Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications

We study superpositions and direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin, Radon measure space, and admitting carr\'e du champ operator. In this case, the representation is only projectively unique.

We say that a set A ⊂ X is µ-trivial if it is µ-measurable and either µA = 0 or µA c = 0.The process M is irreducible if every M-invariant set is µ-trivial.When M is not irreducible, it is natural -in the study of the pathwise properties of M -to restrict our attention to "minimal" M-invariant subsets of X.In the local case, thanks to the quasi-topological characterization of M-invariance, such sets may be thought of as the "connected components" of the space X as seen by M.
This description is in fact purely heuristic, since it may happen that all such "minimal" M-invariant sets are µ-negligible.The question arises, whether these ideas for the study of M-invariance can be made rigorous by resorting to the Dirichlet form (E, D(E)) associated with M.More precisely, we look for a decomposition (E ζ , D(E ζ )) ζ∈Z of (E, D(E)) over some index set Z, and we require that ) is a Dirichlet form on (X, τ ) additionally irreducible (Dfn.3.1) for every ζ ∈ Z; • we may reconstruct (E, D(E)) from (E ζ , D(E ζ )) ζ∈Z in a unique way.
Because of the first property, such a decomposition -if any -would deserve the name of ergodic decomposition of (E, D(E)).
For instance, let us consider the standard Dirichlet form E g on a (second-countable) Riemannian manifold (M, g), i.e. the one generated by the (negative halved) Laplace-Beltrami operator ∆ g and properly associated with the Brownian motion on M .In this case, we expect that Z is a discrete space, indexing the connected components of M , and that where is but the standard form of the connected component of index ζ.This simple example suggests that, in the general case of our interest, we should expect that (E, D(E)) is recovered from the decomposition (E ζ , D(E ζ )) ζ∈Z as a "direct integral", Our purpose is morefold: • to introduce a notion of direct integral of Dirichlet forms, and to compare it with the existing notions of superposition of Dirichlet forms [5, §V.3.1](also cf.[15, §3.1(2 • ), p. 113] and [29]), and of direct integral of quadratic forms [3]; • to discuss an Ergodic Decomposition Theorem for quasi-regular Dirichlet forms, a counterpart for Dirichlet forms to the Ergodic Decomposition Theorems for group actions, e.g.[16,6,9]; • to provide rigorous justification to the assumption -quite standard in the literature about (quasi-) regular Dirichlet forms -, that one may consider irreducible forms with no loss of generality; • to establish tools for the generalization to arbitrary (quasi-regular) Dirichlet spaces of results currently available only in the irreducible case, e.g. the study [22] of invariance under orderisomorphism, cf.[11].
For strongly local Dirichlet forms, our ergodic decomposition result takes the following form.
Plan of the work.Firstly, we shall discuss the notion of direct integral of (non-negative definite) quadratic forms on abstract Hilbert spaces, §2.3, and specialize it to direct integrals of Dirichlet forms, §2.5, via disintegration of measures.In §3.2 we show existence and uniqueness of the ergodic decomposition for regular and quasi-regular (not necessarily local) Dirichlet forms on probability spaces.The results are subsequently extended to strongly local quasi-regular Dirichlet forms on σ-finite spaces and admitting carré du champ operator, §3.3.Examples are discussed in §3.4; an application is discussed in 3.5.
Bibliographical note.Our reference of choice for direct integrals of Hilbert spaces is the monograph [12] by J. Dixmier.For some results we shall however need the more general concept of direct integrals of Banach spaces, after [17,9].For the sake of simplicity, all such results are confined to the Appendix.
2. Direct Integrals.Every Hilbert space is assumed to be separable and a real vector space.
2.1.Quadratic forms.Let (H, • ) be a Hilbert space with scalar product • | • .By a quadratic form (Q, D) on H we shall always mean a symmetric positive semi-definite -if not otherwise stated, densely defined -bilinear form.To (Q, D) we associate the non-relabeled quadratic functional Q : H → R ∪ {+∞} defined by Additionally, for every α > 0, we set For α > 0, we let D(Q) α be the completion of D, endowed with the Hilbert norm Q (iii) Q is lower semi-continuous w.r.t. the strong topology of H; (iv) Q is lower semi-continuous w.r.t. the weak topology of H.
To every closed quadratic form (Q, D(Q)) we associate a non-negative self-adjoint operator −L, with domain defined by the equality D( √ −L) = D(Q), such that 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, and the associated strongly continuous contraction resolvent by G α :=(α − L) −1 , α > 0. By Hille-Yosida Theorem, e.g.[23, p. 27], T t =H-lim α→∞ e tα(αGα−1) . (2.1b) The direct integral of a family of Hilbert spaces is a natural generalization of the concept of direct sum of Hilbert spaces to the case when the indexing set Z is more than countable.In this case, the requirement of ℓ 2 -summability in the definition of direct sum is replaced by a requirement of L 2 -integrability (see below for the precise definitions), which implies that Z should be taken to be a measure space (Z, Z, ν).
When Z is an at most countable discrete space, and ν is the counting measure, then the direct integral of the Hilbert spaces (H ζ ) ζ is isomorphic to their direct sum (2.2).Since their introduction by J. von Neumann in [24, §3], direct integrals have become a main tool in operator theory, and in particular in the classification of von Neumann algebras.
In order to make the definition of direct integral precise, let us first introduce some measure-theoretical notions.
• separated if X separates points in X, i.e. for every x, y ∈ X there exists A ∈ X with x ∈ A and y ∈ A c ; • 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 τ on X so that X coincides with the Borel σ-algebra induced by τ .
For any subset X 0 of a measurable space (X, X ), the trace σ-algebra on X 0 is X ∩X 0 := {A ∩ X 0 : A ∈ X }.
