Extension of monotone operators and Lipschitz maps invariant for a group of isometries

. We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in L p -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.


Introduction
The theory of maximal monotone operators A : X ⇒ X * in Hilbert and reflexive Banach spaces provides a very powerful framework to solve nonlinear equations (see e.g. the review [Bor10]).We recall that an operator A ⊂ X × X * (which we identify with its graph) is said to be monotone if v − w, x − y ≥ 0 for any (x, v), (y, w) ∈ A, while A is said to be maximal monotone if every proper extension of A fails to be monotone.
In the Hilbertian case, the theory can also be applied to differential inclusions of the form d dt x(t) ∈ −Ax(t), x(0) = x 0 (1.1) driven by a maximal monotone operator A and to prove the generation of a semigroup of contractions (see e.g.[Bré73;Bar10]).
The notion of maximality of the (multivalued) operator A plays a crucial role, since by Minty-Browder theorem it is equivalent to the solvability of the resolvent equation where J is the duality map from X to X * [Bar10, Thm.2.2].In the Hilbertian framework, the solution to (1.2) corresponds to the solvability of the Implicit Euler Scheme associated to (1.1) and provides a general condition for the existence of a solution to (1.1).In this respect, an essential tool is the well known fact that every monotone operator A admits a maximal extension [DF64;Bré73], whose domain is contained in the closed convex hull of the domain of A.
Motivated by the study of operators in Bochner-L p spaces X = L p (Ω, B, P; X) (here (Ω, B) is a standard Borel space endowed with a nonatomic probability measure P and X is a reflexive and separable Banach space) which are invariant by measure-preserving transformations of Ω, in this paper we address the general problem of finding maximal extensions of monotone operators which are invariant by a group G of suitable transformations of X × X * .More precisely, let us consider a group G of linear isomorphisms acting on X × X * whose elements U = (U, U ′ ) preserve the duality pairing and the norms in X × X * , i.e. for every (U, U ′ ) ∈ G and every z = (x, v) ∈ X × X * we have Given a monotone operator A ⊂ X × X * which is G-invariant, i.e.
(x, v) ∈ A, (U, U ′ ) ∈ G ⇒ (U x, U ′ v) ∈ A, (1.4) we will prove (see Theorem 2.6) that there exists a maximal extension Â of A preserving the G-invariance; we will also find Â so that its proper domain D( Â) does not exceed the closed convex hull of D(A).Since it is not clear how to adapt to this context the classical proof based on the Debrunner-Flor and Zorn Lemma (see e.g.[Bré73, Thm.2.1 and Cor.2.1, Chap.II]), we will use the powerful explicit construction of [BW09].This is based on kernel averages of convex functionals and on the characterization of monotone and maximal monotone operators via suitable convex Lagrangians on X×X * , a deep theory started with the seminal paper [Fit88] (where the so called Fitzpatrick's function is introduced for the first time) and further developed in a more recent series of relevant contributions (see e.g.[MT01; BS03; Pen04b; Pen04a; Gho08; Vis17] and the references therein).The advantage of this direct approach is that it provides an explicit formula for the extension of A which behaves quite well with respect to the action of the group G.As an intermediate step, which can be relevant also in other applications independently of G-invariance, we will also show (Theorem 2.3) how to modify the construction of [BW09] in order to confine the domain of the extension Â to the closed convex hull of D(A) (see also [Theorem 2.13][BW10] for a partial result in this direction).
As a byproduct, we can adapt the same strategy of [BW10] to prove a version of the Kirszbraun-Valentine extension theorem (see [Kir34;Val43;Val45]) for G-invariant Lipschitz maps in Hilbert spaces (Theorem 2.13): it states that every L-Lipschitz map f : D → H defined in a subset D of an Hilbert space H, whose graph is invariant with respect to the action of a group G of isometries of H, can be extended to a L-Lipschitz function f : H → H which is G-invariant as well.The basic idea here still goes back to Minty: the graphs of nonexpansive maps in Hilbert spaces are in one-to-one correspondence with graphs of monotone maps via the Cayley transformation T : H × H → H × H defined as T (y, w) := 1 √ 2 (y − w, y + w). (1.5) It is worth noticing that such a correspondence allowed the authors of [RS05] to use for the first time the Fitzpatrick's function to prove the Kirszbraun-Valentine theorem (see [Kir34;Val43;Val45]), which states that every 1-Lipschitz continuous map can be extended to the whole H (see also [Bau07] where this approach is improved in order to obtain an extension with an optimal range).The same correspondence, together with the explicit construction of a maximal extension of a monotone operator A in [BW09], is used in [BW10] to provide the first constructive proof of the Kirszbraun-Valentine theorem.
We will also provide in the appendix an alternative proof based on another more recent explicit formula for such kind of extension given by [ALM21] (see also [ALM18]).
These results, besides being interesting by themselves, find interesting applications in the case when X is the L p -space of random variables X = L p (Ω, B, P; X), X * = L p * (Ω, B, P; X * ), p, p * ∈ (1, +∞), 1 p + 1 p * = 1, (1.6) over a space of parametrizations (Ω, B, P), where (Ω, B) is a standard Borel space, P is a nonatomic probability measure, and X is a separable and reflexive Banach space, while G is a group of isomorphisms generated by measure-preserving maps, i.e.B − B measurable maps g : Ω → Ω which are essentially injective and such that g ♯ P = P, where g ♯ P denotes the pushforward of P by g.Every measure-preserving isomorphism g induces an element U g of G whose action on (X, X ′ ) ∈ X × X * is simply given by U g (X, X ′ ) = (X • g, X ′ • g).
The interest for invariance by measure-preserving isomorphisms in X × X * is justified by its link with the stronger property of law invariance: a set A ⊂ X × X * is law invariant if whenever (X, X ′ ) ∈ A then A also contains all the pairs (Y, Y ′ ) ∈ X × X * with the same law of (X, X ′ ), i.e. (Y, Y ′ ) ♯ P = (X, X ′ ) ♯ P. It is clear that law invariant subsets of X × X * are also invariant by measure-preserving isomorphisms; using the results of [BG03], we will show that the converse implication holds for closed sets: therefore for closed sets these two properties are in fact equivalent.Since the graph of a maximal monotone operator is closed, we obtain that a monotone operator in X × X * whose graph is invariant by the action of measure-preserving isomorphisms admits a maximal monotone extension which is law invariant (Theorem 4.5).This framework is exploited in Section 3 (where we study the approximation of transport maps and plans by various classes of measure-preserving isomorphisms) and Section 4.
