Ergodic Decompositions of Dirichlet Forms under Order Isomorphisms

We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.


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 , X i , T i , µ i ) be a locally compact Polish σ-finite Radon-measure space and (T i t ) t≥0 be a sub-Markovian semigroup on L 2 (µ i ). They are intertwined by U : In [13] D. Lenz, M. Schmidt, and the second named author investigated the properties of intertwined sub-Markovian semigroups in the case when the intertwining operator U is additionally an order isomorphism, that is, an invertible linear operator satisfying U f ≥ 0 if and only if f ≥ 0. Suppose that (T i t ) t≥0 is the semigroup of a quasi-regular symmetric Dirichlet form E i , D(E i ) on L 2 (µ i ), and thus it is associated with a symmetric Markov process M i with state space X i . It is then the main result of [13] that M i (quasi-)determines the topology T i , in the following sense.
Theorem 1.1. If the irreducible semigroups (T i t ) t≥0 are intertwined by an order isomorphism, then the corresponding Dirichlet spaces (X i , T i , µ i , E i ) are quasi-homeomorphic.
Here, by a quasi-homeomorphism 1 j : (X 1 , T 1 , µ 1 , E 1 ) → (X 2 , T 2 , µ 2 , E 2 ) we mean a map such that for every ε > 0 there exist closed sets F i ⊂ X i with cap(X i \ F i ) < ε and j restricts to a homeomorphism j F 1 : F 1 → F 2 , cf. [7, §A.4, p. 429]. Importantly, this completely characterizes how a Markov process M identifies the topological properties of its state space (X, T, µ), and constitutes a sharp affirmative result in the spirit of M. Kac' isospectrality question. For example, -as previously established by W. Arendt, M. Biegert, and A. F. M. 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≥0 (or the Markov processes M i ) be irreducible. In terms of the semigroups this is to say that T t 1 A = 1 A T t for some t > 0 ⇐⇒ either A or A c is µ-negligible .
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 E i , D(E i ) is a collection of irreducible Dirichlet spaces E ζ , D(E ζ ) indexed by a measure space (Z, Z, ν) decomposing E, D(E) 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 E i , D(E i ) are intertwined by an order isomorphism U if and only if their ergodic decompositions ζ → E i ζ , D(E i ζ ) are intertwined by an order isomorphism U ζ , in which case U is decomposable over the ergodic decompositions and represented as a direct integral of the operators U ζ .
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 measure-theoretical 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.
• countably separated if there exists a countable family of sets in X separating points in X; • countably generated if there exists a countable family of sets in X generating X as a σ-algebra; • a standard Borel space if there exists a Polish topology T on X so that X coincides with the Borel σ-algebra induced by T.
A σ-finite measure space (X, X , µ) is standard if there exists a µ-conegligible set X 0 ∈ X such that X 0 is a standard Borel space when regarded as a measurable subspace of (X, X ). We denote by (X, X µ ,μ) the (Carathéodory) completion of (X, X , µ). A Hausdorff topological measure space (X, T, X , µ) is a Radonmeasure space if X µ coincides with the µ-completion of the Borel σ-algebra andμ is a Radon measure, i.e. it is locally finite and inner regular with respect to the compact sets.
Let us now recall the notion of a perfect measure space together with some of its properties. A measure space (X, X , µ) is perfect (e.g. [6, 342K]) if whenever f : X → R is measurable and A ∈ X with µA > 0, then there exists a compact set K ⊂ f (A) with f ♯ µ(K) > 0. Here, and everywhere in the following, f ♯ µ is the push-forward measure of µ via f .
Lemma 2.2. Let (X, X , µ) be any measure space. Then, (i) (X, X , µ) is perfect if and only if so is its completion; (ii) if (X, X , µ) is perfect and X 0 is a σ-subalgebra of X , then the restricted measure space (X, X 0 , µ⇂ X0 ) is perfect; (iii) if (X, X , µ) is perfect and λ has a density w.r.t. µ, then (X, X , λ) is perfect; Definition 2.3. Let (X, X , µ) be a σ-finite measure space, and X * ⊂ X be a countably generated σ-subalgebra. We say that: • X is µ-essentially countably generated by X * if for each A ∈ X there is A * ∈ X * with µ(A△A * ) = 0; • X is µ-essentially countably generated if it is so by some X * as above.