We now recall the main definitions concerning direct integrals of separable Hilbert spaces, referring to [12 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 apparent 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, say û in S. In this case, we shall write u = [û] H .When the specification of the variable ζ is necessary, given u ∈ H, resp.û ∈ S, we shall write ζ → u ζ in place of u, resp.ζ → ûζ in place of û.In most cases, the distinction between u and û is however immaterial, similarly to the distinction between the class of a function in L 2 (ν) and any of its ν-representatives.Therefore in most cases we shall simply write u in place of both u and û.
Proof.It suffices to note that L 2 (ν) is separable, e.g.[14, 365X(p)].Then, the proof of [ Appendix.Here, we prefer the original definition in [12], since we shall make a (possibly) different choice of S, more natural when addressing direct integrals of Dirichlet forms.
Let H be a direct integral Hilbert space defined as in (2.4).We now turn to the discussion of bounded operators in B(H) and their representation by measurable fields of bounded operators.

Definition 2.8 (Measurable fields of bounded operators, decomposable operators). A field of bounded operators
ν-measurable vector field with underlying space S for every ν-measurable vector field u with underlying space S. A ν-measurable vector field of bounded operators is called ν-essentially bounded if ν-  (2.5), in which case we write The next statement is readily deduced from e.g.[8,Thm. 2] or [21,Thm. 1.10].For the reader's convenience, a short proof is included.Lemma 2.9.Let H be defined as in (2.4), B ∈ B(H) be decomposable, and D B be the closed disk of radius B op in the complex plane.Then, for every ϕ ∈ C(D B ) the continuous functional calculus ϕ(B) of B is decomposable and  We denote by . the quadratic form defined on H as in (2.4) given by (2.7) Remark 2.12 (Separability).It is implicit in our definition of ν-measurable field of Hilbert spaces that H ζ is separable for every ζ ∈ Z.Therefore, when considering ν-measurable fields of domains as in are ν-measurable fields of bounded operators for every α, t > 0; (iii) Q has resolvent and semigroup respectively defined by (2.8) As a consequence, ζ → D(Q ζ ) α is a ν-measurable field of Hilbert spaces (on Z, with underlying space S Q ) for every α > 0. Thus, it admits a direct integral of Hilbert spaces For α > 0 let (u α n ) n be a fundamental sequence of ν-measurable vector fields for D α and (u n ) n be a fundamental sequence of ν-measurable vector fields for H.  (iii) It suffices to show (2.1) for (Q, D(Q)), G α and T t defined in (2.8).Now, by definition of (Q, D(Q)) one has for every α > 0 By [12,§II.2.3,Cor.,p. 182] and decomposability of G α , one has which concludes the proof of (2.1a) for G α .
Let us show (2.1b) for T t .Define the operators (2.9) strongly in H. On the other hand, by Lemma 2.9 we have that Remark 2.15.Under the assumptions of Proposition 2.13, 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 2.4.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 .
The next proposition completes the picture, by providing a direct-integral representation for the generator of the form (Q, D(Q)) in (2.7).Since we shall not need this result in the following, we omit an account of direct integrals of unbounded operators, referring the reader to [21, §1].Once the necessary definitions are established, a proof is straightforward.Proposition 2.16.Let (Q, D(Q)) be defined as in (2.7).Then, its generator (L, D(L)) has the direct-integral representation Remark 2.17 (Comparison with [3]).As for quadratic forms, (2.11) is understood as a direct-integral representation of the Hilbert space D(L), endowed with the graph norm, by the measurable field of Hilbert 2.4.Dirichlet forms.We recall a standard setting for the theory of Dirichlet forms, following [23].
We shall denote Dirichlet forms by (E, • strongly local if E(f, g) = 0 for every f, g ∈ D(E) with g constant on a neighborhood of supp[f ]; • regular if (X, τ ) is (additionally) locally compact and there exists a core C for (E, 1 -dense in D(E) and uniformly dense in C 0 (τ ).On spaces that are not necessarily locally compact, the interplay between a Dirichlet form (E, D(E)) and the topology τ on X is specified by the following definitions.For an increasing sequence 1 -dense subset of D(E) 1 the elements of which all have an E-quasi-continuous µ-version; (qr 3 ) an E-exceptional set N ⊂ X and a countable family separating points in N c .We refer to [7] or [15, §A.4] for the notion of quasi-homeomorphism of Dirichlet forms.
We say that (E, D(E)) has carré du champ operator (Γ, Finally, let D(E) e be the linear space of all functions u ∈ L 0 (µ) so that there exists an We denote by (D(E) e , E) the space D(E) e endowed with the extension of E to D(E) e called the extended Dirichlet space of (E, D(E)).For proofs of wellposedness in this generality, see [19, p. 693 e is a Hilbert space with inner product E; These definitions are equivalent to the standard ones (e.g.[15, p. 55]) by [15, Thm.s 1.6.2,1.6.3,p. 58], a proof of which may be adapted to the case of spaces satisfying Assumption 2.18.
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 (2.12).To this end, we shall need the concept of a disintegration of measures.

Let now
In general, it does not hold that f ∈ L 2 (µ ζ ) for every ζ ∈ Z, thus we need to adjust the obvious definition of δ(f ) as above in such a way that δ (2.15).It will be shown in Proposition 2.25 below that δ is well-defined as linear morphism mapping µ-classes to H-classes.
Further let A be satisfying (2.17) A is a linear subspace of L 2 (µ), and and the latter is separable, then there exists a countable family Thus for every A as in (2.17) 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 2.4.