The Hilbertian setting when p = p * = 2 and X is an Hilbert space (so that X = L 2 (Ω, B, P; X) is a Hilbert space as well that can be identified with its dual X * ) provides an important case, which we will further exploit in [CSS23].It turns out that maximal dissipative operators B on L 2 (Ω, B, P; X), invariant by measure-preserving isomorphisms, are the Hilbertian counterparts of maximal totally dissipative operators on the Wasserstein space P 2 (X) of laws, where P 2 (X) denotes the space of Borel probability measures with finite second moment endowed with the so-called Kantorovich-Rubinstein-Wasserstein distance W 2 .The results obtained in Section 4 in the framework (1.6) are used in [CSS23] to develop a well-posedness theory for dissipative evolution equations in the metric space (P 2 (X), W 2 ), together with a Lagrangian characterization for the solution of the corresponding Cauchy problem.
Besides the direct application of the general invariance extension result provided in Section 2, in Section 4 we also analyze further properties of Lipschitz functions and maximal dissipative operators on X = L 2 (Ω, B, P; X), which are invariant by measure-preserving isomorphisms.In particular we prove that the effect of a Lipschitz invariant map L : X → X on an element X ∈ X can always be represented as LX(w) = l(X(w), X ♯ P) for a.e.ω ∈ Ω where l : S(X) → X is a (uniqueley determined) continuous map defined in whose sections l(•, µ) are Lipschitz as well, for every µ ∈ P 2 (X).
An important application of these results concerns the resolvent operator, the Moreau-Yosida approximation, and the semigroup associated with a maximal dissipative invariant operator B in L 2 (Ω, B, P; X), for which we obtain new relevant representation formulae (Theorem 4.13).
The above structural characterizations rely on various approximation properties for couplings between probability measures in terms of maps and measure-preserving transformations.We collect them in Section 3, with the aim to present many important results available in the literature (cf.

Extension of monotone operators and Lipschitz maps invariant by a group of isometries
Let X be a reflexive Banach space with norm | • | and let X * be its dual endowed with the dual norm | • | * .We denote by c : X * × X → R, the duality pairing •, • between X * and X and by Z the product space X × X * with dual Z * := X * × X.A (multivalued) operator A : X ⇒ X * (which we identify with its graph, a subset of X × X * ) is monotone if it satisfies v − w, x − y ≥ 0 for every (x, v), (y, w) ∈ A. (2.1) The proper domain D(A) ⊂ X of A is just the projection on the first component of (the graph of) A. A monotone operator A is maximal if any monotone operator in X × X * containing A coincides with A.
In order to address the extension problem of monotone operators A ⊂ X × X * invariant by the action of a group of isometric isomorphisms, it is crucial to have some explicit formula providing a maximal extension of A. In this respect, the characterization of monotone and maximal monotone operators by means of suitable "contact sets" of convex functionals in X×X * , started with the seminal paper [Fit88] and further developed in a more recent series of relevant contributions (see e.g.[MT01; BS03; Pen04b; Pen04a; Gho08; Vis17] and the references therein), and the kernel averaging operation developed by [BW09] provide extremly powerful tools, that we are going to quickly recall in the next section.We will also show how to slightly improve this construction in order to obtain an explicit formula providing a maximal extension of A whose domain is contained in the closed convex hull of D(A).In this connection, we mention that the existence of a maximal extension of A with the desired above-mentioned optimality for the domain can be deduced by [Bau07], thanks to the correspondence revealed by Minty between monotone operators and firmly nonexpansive mappings.Indeed, [Bau07] uses the Fitzpatrick function to prove the Kirszbraun-Valentine extension theorem for firmly non-expansive mappings with optimal range localization.However, part of the proof still relies on Zorn's Lemma and it is not entirely constructive.

Maximal extensions of monotone operators by self-dual Lagrangians
Following the presentation of [BW09] and [Pen04b], given a set A ⊂ X × X * and its indicator function I A , we consider the proper function c A : Notice that c A has an affine minorant if A is monotone, in the sense that where (x 0 , v 0 ) ∈ A is an arbitrary given point.
Recalling that the convex conjugate g * : Z * → (−∞, +∞] of a proper function g : Z → (−∞, +∞] with an affine minorant is defined as with an analogous definition in the case of a function h : Z * → (−∞, +∞], we can introduce the Fitzpatrick function f A : Z → (−∞, +∞] and the convex l.s.c.relaxation p A : 2) It will be often useful to switch the order of the components of elements in X × X * : we will denote by s : X × X * → X * × X the switch map In particular c ⊤ is the duality pairing between X and X * .We collect in the following statement some useful properties.
Theorem 2.1 (Representation of monotone operators). (1) and where ∂f denotes the subdifferential of f .An analogous statement holds for g : (2.6) provides a maximal monotone extension of A.
+∞] is a convex l.s.c.function satisfying the self-duality property (2.9) then f ≥ c ⊤ and the contact set A f is maximal monotone.
Proof.We give a few references and sketches for the proof.
(1) The inclusion in (2.5) follows by [Fit88,Theorem 2.4] and shows in particular that A f is monotone since T f is monotone by [Fit88, Proposition 2.2] .Clearly f ⊤ ≤ c A f so that, passing to the conjugates, the reverse inequality follows.
The previous result suggests a strategy (cf.Theorem 2.2 below) to construct a maximal extension of a given monotone operator A starting from a convex and l.s.c.function f (2.10) Using the kernel average introduced in [BW09] one obtains a self-dual Lagrangian R f which satisfies f A ≤ R f ≤ p ⊤ A , so that the contact set of R f is a maximal monotone extension of A. We introduce a function ψ : Z → [0, +∞) satisfying the simmetry and self-duality condition ψ = ψ ∨ = (ψ * ) ⊤ , (2.11) where ψ ∨ (z) := ψ(−z) for every z ∈ Z.The assumption in (2.11) in particular implies that ψ is continuous, convex, and ψ(0) = ψ * (0) = 0, (2.12) where p, p * ∈ (1, +∞) are given conjugate exponents. (2.13) The following result is an immediate consequence of [BW09, Fact 5.6, Theorem 5.7, Remark 5.8].
(2) The operator Ã defined as the contact set of R f is a maximal monotone extension of A.
