomeomorphism we shall mean a map satisfying the more restrictive conditions in [2, Dfn. 3.1].
A sequence in a Banach space B is called total if the strong closure of its linear span coincides with B.
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Research supported by the Austrian Science Fund (FWF) grant F65 at the Institute of Science and Technology Austria and by the European Research Council (ERC) (Grant agreement No. 716117 awarded to Prof. Dr. Jan Maas). L.D.S. gratefully acknowledges funding of his current position by the Austrian Science Fund (FWF) through the ESPRIT Programme (Grant No. 208). M.W. gratefully acknowledges funding of his current position by the Austrian Science Fund (FWF) through the ESPRIT Programme (Grant No. 156).
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Dello Schiavo, L., Wirth, M. Ergodic decompositions of Dirichlet forms under order isomorphisms. J. Evol. Equ. 23, 9 (2023). https://doi.org/10.1007/s00028022008597
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DOI: https://doi.org/10.1007/s00028022008597