We denote by H the corresponding direct integral of Hilbert spaces Since S is unique, it is in fact independent of A. Indeed, let A 0 , A 1 be satisfying (2.17) and note that A := A 0 ⊕ A 1 satisfies (2.17) as well.Thus, δ(A 0 ), δ(A 1 ) ⊂ S A , and so In particular, for every g, h ∈ H, the fields h + , h − , g ∧ h, and g ∨ h, respectively defined by are ν-measurable fields representing elements of H.In the following, we shall occasionally write -here, 1 H is merely a shorthand - Remark 2.21.For arbitrary measurable spaces, the standard choice for A is the algebra of µintegrable simple functions.If (X, τ, X , µ) were a locally compact Polish Radon measure space, one might take for instance A = C c (τ ), the algebra of continuous compactly supported functions.In fact, for the purposes of the present section, we might as well choose A = L 2 (µ), as the largest possible choice, or A a countable Q-vector subspace of L 1 (µ) ∩ L ∞ (µ), as a smallest possible one.When dealing with direct integrals of regular Dirichlet forms however, the natural choice for A is that of a special standard core C for the resulting direct-integral form.
Remark 2.22 (Comparison with [3]).We note that for every A as in (2.17), [A] µ is a determining class in the sense of [3, p. 402].Conversely, every determining class L 0 is contained in a minimal linear space of functions [A] µ satisfying (2.17).
Remark 2.23 (Caveat).Whereas the space H does not depend on A, in general it does depend on S A , cf.Rmk.2.7.Furthermore, H depends on the chosen pseudo-disintegration too, and thus H need not be isomorphic to L 2 (µ), as shown in the next example.
We show the first assertion in (ii).A proof of the second assertion is similar, and therefore it is omitted.
Argue by contradiction that there exists f ∈ L 2 (µ) with f ≥ 0 µ-a.e., yet such that for all ζ in some B ∈ Z with νB > 0. In particular, since Remark 2.20, the following integral is well-defined and strictly positive Then, by (2.15), and by definition of ι, continuing from the previous inequality, we have that by definition of H.The inequality contradicts the fact, shown in (i), that ι : L 2 (µ) → H is an isometry, and therefore 2.6.Direct integrals of Dirichlet forms.Let (X, X , µ) be σ-finite standard, (Z, Z, ν) be σ-finite countably generated, and ν-measurable field of quadratic forms, each densely defined in L 2 (µ ζ ) with separable domain, and denote by (Q, D(Q)) their direct integral in the sense of Definition 2.11.Note that, if S Q is of the form S A for A ⊂ D(Q) and satisfying (2.17), then S H is of the form S A as well by Remark 2.15.Definition 2.27 (Diagonal restriction).Let (Q, D(Q)) be a direct integral of quadratic forms compatible with a pseudo-disintegration (µ ζ ) ζ∈Z .The form is a closed (densely defined) quadratic form on ι(L 2 (µ)), called the diagonal restriction of (Q, D(Q)).
Remark 2.28 (Comparison with [3]).We note that the form (Q res , D(Q res )) coincides with the form By definition of ι * E and since ι : L 2 (µ) → H is unitary, we have that (ι * E) 1 = ι * E 1 .Therefore, ) is closed on H by Proposition 2.13(i).Furthermore, the Hilbert spaces D(E) 1 and D(ι * E) 1 are intertwined via the unitary isomorphism ι.In the statement of Proposition 2.29 above and everywhere after its proof -with a slight abuse of notation -these two quadratic forms are identified.Again for the sake of clarity, the statement of the proposition equivalently reads as follows: Proof of Proposition 2.29.By e.g.[15, Thm.1.4.1], the closed quadratic form Thus, it suffices to show that T ι • is sub-Markovian if and only if T ζ,• is so for ν-a.e.ζ ∈ Z. Since (E, D(E)) and (ι * E, D(ι * E)) are intertwined by the unitary isomorphism ι, it is not difficult to show that their semigroups T • and T ι • are intertwined as well, viz.
Furthermore, since ι : L 2 (µ) → H is a Riesz homomorphism by Proposition 2.25(ii), where, by definition of the Banach lattice structure on H in Remark 2.20, , where x 2 ranges in Z = R the real line, (X, X , µ x2 ) is again the standard real line for every x 2 ∈ R, and for Leb 1 -a.e.x 2 ∈ R .
Let us now consider • the subspace D of all functions f ∈ L 2 (µ) so that and let us further assume that Then, it is claimed in [5, p. 214] that (2.24) is well-defined and depends only on the µ-class of f , and it Note that we may always choose λ = ν provided that the integral measure µ defined above is σ-finite.
If this is not the case, we may recast the definition by letting ν := λ.In this way, we may always assume with no loss of generality that µ is given, and that (µ ζ ) is a pseudo-disintegration of µ over ν.
Remark 2.34.In fact, [5] requires all functions in (sp 1 )-(sp 2 ) to be Z-measurable, rather than only ν-measurable.Here, we relax this condition to 'ν-measurability' in order to simplify the proof of the reverse implication in the next Proposition 2.35.Our definition of 'superposition' is equivalent to the one in [5].Proof.We only show that (i) implies (ii).A proof of the reverse implication is similar, and it is therefore omitted.For simplicity, set throughout the proof 1 -dense in D, and dense in L 2 (µ) by (sp 3 ).Since (µ ζ ) ζ∈Z is separated by assumption, it follows by Proposition 2.25 that In particular, for the equivalence class h :=[ h] H , we have that h = 0 H .By Proposition 2.25, there exists h ∈ L 2 (µ) representing h ∈ H, and thus satisfying 0 D = [ h] µ ∈ D. On the other hand though, by definition of ĥ.By E 1/2 1 -density of [U] µ in D, the latter implies that h = 0 D , the desired contradiction.
Remark 2.36.If the disintegration in Proposition 2.35 is not separated, then (E, D) is still isomorphic, as a quadratic form, to the diagonal restriction (Q res , D(Q res )) (Dfn.2.27) of the direct integral of quadratic forms (E, D(E)).