We want to show that, for a suitable choice of f as in the previous theorem, we can produce a maximal monotone extension of A with domain included in D := co (D(A)).We claim that such f can be defined as where Theorem 2.3 (A maximal monotone extension with minimal domain).Let A ⊂ X × X * be a monotone operator and let f : Z → (−∞, +∞] be as in (2.16).The following hold: (1) f is a l.s.c. and convex function such that (3) defining R f as in (2.14) and its contact set Ã as in (2.15), D(R f ) ⊂ C = D × X * and Ã provides a maximal extension of A with domain D( Ã) ⊂ D. Proof.
(1) It is clear that f is convex and lower semicontinuous and f ≥ f A .On the other hand, f ⊤ ≤ c A by (2.6) and since Let us compute the conjugate f * of f : by Fenchel-Rockafellar duality theorem, (see e.g.[Roc97, Theorem 16.4]) we have that C ) where cl denotes the lower semicontinuous envelope of a given function (i.e.cl(h) is the largest lower semincontinuous function staying below h) and denotes the inf-convolution (or epigraphical sum) of two functions: if k, j : Z * → (−∞, +∞] then k j is defined as Since I C (x, v) = I D (x), we can easily compute its dual for every z = (v, x) ∈ X * × X where I 0 is the indicator function of the singleton {0} and σ D is the support function of D defined by σ D (v) := sup We thus have (3) By the first claim and Theorem 2.2 with f as in (2.16), we obtain that Ã is a maximal monotone extension of A. We only need to check that D( Ã) ⊂ D = co (D(A)).Since it is clear from the definition of contact set that D x 2 belong to the convex set D by (2) and therefore x ∈ D as well.
2.2.Extension of monotone operators invariant w.r.t. the action of a group of linear isomorphisms of X × X * We will now focus on operators which are invariant with respect to a group G of bounded linear isomorphisms of Z of the form U = (U, U ′ ) : Z → Z.For every z = (x, v) ∈ Z we thus have and we assume that all the maps U ∈ G satisfy the following properties for a fixed self-dual function ψ : Z → [0, +∞) as in (2.11).Notice that the first identity in (2.18) implies with the inverse transpose of U with respect to the duality paring between In particular, we have the formula The following simple result clarifies the relation between G-invariance of monotone operators and G-invariance of the corresponding Lagrangian functions.
(1) We simply have for every where we applied (2.20), the fact that U −1 ∈ G, the G-invariance of f and the fact that {z = U −1 z : z ∈ Z} = Z.
(2) One immediately sees that c A is G invariant thanks to the G-invariance of A and (2.18).The invariance of f A and of p A then follows by the previous claim. ( (4) We first observe that the function thanks to the invariance of f , the invariance of f * from claim (1), and the invariance property of ψ stated in (2.18).Since U is a linear isomorphism we also have Combining the above identities, we immediately get We can now obtain our main result.
Theorem 2.6 (G-invariant maximal monotone extensions).Let A ⊂ X × X * be a G-invariant monotone operator with D := co (D(A)) .Then the function f given by (2.16) is G-invariant and, defining R f as in (2.14) and its contact set Ã as in (2.15), then Ã is a G-invariant maximal monotone extension of A with domain included in co (D(A)).
Proof.Let us first observe that By claim (2) of Proposition 2.5, we deduce that the function f given by (2.16) is G-invariant.
The invariance of Ã then follows by applying Claims (4) and (3) of Proposition 2.5, recalling that R f ≥ c ⊤ by Theorem 2.1(6).We conclude by Theorem 2.3.

Extension of invariant dissipative operators in Hilbert spaces
We now quickly apply the results of the previous section 2.2 to the particular case of dissipative operators in Hilbert spaces.We adopt the dissipative viewpoint in view of applications to differential equations, but clearly all our statements apply to monotone operators as well.The main reference is [Bré73].
We consider a Hilbert space H with norm |•|, scalar product •, • , and dual H * which we identify with We say that a λ-dissipative operator B is maximal if B is maximal w.r.t.inclusion in the class of λ-dissipative operators or, equivalently, if −B λ is a maximal monotone operator.
As an immediate application of Theorem 2.6, we obtain an important result for dissipative operators which are invariant with respect to the action of a group G of isometries.
Theorem 2.8 (Extension of invariant λ-dissipative operators).Let G H be a group of linear isometries of H and let Then there exists a maximal λ-dissipative extension B of B with D( B) ⊂ co (D(B)) which is G-invariant as well.
Proof.We can choose the self-dual kernel and we immediately get that G satisfies (2.18).The statement is then an immediate application of Theorem 2.6 to the G-invariant monotone operator A := −B λ .
Remark 2.9.If B is a G-invariant λ-dissipative operator which is maximal (with respect to inclusion) in the collection of G-invariant λ-dissipative operators, then B is maximal λ-dissipative.
We now derive a few more properties concerning the resolvent, the Yosida regularization and the minimal selection of a maximal λ-dissipative and G-invariant operator B ⊂ H × H.For references on the corresponding definitions and properties we refer to [Bré73], where the theory is developed in detail for the case λ = 0.If λ = 0 analogous statements can be obtained and we refer the interested reader to [CSS23, Appendix A].
Recall that for every 0 < τ < 1/λ + the resolvent The Yosida approximation of B is defined by and the uniqueness property of the resolvent operator.The exponential formula (cf.(2.28)) yields the G-invariance of S t .

Extension of invariant Lipschitz maps in Hilbert spaces
We conclude this general discussion concerning G-invariant sets and maps addressing the problem of the global extension of a G-invariant Lipschitz map defined in a subset of a separable Hilbert space H.As in Theorem 2.8, we consider a group G H of isometric isomorphisms of H inducing the group In addition to Definition 2.4, we also give the following definition.
Definition 2.11.[G H -invariance for H-valued maps] We say that a function f : for every x ∈ D we have U x ∈ D and f (U x) = U f (x), for every U ∈ G H . (2.29) The Val45] states that every Lipschitz function f : D → H defined in a subset D of H with Lipschitz constant L ≥ 0 admits a Lipschitz extension F : H → H whose Lipschitz constant coincides with L. We want to prove that it is possibile to find such an extension preserving G-invariance.