We say that the triples (X i , X i , N i ) are almost isomorphic if there exist sets N i ∈ N i so that the spaces X i \ N i are strictly isomorphic when endowed with the restriction to X i \ N i of the σ-algebra X i ∪ N i and of the σ-ideal N i .
For i ∈ {1, 2} let (X i , X i , µ i ) be measure spaces. We say that they are almost isomorphic if there exist µ i -negligible sets N i ⊂ X i so that the spaces X i \ N i are strictly isomorphic when endowed with the restriction to X i \ N i of the completed σ-algebra (X i ) µ i , of the completed measureμ i , and of the σ-ideal of µ i -negligible sets N µ i .
We always assume that the domain D(Q) = {u ∈ H : Q(u) < ∞} is dense in H and write (Q, 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 α > 0 we set For α > 0, we let D(Q) α be the completion of D(Q), endowed with the Hilbert norm Q 1/2 α . To every closed quadratic form (Q, D(Q)) one can associate a unique non-negative self-adjoint operator −L such that D( √ −L) = D(Q) and Q(u, v) = −Lu | v for all u, v ∈ D(L). We denote the associated strongly continuous contraction semigroup by T t := e tL , t > 0.  . Let (Z, Z, ν) be a σ-finite measure space, (H ζ ) ζ∈Z be a family of separable Hilbert spaces, and F be the linear space F := ζ∈Z H ζ . We say that ζ → H ζ is a ν-measurable field of Hilbert spaces (with underlying space S) if there exists a linear subspace S of F with (a) for every u ∈ S, the function ζ → u ζ ζ is ν-measurable; (c) there exists a sequence (u n ) n ⊂ S such that (u n,ζ ) n is a total sequence 2 in H ζ for every ζ ∈ Z.
Any such S is called a space of ν-measurable vector fields. Any sequence in S possessing property (c) is called a fundamental sequence.
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 ν-equivalence, sayû in S. In this case, we shall write u = [û] H .
We now turn to measurable fields of bounded operators.

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, Z, ν) be a σ-finite countably generated measure space. For ζ ∈ Z let (Q ζ , D ζ ) be a closable (densely defined) quadratic form on a Hilbert space We further denote by 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 ν-measurable field of Hilbert spaces that H ζ is separable for every ζ ∈ Z. As a consequence, (ii) ζ → T ζ,t is ν-measurable fields of bounded operators for every t > 0; (iii) the semigroup associated with Q is given by Under the assumptions of Proposition 2.12, assertion (i) of the same Proposition implies that the space of ν-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.
Direct-integral representation of L 2 -spaces. In order to introduce direct-integral 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 sub-Markovianity property (2.11). To this end, we shall need the concept of a disintegration of measures.
A pseudo-disintegration is • separated if there exists a family of pairwise disjoint sets {A ζ } ζ∈Z ⊂ X µ so that A ζ is µ ζ -conegligible for ν-a.e. ζ ∈ Z, henceforth called a separating family for (µ ζ ) ζ∈Z ; • s-separated if it is separated and there exists a X /Z-measurable map s : X → Z so that s −1 (ζ) ζ∈Z is a separating family; A disintegration of µ over ν is a pseudo-disintegration additionally so that µ ζ is a sub-probability measure for every ζ ∈ Z. A disintegration is • ν-essentially unique if the measures µ ζ are uniquely determined for ν-a.e. ζ ∈ Z.
Direct integrals and disintegrations. Let (X, X , µ) be σ-finite standard, (Z, Z, ν) be σ-finite countably generated, and (µ ζ ) ζ∈Z be a pseudo-disintegration of µ over ν. Denote by • L 0 (µ) the space of µ-measurable real-valued functions (not : µ-classes) on X; • L ∞ (µ) the space of uniformly bounded (not : µ-essentially uniformly bounded) functions in L 0 (µ); For a family A ⊂ L 0 (µ), let [A] µ denote the family of the corresponding µ-classes. (2.6), and δ therefore is well-defined as linear morphism mapping µ-classes to H-classes, see Proposition 2.15 below. Now, assume that Since [A] µ is dense in L 2 (µ) and the latter is separable, then there exists a countable family U ⊂ A so that [U] µ ζ is total in L 2 (µ ζ ) for ν-a.e. ζ ∈ Z. Thus for every A as in (2.8) there exists a unique space of ν-measurable vector fields S = S A containing δ(A), generated by δ(A) in the sense of Proposition 2.6. We denote by H the corresponding direct integral of Hilbert spaces . Let (X, X , µ) be σ-finite standard, (Z, Z, ν) be σ-finite countably generated, and (µ ζ ) ζ∈Z be a pseudo-disintegration of µ over ν. Then, the morphism (i) is well-defined, linear, and an isometry of Hilbert spaces, additionally unitary if (µ ζ ) ζ∈Z is separated; (ii) is a Riesz homomorphism (e.g. [6, 351H]). In particular, 2.6. Dirichlet forms. We recall a standard setting for the theory of Dirichlet forms, following [15].