3. Ergodic decomposition.Everywhere in this section, let (X, τ, X , µ) be satisfying Assumption 2.18.We are interested in the notion of invariant sets for a Dirichlet form.(E, D(E)) be a Dirichlet form on L 2 (µ).We say that A ⊂ X is E-invariant if it is µ-measurable and any of the following equivalent3 conditions holds.3.1.The algebra of invariant sets.Invariants sets of symmetric Markov processes on locally compact Polish spaces are studied in detail by H. Ôkura in [25].In particular, he notes the following.For A ∈ X set Then, one has the following dichotomy, [25, Rmk.1.1(ii)], • As a consequence, when describing an E-invariant set A of a regular Dirichlet form (E, D(E)), we may use interchangeably the E-class [A] E -i.e. the finest object representing A, as far as E is concernedand the µ-class [A] µ representing A in the measure algebra of (X, X , µ).This motivates to allow A in our definition of invariant set to be µ-measurable, rather than only measurable.
We turn now to the study of invariant sets via direct integrals.We aim to show that, under suitable assumptions on µ, a Dirichlet form (E, D(E)) on L 2 (µ) may be decomposed as a direct integral ζ → To this end, we need to construct a measure space (Z, Z, ν) "indexing" E-invariant sets.Let us start with a heuristic argument, showing how this cannot be done naïvely, at least in the general case when (X, X , µ) is merely σ-finite.
Let X 0 be the family of µ-measurable E-invariant subsets of X, and note that X 0 is a σ-sub-algebra of X µ , e.g.[15, Lem.1.6.1,p. 53].Let µ 0 be the restriction of μ to (X, X 0 ).The space (X, X 0 , µ 0 ) -our candidate for (Z, Z, ν) -is generally not σ-finite, nor even semi-finite.For instance, in the extreme case when (E, D(E)) is irreducible and µX = ∞, then X 0 is the minimal σ-algebra on X, the latter is an atom, and thus µ 0 is purely infinite.Since (X, X , µ) is σ-finite, every disjoint family of µ-measurable non-negligible subsets is at most countable [14, 215B(iii)], thus (X, X 0 , µ 0 ) has up to countably many disjoint atoms.However, even in the case when (X, X 0 , µ 0 ) has no atoms, µ 0 might again be purely infinite.This is the case of Example 2.32, where X 0 = {∅, R} ⊗ B(R) Leb 1 is the product σ-algebra of the minimal σ-algebra on the first coordinate with the Lebesgue σ-algebra on the second coordinate, and where µ 0 coincides with the μ-measure of horizontal stripes.This latter example shows that, again even when (X, X 0 , µ 0 ) has no atoms, the complete locally determined version [14, 213D] of (X, X 0 , µ 0 ) is trivial.Thus, in this generality, there is no natural way to make (X, X 0 , µ 0 ) into a more amenable measure space while retaining information on E-invariant sets.
The situation improves as soon as (X, X , µ) is a probability space, in which case so is (X, X 0 , µ 0 ).The reasons for this fact are better phrased in the language of von Neumann algebras.• since (X, X 0 , µ 0 ) is now (semi-)finite, M 0 is unital as well; • by Definition 3.1(d), the algebra M 0 acts by multiplication also on D(E), and the action M 0 L 2 (µ) is compatible with the action M 0 D(E) by restriction.
The next definition, borrowed from [4], encodes a notion of "smallness" of the σ-algebra X w.r.t.µ.

Definition 3.3 ([4, Dfn. A.1]
).Let 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.
By our Assumption 2.18, X is countably generated, thus X 0 is µ 0 -essentially countably generated by X * := X ∩ X 0 .We denote by µ * 0 the restriction of µ 0 to X * .As noted in [4, p. 418], atoms of X * are, in general, larger (in cardinality, not in measure) than atoms of X .It is therefore natural to pass to a suitable quotient space.Following [4, Dfn.A.5], we define an equivalence relation ∼ on X by Further let p : X → Z := X/ ∼ be the quotient map, Z := B ⊂ Z : p −1 (B) ∈ X * be the quotient σalgebra induced by p, and ν := p ♯ µ * 0 be the quotient measure.Similarly to [4, p. 416], it follows by definition of ∼ that every A ∈ X * is p-saturated.In particular: As a consequence X * and Z are isomorphic and thus both are countably generated, since X * is by assumption.Furthermore, (Z, Z) is separable by construction, and thus it is countably separated.
1. Measurable fields.Let C be a special standard core [15, p. 6] for (E, D(E)), and N ⊂ Z be a νnegligible set so that (X, τ, is separable by [23,Prop. IV.3.3(i)], and therefore we can and will assume, with no loss of generality  3. Semigroups.Let T t be the semigroup associated to (E, D(E)) and consider the natural complexification T C t of T t defined on . By [12, §II.2.3, Prop.4(ii), p. 183] and the norm-density of simple functions in L ∞ (ν), it suffices to show that [T t , M B ] = 0 for every B ∈ Z.To this end, recall the discussion [12, p. 165] on ν-measurable structures induced by ν-measurable subsets of Z.
Since B ∈ Z, then A := p −1 (B) ∈ X µ0 0 , and A ∈ X µ as well [14, 235H(c)].Note further that, since H is reconstructed as a direct integral with underlying space S C , for every h ∈ C the representative h ζ of h in H ζ may be chosen so that h ζ = h for every ζ ∈ Z.Thus, for all f, g ∈ C, By density of C in H and since M B is bounded, it follows that M B = 1 A as elements of B(H).
