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Around Strassen's theorems A. G. Kusraeva, S. S. Kutateladzeb a North Caucasus Center for Mathematical Research, Vladikavkaz Scientific Centre of the Russian Academy of Sciences, Vladikavkaz, Russia b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
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    § 1. IntroductionThis paper is dedicated to Vladimir Mikhailovich Tikhomirov on the occasion of his 90th birthday 1.1.The 1965 paper by Strassen1 [1] establishes two fundamental results. The first (Theorem 1 in [1]) asserts that if a bounded linear functional on a separable Banach space is dominated by a continuous sublinear functional defined as the integral with respect to a measurable parameter of a weakly measurable family of continuous sublinear functionals, then it admits an analogous representation as the integral of a weakly measurable family of bounded linear functionals, where each functional is dominated by a sublinear functional in the family for an appropriate value of the parameter. Further advances in this direction led to the creation of duality theory for convex integral functionals and operators in spaces of measurable vector functions. For an account of this theory, together with the corresponding bibliography and historical notes,, we refer to the books [2] by Ioffe and Tikhomirov, [3] by Castaing and Valadier, [4] by Levin and [5] by Ekeland and Temam. 1.2.The second fundamental result of Strassen (see Theorem 7 in [1]) gives necessary and sufficient conditions for two probability Borel measures on complete separable metric spaces to be the marginals on these spaces of a measure from a given convex $\ast$-weakly closed set of probability Borel measures defined on the Cartesian product of these metric spaces. There are various extensions of this result to positive measures with values in ordered vector spaces (see, for example, [6]–[9]). The above results of Strassen’s will be referred to as the disintegration theorem and the marginal theorem. The other results in [1] were various applications of the disintegration theorem (Theorems 2–6 in [1]) and the marginal theorem (Theorems 8–11 in [1]). 1.3.This paper presents a development of some ideas around Strassen’s theorems in the realm of operator convex analysis. In § 2 we formulate the disintegration and marginal theorems and give their interpretations in the language of functional analysis. It is worth noting that Strassen’s original proofs were based on the classical Hahn–Banach, Radon–Nikodým and Alaoglu–Bourbaki theorems. In § 3 we give a brief overview of the corresponding results from operator convex analysis, namely, a dominated extension of linear operators, subdifferentiation formulae and an intrinsic characteristic of subdifferentials; the additional machinery includes Maharam operators and Maharam extension, the Radon–Nikodým theorem for positive operators, the extremal structure of submorphisms, measurable Banach bundles and tensor products of vector lattices, and is introduced below as required. In § 4 we present an abstract version of Strassen’s disintegration theorem and some corollaries to it; § 5 is devoted to disintegration in spaces of sections of measurable Banach bundles. The main result in § 6 is an analogue of Strassen’s marginal theorem for operators in vector lattices. Applications to the operator versions of the Monge–Kantorovich and Choquet theories are presented in § 7 and 8, respectively. For reasons of volume we only sketch the proofs. Theorems 14, 17, 18, 22 and 23 are published for the first time. For necessary facts from subdifferential calculus and the theory of vector lattices, see [10], [11] and [12]–[14], respectively. All vector spaces under consideration are real, and all vector lattices are assumed to be Archimedean. § 2. Strassen’s theorems2.1. DisintegrationLet $X$ be a separable Banach space, $(\Omega,\Sigma, \mu)$ be a complete finite measure space and $p_\omega\colon X\to \mathbb{R}$ be a continuous sublinear functional for each $\omega\in \Omega$. Assume that the function $\omega\mapsto p_\omega(x)$ is measurable for each $x\in X$, and the function $\omega\mapsto\|p_\omega\|$, where $\|p_\omega\|:=\sup\{|p_\omega(x)|\colon \|x\|\leqslant1\}$, is integrable. Then
$$
\begin{equation}
p(x):=\int_\Omega p_\omega(x)\,d\mu(\omega), \qquad x\in X,
\end{equation}
\tag{1}
$$
is a continuous sublinear function $p\colon X\to\mathbb{R}$. Definition 1. The support set or subdifferential (at the origin) $\partial p$ of a sublinear function $p\colon X\to\mathbb{R}$ consists of all $x^\ast\in X^\ast$ satisfying $\langle x,x^\ast\rangle\leqslant p(x)$ for each $x\in X$. Theorem 1 (Theorem 1 in [1]). For each $x^\ast\in \partial p$ there exists a map $\Omega\ni\omega\mapsto x_\omega^\ast\in X^\ast$ such that $\omega\mapsto\langle x,x^\ast_{\omega}\rangle\in L^1(\Omega,\Sigma,\mu)$ for all $x\in X$ and $x^\ast_\omega\in\partial p_\omega$, $\omega\in\Omega$, and 2.2. An interpretation of Theorem 1Under the hypotheses of Theorem 1, let $I_\mu\colon L^1(\mu)\to\mathbb{R}$ denote the Lebesgue integral, that is, and let $P\colon X\to L^1(\mu)$ be the operator defined as follows: $P(x)$ is the equivalence class of the measurable function $\omega\mapsto p_\omega(x)$, $\omega\in\Omega$. The operator $P$ is sublinear, and its subdifferential has the form (see § 4 below)
$$
\begin{equation}
\partial P:=\bigl\{T\in L(X,L^1(\mu))\colon Tx\leqslant P(x) \text{ almost everywhere for}\ x\in E\bigr\}.