Our starting point is the following well known fact, going back to Minty (see also [AA99]).We introduce the isometric Cayley transforms T, T −1 : Lemma 2.12 (Lipschitz and monotone graphs).Let F , A be two subsets of H × H such that A = T (F ).The following two properties are equivalent: (1) F is the graph of a nonexpansive map f defined on the set D given by (2) A is monotone.Moreover, assuming that A is monotone, the following hold (i) A is maximal monotone if and only if Proof.Let us take a pair of elements (x i , v i ) = T (y i , w i ) ∈ A, i = 1, 2; we have (2.33)This proves the first statement.Concerning (i), it is sufficient to recall that the domain D of f coincides with the image of h(A) where h(x, v) := 1 √ 2 (x + v) and we know that A is maximal monotone if and only if such an image coincides with H (cf. [Bré73, Proposition 2.2 (ii)]).The second property (ii) is an immediate consequence of the invertibility of T and of the linearity of the transformations of G so that for every Let us eventually consider claim (iii): we clearly have On the other hand, since both F ′ and F are the graph of a nonexpansive map, F ′ ∩ (D × H) = F ∩ (D × H) and therefore the restriction of f ′ to D coincides with f .We can now state our result concerning the extension of G-invariant Lipschitz maps.
Theorem 2.13 (Extension of G-invariant Lipschitz maps).Let us suppose that f : D → H is L-Lipschitz and G H -invariant according to (2.29).Then there exists a L-Lipschitz map f : H → H extending f which is G H -invariant as well.
Proof.Up to a rescaling, it is not restrictive to assume that L = 1 so that f is nonexpansive.Let F ⊂ H × H be the graph of f and let A := T (F ).By Lemma 2.12 we know that A is a monotone G-invariant operator We can now apply Theorem 2.8 to find a maximal monotone extension Â of A which is still G-invariant.Setting F := T −1 ( Â) we can eventually apply Lemma 2.12 to obtain that F is the graph of a nonexpansive map f : D → H.Moreover, Claim (i) shows that D = H so that f is globally defined, Claim (ii) shows that f is G H -invariant, and Claim (iii) ensures that f is an extension of f .

Borel partitions and almost optimal couplings
In this section, we collect some useful results concerning Borel isomorphisms and partitions of standard Borel spaces.These results, besides being interesting by themselves, will also turn out to be useful in Section 4, where we will deal with a particular group of isometric isomorphisms on Banach spaces of L p -type.We start by fixing the fundamental definitions and notations involved in the statements of the main theorems of this Section, which are presented in Section 3.2.These results concern the approximation of arbitrary couplings between probability measures by couplings which are concentrated on maps, through the action of measure-preserving transformations.In particular, Corollary 3.20 concerns the approximation of bistochastic measures by the graph of measurepreserving maps and it is written here in the general context of a standard Borel space (Ω, B) endowed with a nonatomic probability measure P.This result relies on the analogous property stated for the d-dimensional Lebesgue measure in [BG03, Theorem 1.1].In the same spirit of Corollary 3.20, Corollary 3.21 provides an approximation result for the law of a pair of measurable random variables defined on (Ω, B, P) with values in a pair of separable Banach spaces.A consequence of this result is the key lemma [Car13, Lemma 6.4] (cf.also [CD18, Lemma 5.23 p. 379]), which states that if X and Y are random variables with the same law, then X can be approximated by Y through the action of a sequence of measure-preserving transformations.Finally, we reported a fundamental result in Optimal Transportation Theory, concerning the equivalence between the Monge and the Kantorovich formulations (see [Pra07,Theorem B]).This is the content of Proposition 3.23 where the result is written using the language of random variables (cf.also [GT19, Lemma 3.13]), so as to be easily recalled in Section 4. In order to introduce all the technical tools used to state and prove all these properties, in Section 3.1 we list some well-known facts about standard Borel spaces.Definition 3.1 (Standard Borel spaces and nonatomic measures).A standard Borel space (Ω, B) is a measurable space that is isomorphic (as a measure space) to a Polish space.Equivalently, there exists a Polish topology τ on Ω such that the Borel sigma algebra generated by τ coincides with B. We say that a positive finite measure m on (Ω, B) is nonatomic (also called atomless or diffuse) if m({ω}) = 0 for every ω ∈ Ω (notice that {ω} ∈ B since it is compact in any Polish topology on Ω).
We notice that, being (Ω, B) standard Borel, m is nonatomic if and only if for every B ∈ B with m(B) > 0 there exists Definition 3.2 (Partitions).If (Ω, B) is a standard Borel space and N ∈ N, a family of subsets If (Ω, B) is a standard Borel space endowed with a nonatomic, positive finite measure m, we denote by S(Ω, B, m) the class of B-B-measurable maps g : Ω → Ω which are essentially injective and measure-preserving, meaning that there exists a full m-measure set Ω 0 ∈ B such that g is injective on Ω 0 and g ♯ m = m, where g ♯ m is the push forward of m through g.We recall that, if X and Y are Polish spaces, f : X → Y a Borel map and µ is a nonnegative and finite measure on X, then f ♯ µ is defined by for every ϕ : Y → R bounded (or nonnegative) Borel function.
If A ⊂ B is a sigma algebra on Ω we denote by S(Ω, B, m; A) the subset of S(Ω, B, m) of A − A measurable maps.Finally Sym(I N ) denotes the set of permutations of I N i.e. bijective maps σ : We consider the partial order on N given by where m | n means that n/m ∈ N. We write m ≺ n if m ≺ n and m = n.
Definition 3.3 (Segmentations).Let (Ω, B) be a standard Borel space endowed with a nonatomic, positive finite measure m and let N ⊂ N be an unbounded directed set w.r.t.≺.
We say that a collection of partitions (P N ) N ∈N of Ω, with corresponding sigma algebras (3.3) Corollary 3.7.Let (Ω, B) be a standard Borel space endowed with a nonatomic, positive finite measure m and (Ω ′ , B ′ ) be a standard Borel space.Then for every nonatomic, positive measure µ on (Ω ′ , B ′ ) such that µ(Ω ′ ) = m(Ω), there exists a measurable map Lemma 3.8 (Existence of N-segmentations).For any standard Borel space (Ω, B) endowed with a nonatomic, positive finite measure m and any unbounded directed set N ⊂ N w.r.t.≺, there exists a N-segmentation of (Ω, B, m).
where the subscript I denotes the restriction to ∪ k∈I Ω N,k or ∪ k∈I Ω ′ N,k .
Proof.Applying Theorem 3.6 to the standard Borel spaces (Ω {k} , B {k} ) and (Ω ′ {k} , B ′ {k} ) endowed, respectively, with the nonatomic, positive finite measures m {k} and m ′ {k} for every k ∈ I N , we obtain the existence of measurable functions Corollary 3.11 (Lifting permutations to isomorphisms).Let (Ω, B, m) be a standard Borel space endowed with a nonatomic, positive finite measure m and let there exists a measure-preserving isomorphism g ∈ S(Ω, B, m; σ(P N )) such that where g k is the restriction of g to Ω N,k .