Thus, [T t , M B ] = [T t , 1 A ] = 0 for every t > 0 by Definition 3.1(a), since A is E-invariant.We claim that

By the characterization of decomposable operators via diagonalizable operators
By [23, p. 27], by Fatou's Lemma.It is readily checked that, since T ζ,t op ≤ 1, we may exchange the order of both integration and H ζ -scalar products by Fubini's Theorem.Thus, By the representation of Finally, by (2.14), [23, p. 27] and (2.1c), 1 -dense for ν-a.e.ζ ∈ Z. Argue by contradiction that there exists a ν-measurable non-negligible set B so that the inclusion 1 -dense for every ζ ∈ B, and let C ⊥ ζ be the defined by (2.7).We claim that ( Ẽ, D( Ẽ)) = (E, D(E)).This is a consequence of Proposition 2.13(iii), since (2.8) was shown in Step 3 for T t .Definition 2.26 holds with A = C by construction.

Forms: irreducibility
With no loss of generality, we may and will assume that A ζ ∈ X .Up to removing a ν-negligible set of ζ's, we have that A ζ ⊂ s −1 (ζ), by strong consistency of the disintegration.Thus, by (2.14), and so A ζ is E-invariant, and thus In the statement of Theorem 3.4, we write that each with underlying space (X, τ, X , µ ζ ) to emphasize that the topology of the space is the given one.As it is well-known however, in studying the potential-theoretic and probabilistic properties of a Dirichlet form (E, D(E)) on L 2 (µ), one should always assume that µ has full support, which is usually not the case for µ ζ on (X, τ ).In the present case, the restriction of Remark 3.5.As anticipated in §1, if (E, D(E)) is regular and strongly local, then every invariant set admits an E-quasi-clopen µ-modification [15,Cor. 4.6.3,p. 194].This suggests that, at least in the local case, one may treat E-invariant sets as "connected components" of X.Our intuition can be made rigorous by noting that E-invariant subsets of X are in bijective correspondence to compact open subsets of the spectrum spec(M 0 ) of the von Neumann algebra M 0 (cf.Rmk.3.2), endowed with its natural weak* topology.In particular, spec(M 0 ) coincides with the Stone space of the measure algebra of (X, X 0 , µ 0 ), and is thus a totally disconnected Hausdorff space.Its singletons correspond to the "minimal connected components" sought after in §1.At this point, we should emphasize that (Z, Z) and spec(M 0 ) are different measure spaces, the points of which index "minimal invariant sets" in X.However, points in Z indexvia s -sets in X * , whereas points in spec(M 0 ) index sets in X 0 .In this sense at least, Z is minimal with the property of indexing such "minimal invariant sets", while spec(M 0 ) is maximal.For this reason, one might be tempted to use spec(M 0 ) in place of (Z, Z) in Theorem 3.4.The issue is that spec(M 0 ) is nearly always too large for the disintegration to be strongly consistent with the indexing map.
In the next result we show that the regularity of the Dirichlet form (E, D(E)) in Theorem 3.4 may be relaxed to quasi-regularity.As usual, a proof of this result relies on the so-called transfer method.
Proof.By the general result [7,Thm. 3.7], there exist a locally compact Polish, Radon probability space (X ♯ , τ ♯ , X ♯ , µ ♯ ) and a quasi-homeomorphism where 1. Forms.In the following, whenever (F k ) k is a nest, let us set F := k F k .With no loss of generality by [15, Lem.2.1.3,p. 69], we may and will always assume that every nest is increasing, and regular w.r.t.
a measure apparent from context.
Let (F k ) k , resp.F ♯ k k , be an E-, resp.E ♯ -, nest, additionally so that j : F → F ♯ restricts to a homeomorphism j : ) is regular by Theorem 3.4.Let X ∂ := X ∪ {∂}, where ∂ is taken to be an isolated point in X ∂ .Since j may be not surjective, in the following we extend j −1 on X ♯ \ j(F ) by setting j −1 (x ♯ ) = ∂.Note that this extension is X ♯ -to-X -measurable (having care to extend X on X ∂ in the obvious way).Since j ♯ µ = µ ♯ the set N 2 :={ζ ∈ Z : µ ♯ ζ j(F ) < 1} is ν-measurable, since j is measurable on F , and thus it is ν-negligible.In particular, j −1 ♯ µ ♯ {∂} = 0, and and note that, again by [ ], 3. Direct integral representation.By (2.8) for the resolvent applied to (E ♯ , D(E ♯ )), Thus, by a further application of [12, §II.2.3, Prop.3, p. 182], Cancelling j * by its inverse (j −1 ) * , this yields the direct-integral representation of By (2.8) for the resolvent, this shows 3.3.Ergodic decomposition of forms: σ-finite measure case.Under some additional assumptions, we may now extend the results in Theorem 3.4 to the case when µ is only σ-finite.The main idea -borrowed from [6] -is to reduce the σ-finite case to the probability case.