Assumption 2.16. The quadruple (X, T, X , µ) is so that (X, T) is a metrizable Luzin space with Borel σ-algebra X andμ is a Radon measure on (X, T, X µ ) with full support.
By [6, 415D(iii), 424G] any space (X, X , µ) satisfying Assumption 2.16 is σ-finite standard. The support of a (µ-)measurable function f : X → R (possibly defined only on a µ-conegligible set) is defined as the support of the measure |f | · µ. Every such f has a support, independent of the µ-representative of f , cf. [15, p. 148].
A closed positive semi-definite quadratic form (Q, D(Q)) on L 2 (µ) is a (symmetric) Dirichlet form if We shall denote Dirichlet forms by (E, D(E)). A Dirichlet form (E, D(E)) is regular if (X, T) is (additionally) locally compact and D(E) ∩ C 0 (X) is both E 1/2 1 -dense in D(E) and uniformly dense in C 0 (X). Finally, we are interested in the notion of invariant sets of a Dirichlet form (E, D(E)) on L 2 (µ). We say that A ⊂ X is E-invariant if it is µ-measurable and any of the following equivalent conditions holds.
e. for any f ∈ L 2 (µ) and t > 0; The form (E, D(E)) is irreducible if every invariant set is either negligible or conegligible.
Proposition 2.18 motivates the following definition.
Definition 2.19. A quadratic form (E, D(E)) on L 2 (µ) is a direct integral of Dirichlet forms ζ → (E ζ , D(E ζ )) if it is a direct integral of quadratic forms ζ → (E ζ , D(E ζ )), for each ζ the form (E ζ , D(E ζ )) is a Dirichlet form on L 2 (µ ζ ), and the direct integral is additionally compatible with the separated pseudodisintegration (µ ζ ) ζ 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 (Order-preserving operators, order isomorphisms). For i = 1, 2 let (X i , X i , µ i ) be σ-finite countably generated and countably separated. A linear operator U : ). An order isomorphism is an invertible order-preserving linear operator with order-preserving inverse.
The structure of order isomorphisms between L 2 -spaces is characterized by the following Banach-Lamperti-type theorem.
Proposition 3.2 (Order isomorphism as weighted composition operator). If U : L 2 (X 1 , µ 1 ) → L 2 (X 2 , µ 2 ) is an order isomorphism, then there exist a measurable map h : X 2 → (0, ∞) and an X 2 /X 1measurable almost isomorphism τ : X 2 → X 1 such that The maps h and τ are unique up to equality almost everywhere.
Proof. See [16, Thm. 5.1] for existence in the case of finite measures µ i . For the general case argue as follows. Since (X i , X i , µ i ) is σ-finite standard, there exists a function f i satisfying f i > 0 µ i -a.e., and f i L 2 (µ i ) = 1. Thus, the multiplication operator M i : is an order isomorphism as well. Applying the assertion for U ′ yields maps h ′ and τ ′ . Finally, letting h := f −1/2 2 • τ · f 1/2 1 · h ′ and τ := τ ′ yields the assertion in the σ-finite case. For uniqueness, one can similarly reduce to the case of finite measure. Then h = U 1 is determined uniquely up to equality a.e. Moreover, if τ 1 , τ 2 : The maps h and τ 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 τ the transformation associated with U . . Let U : L 2 (X 1 , µ 1 ) → L 2 (X 2 , µ 2 ) be an order isomorphism with associated scaling h and transformation τ . Then τ ♯ µ 2 and µ 1 are mutually absolutely continuous. Moreover, the adjoint of U is given by and

Two families of operators
The following is not difficult to show. 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 , X i , µ i ) be σ-finite standard, T (i) t t≥0 be a sub-Markovian semigroup on L 2 (µ i ) with corresponding Dirichlet form (E i , D(E i )), and U : L 2 (µ 1 ) → L 2 (µ 2 ) be a unitary order-isomorphism intertwining T (1) t t≥0 and T (2) t t≥0 . Then, and Proposition 3.7. For i = 1, 2 let (X i , X i , µ i ) be σ-finite standard, (Z, Z, ν) be σ-finite countably generated, and µ i ζ ζ∈Z be a separated pseudo-disintegration of µ i over ν. Further let ζ → U ζ be a νmeasurable field of bounded operators U ζ : L 2 (µ 1 ζ ) → L 2 (µ 2 ζ ) in the sense of Definition 2.9, and set Then, U is a bounded operator U : L 2 (µ 1 ) → L 2 (µ 2 ) and (i) U : L 2 (µ 1 ) → L 2 (µ 2 ) is unitary if and only if U ζ is so for ν-a.e. ζ ∈ Z, in which case (ii) U : L 2 (µ 1 ) → L 2 (µ 2 ) is order-preserving if and only if U ζ is so for ν-a.e. ζ ∈ Z.