Proof.Applying Corollary 3.11 to the standard Borel space (Ω 0 ∪ Ω 1 , B| Ω 0 ∪Ω 1 ) endowed with the nonatomic, positive finite measure m| Ω 0 ∪Ω 1 with the 2-Borel partition P 2 = {Ω k } k=0,1 and σ sending 0 to 1, we obtain the existence of a measure-preserving isomorphism g ∈ S( where gi is the restriction of g to Ω k , k = 0, 1.It is then enough to define g : Ω → Ω as The next result is a particular case of Doob's Martingale Convergence Theorem for Banachvalued maps, see [Str11, Theorem 6.1.12].We recall that a filtration on (Ω, B) is a sequence (F n ) n∈N of sub-sigma algebras of B such that F n ⊂ F n+1 .
Theorem 3.13.Let (Ω, B) be a standard Borel space endowed with a nonatomic, positive finite measure m, (F n ) n∈N be a filtration on (Ω, B) such that σ ({F n | n ∈ N}) = B, let X be a separable Banach space and let p ∈ [1, ∞).Then, given X ∈ L p (Ω, B, m; X), the X-valued martingale both m-a.e. and in L p (Ω, B, m; X).
In general, the collection of sigma-algebras (B N ) N ∈N associated with a segmentation according to Definition 3.3 is not a filtration since it fails to be ordered by inclusion (recall Remark 3.4).However, it is always possible to extract from (B N ) N ∈N a filtration still satisfying item (4) in Definition 3.3.More precisely we have the following result.
Lemma 3.14 (Cofinal filtrations).Let N ⊂ N be an unbounded directed subset w.r.t.≺.Then there exists a totally ordered cofinal sequence • for every N ∈ N there exists n ∈ N such that N | b n .In particular, for every N-refined standard Borel measure space (Ω, B, m, (P N ) N ∈N ) it holds that (B bn ) n∈N is a filtration on (Ω, B), For every p ∈ [1, +∞) and every separable Banach space X, we thus have that In the next Lemma we show that, given two distinct points ω, ω ′′ , they can always be separated by some partition P N for N ∈ N sufficiently large.Lemma 3.15 (Separation property).Let (Ω, B, m, (P N ) N ∈N ) be a N-refined standard Borel measure space.Then there exists Ω 0 ∈ B with m(Ω 0 ) = 0 such that for every ω Proof.Let (b n ) n ⊂ N be a totally ordered cofinal sequence as in Lemma 3.14 and let τ be a Polish topology on Ω such that B coincides with the Borel sigma algebra generated by τ .By [Bog06, Proposition 6.5.4] there exists a countable family F of τ -continuous functions f : Ω → [0, 1] separating the points of Ω, meaning that for every ω ′ , ω ′′ ∈ Ω, ω ′ = ω ′′ there exists f ∈ F such that f (ω ′ ) = f (ω ′′ ).Since F ⊂ L 2 (Ω, B, m; R), by Theorem 3.13 with F n := B bn , for every f ∈ F there exists a m-negligible set Ω f such that we conclude that the points ω ′ and ω ′′ belong to different elements of P M , and therefore they also belong to different elements of P N for every N ∈ N multiple of M .Proposition 3.16 (Segmentation preserving isomorphisms).Let (Ω, B, m, (P N ) N ∈N ) and (Ω ′ , B ′ , m ′ , (P ′ N ) N ∈N ) be N-refined standard Borel measure spaces such that m(Ω) = m ′ (Ω ′ ).Then there exist two measurable functions ϕ : Ω → Ω ′ and ψ : Ω ′ → Ω such that for every N ∈ N and every I ⊂ I N it holds where the subscript I denotes the restriction to Proof.By Lemma 3.14, it is enough to prove the statement in case N = (b n ) n , where (b n ) n ⊂ N is strictly ≺-increasing sequence and (Ω ′ , B ′ , m ′ , (P ′ N ) N ∈N ) is ([0, 1), B([0, 1)), λ c , (I N ) N ∈N ) as in Example 3.5 with c = m(Ω).By Lemma 3.10, we can find for every n ∈ N two measurable maps ϕ n : Ω → [0, 1) and ψ n : [0, 1) → Ω satisfying the thesis of Lemma 3.10 for the standard Borel spaces (Ω, B) and ([0, 1), B([0, 1)) endowed with nonatomic, positive and finite measures m and λ c respectively and the m − λ c compatible b n -partitions of (Ω, B) and ([0, 1), B([0, 1))) given by P bn and I bn , where we recall from Example 3.5 that I bn = (I bn,k ) k∈I bn with ) is Cauchy, hence converges.We thus have the existence of a measurable map ϕ : Ω → [0, 1) such that since for m sufficiently large (ϕ m ) ♯ m| Ω bn,k = λ c | I bn,k by Lemma 3.10.This shows that ϕ ♯ m| Ω bn,k = λ c | I bn,k for every k ∈ I bn and every n ∈ N. To conclude it is enough to show that ϕ is m-essentially injective.Let Ω 0 ⊂ Ω be the m-negligible subset of Ω given by Lemma 3.15 and let Ω 1 := ϕ −1 (J), where Then there exists M ∈ N such that ω ′ and ω ′′ belong to different elements of P bn for every n ≥ M .By (3.4) and Lemma 3.15, we can find k it must be that ϕ(ω ′ ) = ϕ(ω ′′ ).

Approximation of couplings by using measure-preserving isomorphisms
If X is a Polish space, we denote by P(X) the space of Borel probability measures on X which is endowed with the weak (or narrow) topology: a sequence (µ n ) n ⊂ P(X) converges to µ ∈ P(X) if lim for every ϕ : X → R continuous and bounded.In this case, we write µ n → µ in P(X).
If X, Y are Polish spaces and (µ, ν) ∈ P(X) × P(Y ), we define the set of admissible transport plans Γ(µ, ν) where π i , i = 1, 2, denotes the projection on the i-th component and we call π i ♯ γ the i-th marginal of γ.
The following result is an application of [BG03, Theorem 1.1].We will use the following notation: if X 1 and X 2 are sets and X 1 : Theorem 3.18 (Approximation of bistochastic couplings).Let (Ω, B, P, (P N ) N ∈N ) be a Nrefined standard Borel probability space.Then for every γ ∈ Γ(P, P) there exist a totally ordered strictly increasing sequence (N n ) n ⊂ N and maps g n ∈ S(Ω, B, P; B Nn ) such that, for every separable Banach spaces Z, Z ′ and every Z ∈ L 0 (Ω, B, P; Z), Z ′ ∈ L 0 (Ω, B, P; Z ′ ), it holds (3.12) Proof.By Lemma 3.14 it is not restrictive to assume that N = (b n ) n for a totally ordered strictly increasing sequence (b n ) n , n ∈ N. We divide the proof in several steps.