Assume further that (E, D(E)) has carré du champ (Γ, D(E)).Let ϕ ∈ D(E), with ϕ > 0 µ-a.e. and ϕ L 2 (µ) = 1, and set µ ϕ := ϕ 2 • µ.Here, we understand ϕ as a fixed E-quasi-continuous representative of its µ-class.Note that (X, X , µ ϕ ) is a probability space and that µ ϕ is equivalent to µ.Therefore, µ-classes and µ ϕ -classes coincide.On L 2 (µ ϕ ) we define a bilinear form Then, we may apply Theorem 3.4 to obtain • a probability space (Z ϕ , Z ϕ , ν ϕ ) and a measurable map s ϕ : X → Z; • a ν ϕ -essentially unique disintegration µ be a measure on (X, X ) and suppose further that is a (closed) regular Dirichlet form on L 2 (µ ζ ) for ν ϕ -a.e.ζ ∈ Z.Then, finally, we may expect to have a direct-integral representation As it turns out, the properties of the Girsanov-type transformation (3.12) are quite delicate.Before discussing the technical details, let us note here that, provided we have shown the direct-integral representation in (3.15), it should not be expected that the latter is (essentially) unique, but rather merely essentially projectively unique -as it is the case for other ergodic theorems, e.g.[6,Thm. 2].In the present setting, projective uniqueness is understood in the following sense.Definition 3.7.We say that the direct integral representation (3.15) is essentially projectively unique if, for every ϕ, ψ as above: (a) the measurable space (Z, Z) :=(Z ϕ , Z ϕ ) = (Z ψ , Z ψ ) is uniquely determined; (b) the measures ν ϕ , ν ψ are equivalent (i.e., mutually absolutely continuous); (c) the forms E ζ∈Z is merely a pseudo-disintegration (as opposed to: a disintegration).Thus, for every measurable g : Z → (0, ∞), Since g is defined on Z, the pullback function f :=(s ϕ ) * g is X 0 -measurable, i.e. all its level sets are E-invariant; by strong locality of (E, D(E)), f is E-quasi-continuous, and therefore an element of the extended domain F e of (E, D(E)).As soon as f ∈ L 2 (µ), then we have the direct-integral representation Proofs' summary.The Girsanov-type transformations (3.12) are thoroughly studied by A. Eberle in [13], where Proof.Note: In this proof we shall make use of results in [5].We recall that a regular form is 'strongly local' in the sense of [15, p. 6] if and only if it is 'local' in the sense of [5, Dfn.I.V. 1.2, p. 28].This is noted e.g. in [28, §2, p. 78], after [26,Prop. 1.4].In this respect, we always adhere to the terminology of [23,15].Let The same holds for (E, D(E)).Now, argue by contradiction that there exists a ν-measurable non-negligible set B ⊂ Z so that the and νB > 0, there exists some fixed n * so that νB n * > 0. Without loss of generality, up to relabeling, we may choose n * = 1.
Analogously to the proof of the Claim in Step 3 of Theorem 3.4(iii), set A := p −1 (B 1 ) and note that it is E-invariant.Thus, finally, 1 A u 1 ∈ D(E) and which contradicts the strong locality of (E, D(E)).
Remark 3.10.The converse implications to Lemmas 3.8, 3.9 are true in a more general setting, viz.We are now ready to prove the main result of this section.
It follows that the σ-ideal N := N ϕ of ν ϕ -negligible sets in Z does not in fact depend on ϕ.In the following, we write therefore "N -negligible" in place of "ν ϕ -negligible" and "N -a.e." in place of "ν ϕ -a.e.".(ii) (E, D(E)) is a regular Dirichlet form on L 2 (µ) with core C ⊗ C 0 (Z) and semigroup As a further example, we state here the ergodic decomposition theorem for mixed Poisson measures on the configuration space over a connected Riemannian manifold.We refer the reader to [2] for the main definitions.
Example 3.13 (Mixed Poisson measures, [2]).Let (M, g) be a Riemannian manifold with infinite volume, and σ = ρ • vol g be a non-negative Borel measure on M with density ρ > 0 vol g -a.e., and satisfying ρ 1/2 ∈ W 1,2 loc (M ).Let further Γ M be the configuration space over M , endowed with the vague topology and the induced Borel σ-algebra, and denote by π σ the Poisson measure on Γ M with intensity measure σ.Let now λ be a Borel probability measure on R + :=(0, ∞) with finite second moment.The mixed Poisson measure with intensity measure σ and Lévy measure λ is the measure µ λ,σ := R+ π sσ dλ(s) .
Remark 3.14.Other examples are given by [1,Thm. 3.7] and [3], both concerned with strongly local Dirichlet forms on locally convex topological vector spaces, and by [10], concerned with a particular quasi-regular Dirichlet form on the space of probability measures over a closed Riemannian manifold.
3.5.Some applications.We collect here some applications of the direct-integral decomposition discussed in the previous sections.
Transience/recurrence.Let (X, τ, X , µ) be satisfying Assumption 2.18.For an invariant set A ∈ X 0 , we denote by µ A the restriction of µ to A, and by (E A , D(E A )) the Dirichlet form Corollary 3.15 (Ergodic decomposition: transience/recurrence).Under the assumptions of Theorem 3.6, there exist E-invariant subsets X c , X d , and a properly E-exceptional subset N of X, so that As an application, we have the following proposition.Similarly to Remark 3.10, some implications hold for superpositions of arbitrary Dirichlet forms.Proof.Analogously to the proof of Theorem 3.6 we may restrict to the regular case by the transfer method.Thus we can and will assume with no loss of generality that (X, τ, X µ , μ) is a locally compact Polish Radon measure space with full support, and enlargement of C, we may assume that C is special standard (e.g., [15, p. 6]).In particular, C is a lattice.Since X 0 is µ-essentially countably generated by X * , we may and shall assume without loss of generality that X d ∈ X * , so that B := s(X d ) ∈ Z. Since µX d > 0, we have νB > 0. It is not difficult to show that the direct-integral decomposition of L 2 (µ) splits as a direct sum of Hilbert spaces Ergodic decomposition of measures.Let (X, τ, X , µ) be a locally compact Polish probability space.Since (X, X ) is a standard Borel space, the space M of all σ-finite measures on (X, X ) is a standard Borel space as well when endowed with the σ-algebra generated by the family of sets {η ∈ M : a 1 < ηA < a 2 } , a 1 , a 2 ∈ R + , A ∈ X .
The semigroup T • of (E, D(E)) is thus well-defined on bounded Borel measurable functions, by letting Definition 3.17.We say that a σ-finite measure η on (X, X ) is T An invariant measure η is T • -ergodic if every E-invariant set is either η-negligible or η-conegligible.We denote by M inv , resp.M erg , the set of all σ-finite T • -invariant, resp.T • -ergodic, measures.