\end{equation}
\tag{2}
$$
By (1) we have $p=I_\mu\circ P$, and therefore from Strassen’s disintegration theorem (Theorem 1) we obtain the representation Conversely, if (3) holds, then any functional in $\partial p$ has a representation as in Theorem 1. Indeed, if $x^\ast\in\partial p$, then by (2) and (3) there exists an operator $T\in L(X,L^1(\mu))$ such that $Tx\leqslant P(x)$ and $\displaystyle \langle x,x^\ast\rangle= \int_\Omega Tx\,d\mu$ for each $x\in X$. Under the assumptions of Theorem 1 there exists a map $\Omega\ni\omega\mapsto x^\ast(\omega)\in X^\ast$ such that $T$ has a representation $Tx=\langle x,x^\ast(\,{\cdot}\,)\rangle$, where $\langle x,x^\ast(\,{\cdot}\,)\rangle$ is the equivalence class of the measurable function $\omega\mapsto\langle x,x^\ast(\omega)\rangle$ and $x^\ast(\,{\cdot}\,)$ is a selection of the set-valued mapping $\omega\mapsto\partial p_\omega$, that is, $x^\ast(\omega)\in\partial p_\omega$ for all $\omega\in\Omega$; see § 4.5.8 in [11]. 2.3. Measures with given marginalsNow let $S$ and $T$ be complete separable metric spaces, and $\mathcal{B}_S$, $\mathcal{B}_T$ and $\mathcal{B}_{S\times T}$ be Borel $\sigma$-algebras of the spaces $S$, $T$ and $S\times T$, respectively. Let $Y$ be the order ideal in the lattice $C(S)$ of all continuous functions on $S$ that is generated by a fixed strictly positive function $\widehat{y}\in C(S)$; this ideal is equipped with the $AM$-norm $\|y\|:=\sup\{|y(s)|/\widehat{y}(s)\colon s\in S\}$. One defines similarly $Z\subset C(T)$ and $X\subset C(S\times T)$, where the strictly positive function $\widehat{z}\in C(T)$ is arbitrary and $\widehat{x}\in C(S\times T)$ has the form $x(s,t):=y(s)+z(t)$. (Imposing some limitations on generality but without impairing the conceptual aspect, we can assume that $\widehat{x}\equiv1$, $\widehat{y}\equiv1$ and $\widehat{z}\equiv1$, that is, $Y$, $Z$ and $X$ are the Banach spaces of all bounded continuous functions on $S$, $T$ and $S\times T$, respectively.) Consider a nonempty convex $\ast$-weakly closed set $\Lambda$ of probability measures on $\mathcal{B}_{S\times T}$, and consider two fixed probability measures $\mu$ and $\nu$ on $\mathcal{B}_S$ and $\mathcal{B}_T$, respectively. We can assume that $\widehat{y}$ and $\widehat{z}$ are integrable with respect to $\mu$ and $\nu$, respectively, and $\widehat{x}$ is integrable with respect to each $\lambda\in\Lambda$. Definition 2. Two measures $\mu\colon \mathcal{B}_{S}\to\mathbb{R}$ and $\nu\colon \mathcal{B}_{T}\to\mathbb{R}$ are the marginals of some measure $\lambda\colon \mathcal{B}_{S\times T}\to\mathbb{R}$ on $\mathcal{B}_S$ and $\mathcal{B}_T$, respectively, if $\mu(A)=\lambda(A\times T)$ and $\nu(B)=\lambda(S\times B)$ for all $A\in\mathcal{B}_S$ and $B\in\mathcal{B}_T$. Theorem 2 (Theorem 7 in [1]). A necessary and sufficient condition ensuring that there exist a probability measure $\lambda\in\Lambda$ with prescribed marginals $\mu$ and $\nu$ on $\mathcal{B}_S$ and $\mathcal{B}_T$, respectively, is that for all $y\in Y$ and $z\in Z$ 2.4. An interpretation of Theorem 2Given $y\in Y$ and $z\in Z$, set
$$
\begin{equation*}
\begin{gathered} \, f(y)=\int_{S}y(s)\,d\mu(s), \qquad g(z)=\int_{T}z(t)\,d\nu(t) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
d_\lambda(y,z)=\int_{S\times T}y(s)z(t)\,d\lambda(s,t), \qquad \mathcal{D}\ni d_\lambda \longleftrightarrow \lambda\in \Lambda.