Corollary 3.20.Let (Ω, B, P, (P N ) N ∈N ) be a N-refined standard Borel probability space.Then for every γ ∈ Γ(P, P) there exist a totally ordered strictly increasing sequence (N n ) n ⊂ N and maps g n ∈ S(Ω, B, P; B Nn ) such that, for every Polish topology τ on Ω generating B, it holds where τ ⊗ τ is the product topology on Ω × Ω.
Proof.By Theorem 3.18 and Remark 3.19, we have the existence of a strictly increasing sequence (N n ) n ⊂ N and maps g n ∈ S(Ω, B, P; B Nn ) such that, choosing the separable Hilbert space R, we get . Since the range of ϕ i is bounded and thus relatively compact, by the P(R 2 ) convergence we get that for every continuous function h : R 2 → R where γ n = (i Ω , g n ) ♯ P, n ∈ N. Choosing h(x, y) := xy, we get that τ ) be a unital subalgebra whose induced initial topology on Ω coincides with τ (e.g. the subset of d-Lipschitz continuous and bounded functions for a complete distance d inducing τ ).It is easy to check that is a unital subalgebra whose induced initial topology on Ω × Ω coincides with τ ⊗ τ .By (3.15) we thus have that We conclude by [SS22, Lemma 2.3].
The second part of the following corollary represents a sort of extension of the known result in [Car13, Lemma 6.4] (cf.also [CD18, Lemma 5.23 p. 379]) to the class of pairs (X, Y ) of random variables, where X and Y take values on possibly different separable Banach spaces, with possibly different p-integrability.Whenever the joint distribution of two pairs (X, Y ), (X ′ , Y ′ ) belonging to such a class is equal, we are able to prove the existence of a sequence of measure-preserving maps giving the desired strong approximation result for both the components.
Corollary 3.21.Let (Ω, B, P, (P N ) N ∈N ) be a N-refined standard Borel probability space, let Z and Z ′ be separable Banach spaces and let Z ∈ L 0 (Ω, B, P; Z), Z ′ ∈ L 0 (Ω, B, P; Z ′ ).Then for every µ ∈ Γ(Z ♯ P, Z ′ ♯ P) there exist a totally ordered strictly increasing sequence (N n ) n ⊂ N and maps g n ∈ S(Ω, B, P; B Nn ) such that (3.16) In particular, if X and Y are separable Banach spaces, X, X ′ ∈ L p (Ω, B, P; X), Y, Y ′ ∈ L q (Ω, B, P; Y), p, q ∈ [1, +∞), and (X, Y ) ♯ P = (X ′ , Y ′ ) ♯ P, then there exist a totally ordered strictly increasing sequence (N n ) n ⊂ N and maps g n ∈ S(Ω, B, P; B Nn ) such that Proof.Let µ := Z ♯ P and µ ′ := Z ′ ♯ P; let us first observe that there exists γ ∈ Γ(P, P) such that (Z ⊗ Z ′ ) ♯ γ = µ.In fact, we can disintegrate P w.r.t.Z and µ (see e.g.[AGS08, Theorem 5.3.1]for details on the disintegration theorem), obtaining a Borel family of measures (θ z ) z∈Z ⊂ P(Ω) such that θ z is concentrated on Z −1 (z) for µ-a.e.z ∈ Z and P = θ z dµ(z); similarly, we can also find (θ Notice also that it is enough to verify that (Z, Z ′ ) ♯ P and γ have the same integral for every function of the form ϕ 1 ⊗ ϕ 2 as above to conclude that the two measures coincide (see e.g. the proof of the above Corollary 3.20).We can then apply (3.12) and obtain (3.16).
Let us show the last part of the statement: we take Z = Z ′ := X × Y and, by (3.16) with We thus have that (X, ).As a consequence (see e.g.[AGS08, Proposition 7.1.5,Lemma 5.1.7]),we get The proof for Y and Y ′ is identical.
Remark 3.22.In the same setting of Corollary 3.21 and similarly to Remark 3.19, if Z = Z ′ = X for a separable Banach space X and X, X ′ ∈ L p (Ω, B, P; X), p ∈ [1, +∞), then (3.11) gives that the convergence in (3.16) holds in P p (X 2 ), As a byproduct, we recover the following important result (see e.g.[GT19, Lemma 3.13] for a statement in case p = 2 and X = R d ), which is also related to the equivalence between the Monge and the Kantorovich formulations of Optimal Transport problems [Amb03, Theorem 2.1, Theorem 9.3], [Pra07, Theorem B].
Proof.Let us consider the N-refined standard Borel probability space (Ω, B, P, (P N ) N ∈N ) with N = (2 k ) k∈N ; let µ ∈ Γ o (µ, ν) and let X ′ ∈ L p (Ω, B, P; X) be such that X ′ ♯ P = ν.By Corollary 3.21 and Remark 3.22, there exist a strictly increasing sequence (N n ) n ⊂ N and maps g n ∈ S(Ω, B, P; B Nn ) such that (X, Thus given ε > 0 it is always possible to find n ∈ N sufficiently large such that Y := X ′ • g n satisfies the thesis. 4. Monotone-dissipative operators and Lipschitz maps in L p -spaces invariant by measure-preserving transformations Let (Ω, B) be a standard Borel space endowed with a nonatomic probability measure P (see Definition 3.1).We denote by S(Ω) the class of B-B-measurable maps g : Ω → Ω which are essentially injective and measure-preserving, i.e. there exists a full P-measure set Ω 0 ∈ B such that g is injective on Ω 0 and g ♯ P = P.If g ∈ S(Ω), there exists g −1 ∈ S(Ω) (defined up to a P-negligible set) such that g −1 • g = g • g −1 = i Ω P-a.e. in Ω.Consider two separable Banach spaces X, Y and fix exponents p, q ∈ [1, +∞).We set X := L p (Ω, B, P; X), Y := L q (Ω, B, P; Y). (4.1) Notice that for every g ∈ S(Ω) the pullback transformation g * : X → X sending X to X • g is a linear isometry of X: in particular X n → X weakly in X =⇒ g * X n → g * X weakly in X. (4.3) The aim of this section is to study properties of maps and sets/operators, defined on these particular spaces, which are invariant by measure-preserving transformations.We will also apply the results of Section 2 to this particular setting.The interest on such kind of properties is made evident by the implications in the study of dissipative evolutions in Wasserstein spaces via Lagrangian representations (cf.[CSS23]).