The formulation of the following result is adapted from [6,Thm. 1].In light of Corollary 3.15, we may restrict to the case of recurrent Dirichlet forms.
4. Appendix.The theory of direct integrals of Banach spaces is inherently more sophisticated than the corresponding theory for Hilbert spaces.We discuss here an irreducible minimum after [17, and especially [9, §3].For simplicity, we restrict ourselves to the case of σ-finite (not necessarily complete) indexing spaces (Z, Z, ν).

2. 3 .
Direct integrals of quadratic forms.The main object of our study are direct integrals of quadratic forms.Before turning to the case of Dirichlet forms on concrete Hilbert spaces (L 2 -spaces), we give the main definitions in the general case of quadratic forms on abstract Hilbert spaces.Definition 2.11 (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 H ζ .We say that ζ → (Q ζ , D ζ ) is a ν-measurable field of quadratic forms on Z if (a) ζ → H ζ is a ν-measurable field of Hilbert spaces on Z with underlying space S H ; (b) ζ → D(Q ζ ) 1 is a ν-measurable field of Hilbert spaces on Z with underlying space S Q ; (c) S Q is a linear subspace of S H under the identification of D(Q ζ ) with a subspace of H ζ granted by Lemma 2.1.

Definition 2 . 2 1
11(b), D(Q ζ ) 1 is taken to be (Q ζ ) 1/-separable by assumption.Proposition 2.13.Let (Q, D(Q)) be a direct integral of quadratic forms.Then, (i) Q, D(Q) is a densely defined closed quadratic form on H; spaces ζ → D(L ζ ), each endowed with the relative graph norm.The set-wise identification of D(L) as a linear subspace of H as in (2.4) is already shown in [3, Prop.1.6].

1 )
(e) 1 A f ∈ D(E) e for any f ∈ D(E) e and (3.1) holds for any f, g ∈ D(E) e .The form (E, D(E)) is irreducible if, whenever A is E-invariant, then either µA = 0 or µA c = 0.As shown by Example 2.32, the form (E, D(E)) constructed in Proposition 2.29 is hardly ever irreducible, even if (E ζ , D(E ζ )) is so for every ζ ∈ Z.
As a consequence, by Proposition 2.4 there exists a unique ν-measurable field of Hilbert spaces ζ → L 2 (µ ζ ) making ν-measurable all functions of the form ζ → µ ζ f with f ∈ C. We denote by S C the underlying space of ν-measurable vector fields.Everywhere in the following, we identify [f ] µ ζ with a fixed continuous representative f ∈ C, thus writing f in place of δ(f ).
of µ ϕ over ν ϕ , strongly consistent with s ϕ ; • a family of regular strongly local Dirichlet forms E each other for ν ϕ -(hence ν ψ -)a.e.ζ ∈ Z.As it is clear, the definition only depends on the σ-ideal of ν ϕ -negligible sets in Z.By condition (b), this ideal does not, in fact, depend on ν ϕ , hence the omission of the measure in the designation.The lack of uniqueness is shown as follows.Since µ [ϕ] ζ is merely a σ-finite (as opposed to: probability) measure, the family µ [ϕ] ζ

Lemma 3 . 9 .
) is proved.We shall therefore start by showing (b) above, Lemma 3.8 below.Informally, in the setting of Theorem 3.4, if (E, D(E)) has carré du champ Γ, then Γ = ⊕ Z Γ ζ dν(ζ) , (3.16) where Γ ζ is the carré du champ of (E ζ , D(E ζ )).Since the range of Γ is a Banach (not Hilbert) space, we shall need the concept of direct integrals of Banach spaces.In particular, we shall need an analogue of Proposition 2.25 for L 1 -spaces, an account of which is given in the Appendix, together with a proof of the next lemma.Lemma 3.8.Under the assumptions of Theorem 3.4 suppose further that (E, D(E)) admits carré du champ operator (Γ, D(E)).Then, (E ζ , D(E ζ )) admits carré du champ operator (Γ ζ , D(E ζ )) for νa.e.ζ ∈ Z.Under the assumptions of Theorem 3.4 suppose further that (E, D(E)) is strongly local.Then, (E ζ , D(E ζ )) is strongly local for ν-a.e.ζ ∈ Z.

ζ
dν ϕ (ζ) , which establishes the direct integral representation (3.15), with underlying space S C .The regularity of the forms E ζ for N -a.e.ζ ∈ Z. Argue by contradiction that there exist B ∈ Z \ N and a family (A ζ ) ζ∈B with A ζ ∈ X and, without loss of generality, µ (ϕ) ζ A ζ > µ (ψ) ζ A ζ = 0 for all ζ ∈ B. Set further Ã := ∪ ζ∈B A ζ and let A ∈ X be its measurable envelope [14, 132D].Then, by (2.14) and strong Example 3.12 (Ergodic decomposition of forms on product spaces).Let X = Y × Z be a product of locally compact Polish spaces endowed with a probability (hence Radon) measure µ, and (µ ζ ) ζ∈Z be a disintegration of µ over ν := pr Z ♯ µ strongly consistent with the standard projection pr Z : X → Z.This includes the setting of Example 2.32.Indeed, let (E ζ , D(E ζ )) be regular irreducible Dirichlet forms on L 2 (µ ζ ), all with common core C ⊂ C 0 (Y ), and assume that ζ → (E ζ , D(E ζ )) is a ν-measurable field of quadratic forms in the sense of Definition 2.11 with underlying ν-measurable field S = S C .Then, it is readily verified that (i) the direct integral (E, D(E)) of quadratic forms ζ → (E ζ , D(E ζ )) is a direct integral of Dirichlet forms;

well-defined on L 2
(µ A ) as a consequence of Definition 3.1(d).The next result is standard.In the generality of Assumption 2.18, a proof is readily deduced from the corresponding result for µ-tight Borel right processes, shown with different techniques by K. Kuwae in [20, Thm.1.3], in the far more general setting of quasi-regular semi-Dirichlet forms.