Proof. By definition of U , we have U : Let i = 1, 2. Since the disintegration (µ i ζ ) ζ∈Z is separated, by Proposition 2.15 (i) there exists a unitary order-isomorphism ι (i) satisfying (2.10). Since ι (i) is as well an order isomorphism by Proposition 2.15 (ii), in the rest of the proof we may identify U with ι (2) which concludes the proof.

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 non-trivial 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).
Remark 4.2. Since the pseudo-disintegration (µ ζ ) ζ∈Z in Definition 4.1 (ii) is s-separated, the map s is implicitly always assumed to be surjective. (a) in [3] under the additional assumptions that either µ be a finite measure, or that the form (E, 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 µ.
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 non-irreducible case) has already 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, X , µ) be a locally compact Polish Radon-measure space, and (E, D(E)) be a regular Dirichlet form on L 2 (µ). Further assume that (E, D(E)) admits Then, there exists an almost isomorphism ̺ : Proof. Let X 0 be the family of µ-measurable E-invariant subsets of X, and note that X 0 is a σsubalgebra of X µ , e.g. [7, Lem. 1.6.1, p. 53]. Let µ 0 be the restriction ofμ to (X, X 0 ). By our Assumption 2.16, X is countably generated, thus X 0 is µ 0 -essentially countably generated by X * := X ∩ X 0 .
Further let C be a special standard core for (E, D(E)) witnessing the compatibility of the direct integral representation, i.e. so that C = A as in Definition 2.17. Note that for every u ∈ C we may choose u ζ ≡ u as a µ ζ -representative of the diagonal embedding δ(u) ζ .
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 ∈ Z the set A := s −1 (B) is E-invariant. Since (µ ζ ) ζ∈Z is s-separated, (X ζ ) ζ∈Z , with X ζ := s −1 (ζ), is a separating family in the sense of Definition 2.13. The multiplication operator M X ζ := M 1 X ζ : L 2 (µ ζ ) → L 2 (µ ζ ) satisfies M X ζ = id L 2 (µ ζ ) , hence T ζ,t M X ζ = M X ζ T ζ,t . As a consequence, Claim II: for every E-invariant A ∈ X 0 there exists B ∈ Z with µ 0 (A△s −1 (B)) = 0. It suffices to show the statement for A ∈ X * . Since (Z, Z) is separable countably generated, it is countably separated, hence by [10,Prop. 12.1(iii)] there exists a separable metrizable topology T Z on Z so that Z is the Borel σ-algebra generated by T Z . As a consequence, s : X → Z is Borel, and therefore s(A) is analytic by [6, 423G(b)], thus Z-universally measurable by [6, 434D(c)], hence finally ν-measurable. In particular, there exist B 0 ⊂ s(A) ⊂ B 1 with B 0 , B 1 ∈ Z and ν(B 1 \ B 0 ) = 0. By the previous claim, we have Claim III: s * Z := s −1 (B) : B ∈ Z is µ 0 -essentially countably generating X 0 . It suffices to combine the previous two claims with the fact that Z is countably generated by assumption.
Finally, since both ν and σ are finite measures, ν has a density w.r.t. σ by the Radon-Nikodym Theorem, and therefore (Z, Z, ν) is perfect since so is (Z, Z, σ).