Definition 4.1 (Invariant sets and maps).We say that a set B ⊂ X× Y is invariant by measurepreserving isomorphisms if for every g ∈ S(Ω) it holds where, with a slight abuse of notation, we denote with the same symbol g * both the pullback transformation induced by g on X and on Y.
A (single valued) operator L : D(L) ⊂ X → Y is invariant by measure-preserving isomorphisms (resp.law invariant) if its graph in X × Y is invariant by measure preserving isomorphisms (resp.law invariant).
It is easy to check that a law invariant set or operator is also invariant by measure-preserving isomorphisms.It is also immediate to check that an operator L : D(L) ⊂ X → Y is invariant by measure-preserving isomorphisms if for every g ∈ S(Ω) and every X ∈ D(L) it holds Remark 4.2.Notice that, when Y = X * are reflexive, p > 1, and q = p * is the conjugate exponent of p, then Y = X * and the notion of invariance by measure-preserving isomorphisms for B ⊂ X × Y coincides with the G-invariance of Definition 2.4, G being the group of isometric isomorphisms induced by S(Ω) via g * : X × X * → X × X * with g * (X, V ) = (X • g, V • g) for every (X, V ) ∈ X × X * and every g ∈ S(Ω).
Let us denote by ι : X → P p (X) the push-forward operator, ι(X) := X ♯ P (cf.(3.1)).We frequently use the notation ι X = ι(X).The map ι induces a one-to-one correspondence between subsets in P p (X) (see (3.9)) and law invariant subsets of X.
As a first result we show that for closed sets and continuous operators the two notions of invariance are in fact equivalent.As an application of the results in Section 2, we obtain the following extension results.
Theorem 4.5 (Maximal extensions of monotone operators invariant by measure-preserving isomorphisms).Let X be a separable and reflexive Banach space, p ∈ (1, ∞), X := L p (Ω, B, P; X).If A ⊂ X × X * is a monotone operator invariant by measure-preserving isomorphisms, then there exists a maximal monotone extension of A which is invariant by measure-preserving isomorphisms (and therefore also law invariant) whose domain is contained in co (D(A)) .
Proof.The thesis follows by applying Theorem 2.6 and Remark 4.2.Recall also that if A is maximal monotone, then it is closed (see e.g.[Bar10, Proposition 2.1]) hence we can apply Proposition 4.3.
Theorem 4.6 (Extension of Lipschitz and λ-dissipative invariant graphs).Assume p = 2 and let X be a separable and Hilbert space.The following hold: (1) if L : D(L) ⊂ X → X is a L-Lipschitz function invariant by measure-preserving isomorphisms, then there exists a L-Lipschitz extension L : X → X, defined on the whole X which is invariant by measure-preserving isomorphisms; (2) if B ⊂ X×X is a λ-dissipative operator which is invariant by measure-preserving isomorphisms, then there exists a maximal λ-dissipative extension B of B which is invariant by measure-preserving isomorphisms (and therefore also law invariant) whose domain is contained in co (D(B)) .
Proof.The assertion is an immediate application of Theorems 2.13, 2.8, choosing H := X and G H as the group of isometric isomorphisms induced by S(Ω) via g * : X → X with g * X = X • g for every X ∈ X and every g ∈ S(Ω) (cf.Remark 4.2).Notice that B is closed being maximal dissipative hence law invariant by Proposition 4.3.
We conclude this section with some useful representation results for various classes of law invariant transformations.Given a set D ⊂ P p (X), we set and we just write S(X) = S(X, P p (X)).The set S(X) is of G δ type (see [FSS23,Formula (4.3)]) so that S(X, D) is a Borel set, if D is Borel.
We state a first result on uniqueness of a representation of a single-valued operator L : X → Y by a map from S(X) to Y.We then state existence (and uniqueness) of such map-representation when L is Lipschitz continuous and invariant, showing also further properties inherited by such map-representation.Finally, we will go back to the case of (possibly multivalued) maximal λ-dissipative operators in the final Theorem 4.13.• for every X ∈ D(L) LX(w) = f i (X(w), X ♯ P) for a.e.ω ∈ Ω, then f 1 = f 2 .
Since X(Ω 0 ) is dense in supp(µ) we can find (ω n ) n ⊂ Ω 0 such that X(ω n ) → x.Using the continuity of f i (•, µ) we can write the above equality for ω = ω n and then pass to the limit as n → +∞ obtaining that f 1 (x, µ) = f 2 (x, µ).
In the following Theorem 4.8, we provide an important structural representation of invariant Lipschitz maps from X to Y.A similar kind of problems have been considered in [CD18, Proposition 5.36].
Theorem 4.8 (Structure of invariant Lipschitz maps).Let X, Y be separable Banach spaces and let X, Y be as in (4.1).
(1) Let L : X → Y be a L-Lipschitz map, invariant by measure-preserving isomorphisms.
Then there exists a unique continuous map f L : S(X) → Y such that for every X ∈ X, LX(ω) = f L (X(ω), X ♯ P) for a.e.ω ∈ Ω. (4.9) (2) Let X be an Hilbert space and p = 2, let D(L) ⊂ X and let L : D(L) → X be a L-Lipschitz map, invariant by measure-preserving isomorphisms.Then there exists a unique continuous map f L : S(X, ι(D(L)) → X such that for every X ∈ D(L), LX(ω) = f L (X(ω), X ♯ P) for a.e.ω ∈ Ω. (4.10) Proof.We prove item (1).Let N := {2 n | n ∈ N} and let (P N ) N ∈N be a N-segmentation of (Ω, B, P) as in Definition 3.3, whose existence is granted by Lemma 3.8.Let us define We divide the proof in several steps.(a) If X ∈ X m for some m ∈ N, then (there exists a unique representative of ) LX (that) belongs to Y m and Let Ω ′ ⊂ Ω be a full P-measure subset of Ω where both (3.5) and Lemma 3.15 hold for the L q (Ω, B, P; Y) function LX.