Z
= {B ⊂ Z : B ∩ Z α ∈ Z for all α ∈ A} and νB = α∈A ν(B ∩ Z α ) , B ∈ Z .Definition 4.1 (Measurable fields, cf.[17, §6.1, p. 61f.] and [9, §3.1]).Let (Z, Z, ν) be a σ-finite measure space, and V be a real linear space.A ν-measurable family of semi-norms on V is a family( • ζ ) ζ∈Z so that • • ζ is a semi-norm on V for every ζ ∈ Z; • the map ζ → v ζ is ν-measurable for every v ∈ V .Letting Y ζ denote the Banach completion of V / ker • ζ , we say that a vector field u ∈ ζ∈Z Y ζ is νmeasurable if, for each B ∈ Z with νB < ∞, there exists a sequence (u n ) n of simple V -valued vector fields on B so that lim n u ζ − u n,ζ ζ = 0 ν-a.e. on B. A family (Y ζ ) ζ∈Z of Banach spaces Y ζ is a ν-measurablefield of Banach spaces if there exist • a decomposition (Z α ) α∈A of (Z, Z, ν) consisting of sets of finite ν-measure;• a family of real linear spaces (Y α ) α∈A ;• for each α ∈ A, a ν-measurable family of norms • ζ on Y α , so that, for each α ∈ A and each ζ ∈ Z α , the space Y ζ is the completion of (Y α , • ζ ).Extending the above definition of ν-measurability, we say that u ∈ ζ∈Z Y ζ is ν-measurable if (and only if) the restriction of u to each Z α is ν-measurable.
1c) 2.2.Direct integrals.Let (H ζ ) ζ∈Z be a family of Hilbert spaces indexed by some index set Z. If Z is at most countable, the direct sum of the Hilbert spaces H ζ is defined as , § §II.1, II.2] for a systematic treatment.Definition 2.3 (Measurable fields, [12, §II.1.3,Dfn. 1, p. 164]).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 12, §II.1.6,Cor., p. 172] applies verbatim.Remark 2.7.In general, the space H in (2.4) depends on S, cf.[12, p. 169, after Dfn.3].It is nowadays standard to define the direct integral of ζ → H ζ as the one with underlying space S generated (in the sense of Proposition 2.4) by an algebra S of 'simple functions', see e.g.[17, §6.1, p. 61] or the family of Hilbert spaces by Definition 2.11(a), the map ζ → u ζ ζ is ν-measurable for every u ∈ S H by Definition 2.3(a).Analogously, the map ζ → Q injective and non-expansive 2 for every ζ ∈ Z by Lemma 2.1.By Definition 2.11, D α and H are defined on the same underlying space S. Example, p. 182] and the injectivity of ι ζ,α for every ζ ∈ Z and every α > 0, the operator ι α : D α → H is injective.In particular, the composition of ι 1 with the inclusion of D(Q) Thm.1.2].As a consequence, at least in this case, the closability of E in [3, Thm.1.2]followsfromourProposition 2.13.Our first result on direct integrals of concrete quadratic forms is as follows.Remark 2.30.Since (µ ζ ) ζ∈Z is separated, the isometry ι in Proposition 2.25 is a unitary operator, thus there exists ι −1 : H → L 2 (µ), where H is as in(2.19).For the sake of clarity, only in the proof of Proposition 2.29 below, we distinguish between the quadratic form (E, D(E)) on H and the quadratic form Proposition 2.29.Let (X, X , µ) be σ-finite standard, (Z, Z, ν) be σ-finite countably generated, and (µ ζ ) ζ∈Z be a separated pseudo-disintegration of µ over ν.Further let (E, D(E)) be a direct inte-gral of closed quadratic forms ζ → (E ζ , D(E ζ )) compatible with (µ ζ ) ζ∈Z .Then, (E, D(E)) is a Dirichlet form on L 2 (µ) if and only if (E ζ , D(E ζ )) is so on L 2 (µ ζ ) for ν-a.e.ζ ∈ Z.
be a fundamental sequence of ν-measurable vector fields for H, and recall that (u n,ζ ) n is total in L 2 (µ ζ ) for every ζ ∈ Z by Definition 2.3(c).Applying (2.23) to each element u n of this sequence proves the assertion.Proposition 2.29 motivates the next definition.A simple example follows.Definition 2.31.A quadratic form (E, D(E)) on L 2 (µ) is a direct integral of Dirichlet forms ζ → (E ζ , D(E ζ )) on L 2 (µ ζ ) if it is a direct integral of the Dirichlet forms ζ → (E ζ , D(E ζ )) additionally compatible with the separated pseudo-disintegration (µ ζ ) ζ in the sense of Definition 2.26.Example 2.32.Let X = R 2 with standard topology, Borel σ-algebra, and the 2-dimensional [12, §II.2.5, Thm. 1, p. 187], T t is decomposable, and represented by a ν-measurable field of contraction operators ζ → T ζ,t .Finally, in light of [12, §II.2.3, Prop.4, p. 183], it is a straightforward verification that T ζ,t , t > 0, is a strongly continuous symmetric contraction semigroup on H ζ for ν-a.e.ζ ∈ Z, since so is T t .Analogously to the proof of Proposition 2.29, T ζ,t is sub-Markovian for ν-a.e.ζ ∈ Z, since so is T t .
nest witnessing the (quasi-)regularity of the form, i.e. verifying [7, Dfn.2.8].With no loss of generality, up to intersecting F ♯ ζ,k with F ♯ h if necessary, we may assume that for every k there exists h