Step 2: Almost isomorphism. Throughout this step, let i = 1, 2. By construction, the measure algebra (Z i ,σ i ) of (Z, Z i , σ i ) is isomorphic to the measure algebra of (X, s * i Z i , λ i ). Furthermore, by Step 1: Claim III the σ-algebra Z i is µ 0 -essentially countably generating the algebra X 0 of E-invariant sets. Thus, Z i is as well λ 0 := λ ′ ⇂ X * -essentially countably generating X 0 , and therefore As a consequence, since the measure algebra of a measure space coincides with that of its completion, the measure algebras (Z 1 ,σ 1 ) and (Z 2 ,σ 2 ) are isomorphic. By all of the above, the spaces (Z i , (Z i ) σi ,σ i ) are countably separated perfect complete probability spaces, with isomorphic measure algebras. Therefore, the corresponding measure spaces (Z i , Z i , σ i ) are almost isomorphic by [6, 344C], via some almost isomorphism ̺ : Z 1 → Z 2 satisfying ̺ ♯ σ 1 = σ 2 . Since σ i ∼ ν i by the proof of Step 1 Claim IV, we conclude that ̺ ♯ ν 1 ∼ ν 2 .
Furthermore, exchanging the roles of ν 1 and ν 2 in Step 2 we have that ̺ ♯ ν 1 ∼ ν 2 and ̺ −1 ) is a ν 2 -measurable field of quadratic forms, for every u, v ∈ C the map ) is a ν 1 -measurable field of quadratic forms on L 2 (µ 2 ̺(ζ) ). The corresponding direct integral form E, D( E) is compatible with the disintegration in (4.1) in the sense of Definition 2.17 with underlying space S C , since C is a core for for ν 1 -a.e. ζ ∈ Z 1 . Finally, let us show that the form E, D( E) coincides with (E, D(E)). It suffices to note that, for all u, v ∈ C we have

4.2.
Intertwining. We are now ready to state our main result.
Step 2: Direct-integral representations of L 2 -spaces. Since µ 1 ζ ζ∈Z is separated by assumption and since µ 2 ζ ζ∈Z is separated by (i), Proposition 2.15 yields the direct-integral representations for every space of ν-measurable vector fields S i = S A i induced by A i as in (2.8). In the following, we shall always choose A 1 := C to be a special standard core for the (regular) form (E 1 , D(E 1 )), and set A 2 := h · f • τ : f ∈ A 1 . In light of Proposition 3.6 we have A 2 ⊂ 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 (µ 1 ) → L 2 (µ 2 ) is decomposable, and represented by a ν-measurable field of operators ζ → U ζ , then U ζ : L 2 (µ 1 ζ ) → L 2 (µ 2 ζ ) for ν-a.e. ζ ∈ Z.
Step 3: Decomposability of U . For every A ∈ (X 1 ) µ 1 , and every f ∈ L 2 (µ 1 ), Let (X, T, X , µ) be a locally compact Polish Radon-measure space, (Z, Z, ν) be a separable countably generated probability space, and s : (X, X µ ) → (Z, Z) be a measurable map. For i = 1, 2 further let (µ i ζ ) ζ∈Z be an s-separated pseudo-disintegration of µ over (Z, Z, ν), and ζ → (E i ζ , D(E i ζ )) be a ν-measurable field of Dirichlet forms compatible with the disintegration. Further assume that there exists a ν-measurable field ζ → U ζ of unitary order isomorphisms U ζ : L 2 (µ ζ ) → L 2 (µ ζ ) intertwining the semigroups of E i ζ , D(E i ζ ) as in (4.3). Then, the direct integral forms E i := ⊕ Z E i ζ dν(ζ) are intertwined by the unitary order isomorphism By the transfer method for quasi-regular Dirichlet spaces, e.g. [2], it is possible to extend all the previous results to the quasi-regular case. The Definition 4.1 of ergodic decomposition is readily adapted by letting (X, T, X , µ) 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. Remark 4.10 (On quasi-homeomorphisms). It is shown in [13,Thm. 3.11] that the transformation τ associated to any order isomorphism U intertwining two irreducible regular Dirichlet forms has a versionτ which is additionally a quasi-homeomorphism. This result has been extended by L. Li and H. Lin in [14] to the case of up-to-countable 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.