Let us fix k ∈ I m := {0, . . ., 2 m − 1} and show that (a representative of) LX is constant on for every n ∈ N (see Definition 3.3), by Corollary 3.12 we can find a measure-preserving isomorphism g n ∈ S(Ω) such that and g n is the identity outside Ω n,k(n;ω ′ ) ∪ Ω n,k(n;ω ′′ ) .By (4.6) and by Lipschitz continuity of L, we have 0 for every integer n sufficiently large, since X is constant on the whole Ω m,k .This implies that By definition of conditional expectation, this means that Passing to the limit as n → +∞ we get by (3.5) that LX(ω ′ ) = LX(ω ′′ ).This proves that LX is P-almost everywhere constant on Ω m,k ; being k ∈ I m arbitrary, we can find a representative of LX belonging to Y m .If ω ′ , ω ′′ ∈ Ω and ω ′ ∈ Ω m,i , ω ′′ ∈ Ω m,j , i, j ∈ I m we choose as g ∈ S(Ω) a measure-preserving isomorphism induced by the permutation σ ∈ Sym(I m ) that swaps i and j (see Corollary 3.11), so that we get by Lipschitz continuity of L that 1 (b) For every X ∈ X there exists a unique L-Lipschitz map f X : supp(X ♯ P) → Y such that LX(ω) = f X (X(ω)) for P-a.e. ω.Let X ∈ X; setting X n := E[X|B n ] ∈ X n , by Theorem 3.13 we have X n → X (hence also LX n → LX).Let us consider two representatives of LX and X, a full measure set Ω 0 ⊂ Ω and a subsequence (X n k ) k s.t.X n k (ω) → X(ω) and LX n k (ω) → LX(ω) for every ω ∈ Ω 0 .By (4.11) we have Passing to the limit in the above inequality for every couple (ω, ω ′ ) ∈ Ω 2 0 , we obtain that This gives the existence of a (unique) L-Lipschitz function f In particular, we can define f L (•, µ) := f X (•) whenever µ = X ♯ P. By hypothesis we have (X, LX) ♯ P = (X ′ , LX ′ ) ♯ P. (4.12) By the previous claim, the disintegration (see e.g.[AGS08, Theorem 5.3.1]for details on the disintegration theorem) of the common measure µ := (X, LX) ♯ P with respect to its first marginal µ is given by δ f X (•) which should coincide with δ f ′ X (•) µ-a.e. in X.Since f X and f X ′ are both Lipschitz continuous and coincide µ-a.e., they coincide on supp(µ).
(d) The map f L is continuous on S(X).Let us consider a sequence (x n , µ n ) n in S(X) converging to (x, µ) ∈ S(X) and let us prove that there exists an increasing subsequence k By Proposition 3.23, we can find a limit map X ∈ X and a sequence X n ∈ X converging to X such that (X n ) ♯ P = µ n , X ♯ P = µ.Since LX n → LX, we can then extract a subsequence k → n(k) and find a set of full measure Ω Let us fix ε > 0; since x ∈ supp(µ) and X(Ω 0 ) ∩ supp(µ) is dense in supp(µ), we can find ω ∈ Ω 0 such that |X(ω) − x| X ≤ ε.We then obtain Taking the lim sup as k → ∞ we get lim sup and, since ε > 0 is arbitrary, we obtain the convergence.
We prove item (2): by Theorem 4.6, it is enough to prove the statement in case the map L is defined on the whole X.We can then apply the previous claim with Y = X and p = q = 2.
In the particular case when p = q = 2 and X = Y is a Hilbert space, also λ-dissipativity (2.23) is inherited from L to its representative map f L .Proposition 4.9 (λ-dissipative representations).Let us suppose that p = 2, X is a separable Hilbert space, and L : X → X is an invariant Lipschitz λ-dissipative operator.Then, for every µ ∈ P 2 (X), the map f L (•, µ) of Theorem 4.8 is (pointwise) λ-dissipative, i.e.
Then L is law invariant and, for every µ ∈ ι(D), there exists a map f [µ] ∈ L q (X, µ; Y) such that, for every X ∈ D there exists an increasing subsequence k → n k such that Proof.First of all notice that L is law invariant, being pointwise limit of law invariant maps.Let µ ∈ ι(D) be fixed.If X ∈ D is such that X ♯ P = µ, we can find an increasing subsequence k → n k such that L n k X → LX P-a.e. in Ω and define where E ⊂ X is the Borel set of points in X where lim k→+∞ f n k (x, µ) exists.Let us now show that f X represents L, i.e.
We know that for every k ∈ N L n k X(ω) = f n k (X(ω), X ♯ P) for a.e.ω ∈ Ω, so that, choosing representatives of X, LX, L n k X, we can find a full P-measure set Ω 0 ⊂ Ω such that LX(ω) = lim k→+∞ L n k X(ω) = lim k→+∞ f n k (X(ω), X ♯ P) for every ω ∈ Ω 0 .
Thus, for every ω ∈ Ω 0 , X(ω) ∈ E and then LX(ω) = f X (X(ω)).If now X ′ ∈ D is such that X ′ ♯ P = µ, arguing as in claim (c) of the proof of Theorem 4.8, it is easy to see that f X = f X ′ µ-almost everywhere.
We eventually apply the previous results to (possibly multivalued) maximal λ-dissipative operators B ⊂ X × X (with p = q = 2 and X = Y) which are invariant by measure-preserving isomorphisms.We show that the invariance property is inherited by the associated resolvent operator, Yosida approximation, minimal selection and semigroup which also enjoy a map-representation property.
We denote by ι D(B) := X ♯ P : X ∈ D(B) the image in P 2 (X) of the domain of B.
Claim 2: the map g := co(g) is G H -invariant as well.
It is sufficient to observe that for every U ∈ G H , N ∈ N, and α i ≥ 0 with N i=1 α i = 1, a collection {(x i , y i )} N i=1 ∈ (H × H) N satisfies N i=1 α i (x i , y i ) = (x, y) if and only if N i=1 α i (U x i , U y i ) = (U x, U y).Using (A.1) and the invariance of g we thus obtain g(U x, U y) = g(x, y) for every x, y ∈ H. Claim 3: the map F := ∇ y g(•, 0) is G H -invariant. Since we know that g is Frechét differentiable, we observe that z = F (x) if and only if g(x, y) = g(x, 0) + z, y + o(|y|) as y → 0.

. 7 )
Proof.Since N is unbounded and directed, for every finite subset M ⊂ N the quantity succ(M) := min {N ∈ N | M ≺ N ∀M ∈ M} is well defined.Let (a n ) n ⊂ N be an enumeration of N and consider the following sequence defined by induction b 0 = a 0 , b n+1 = succ ({a n+1 , b n }) , n ∈ N. Then b n ≺ b n+1 for every n ∈ N and (3.6) holds for (b n ) n and any N-refined standard Borel measure space (Ω, B, m, (P N ) N ∈N ).