\end{equation*}
\notag
$$
Let $W$ be the set of all bounded bilinear forms on $Y\times Z$. We have $f\in Y'$, $g\in Z'$ and $d_\lambda\in W$. In addition, the map $\lambda\to d_\lambda$ is an affine bijection $\Lambda$ onto the convex $\sigma(W,Y\otimes Z)$-closed set $\mathcal{D}\subset W$. Strassen’s marginal theorem (Theorem 2) asserts that, given $f\in Y'$, $g\in Z'$ and $\mathcal{D}\subset W$, the following properties are equivalent:
$$
\begin{equation}
\exists\, b\in\mathcal{D}\colon \quad f(\,{\cdot}\,)=b(\,{\cdot}\,,1_T), \quad g(\,{\cdot}\,)=b(1_S,\,{\cdot}\,),
\end{equation}
\tag{4}
$$
and
$$
\begin{equation}
f(y)+g(z)\leqslant\sup_{d\in\mathcal{D}}\bigl\{d(y,1_T)+d(1_S,z)\bigr\}, \qquad y\in Y, \quad z\in Z.
\end{equation}
\tag{5}
$$
Below, in §§ 4–6 we establish general operator versions of representation (3) and of an equivalence of the form (4) $\Leftrightarrow$ (5). § 3. Hahn–Banach–Kantorovich theoremWe start our presentation of the machinery required below with some versions of and corollaries to the Hahn–Banach theorem. 3.1. Dominated extension of linear operatorsLet $X$ be a real vector space, $E$ be an ordered vector space and $L(X,E)$ be the space of all linear operators from $X$ to $E$. Definition 3. An operator $P\colon X\to E$ is sublinear if, for all $0\leqslant\lambda\in\mathbb{R}$ and $x,y\in X$, and Let $\operatorname{Sbl}(X,E)$ denote the set of sublinear operators from $X$ to $E$. Definition 4. The support set or subdifferential (at the origin) $\partial P$ of a sublinear operator $P$ is the set of all linear operators dominated by the operator $P$: Consider a subspace $X_0\subset X$, a linear operator $T_0\colon X_0\to E$ and a sublinear operator $P\colon X\to E$, where $T_0$ is dominated by the restriction of the operator $P$ to $X_0$, that is, $T_0x\leqslant P(x)$ for each $x\in X_0$. Definition 5. A space $E$ admits a dominated extension of linear operators if for all $X$, $X_0$, $T_0$ and $P$ as above there exists a linear operator $T\colon X\to E$ that extends $T_0$ and preserves the dominance, that is, The following result characterizes the ordered vector spaces admitting a dominated extension of linear operators. Theorem 3. An ordered vector space admits a dominated extension of linear operators if and only if each set in this space that is order bounded above has a least upper bound. For details of the proof, see § 1.4 of [11]. Definition 6. An ordered vector space is a vector lattice2 if each pair of elements of this space has a least upper and a greatest lower bound. A Kantorovich space (a $K$-space for short) is a vector lattice where each subset order bounded above has a least upper bound. 3.2. Some corollariesWe need two corollaries to Theorem 3, where we assume that $E$ is a Kantorovich space. For other corollaries and applications, see the remarks below. Corollary 1. For each sublinear operator $P\colon X\to E$, where the supremum on the right is attained for each $x\in X$. Corollary 2. A sublinear operator $P$ from an ordered vector space $X$ to $E$ is increasing if and only if its support set $\partial P$ consists of positive operators: Here and in what follows $\operatorname{Sbl}^+(X,E)$ is the part of $\operatorname{Sbl}(X,E)$ consisting of increasing operators $(x_1\leqslant x_2\Rightarrow P(x_1)\leqslant P(x_2))$; we also set $L^+(X,E):=L(X,E)\cap\operatorname{Sbl}^+(X,E)$. Remark 1. (1) The field of real numbers $\mathbb{R}$ (a one-dimensional ordered vector space) admits a dominated extension of linear functionals. This is the classical Hahn–Banach theorem, established by Hahn (1927) and Banach (1929). The first version of this theorem is due to Helly (1912); see the historical notes by Hochstadt [15]. (2) Any Kantorovich space admits a dominated extension of linear operators. This fact, established by Kantorovich [16] in 1935, is usually called the Hahn–Banach–Kantorovich theorem. An equivalent result (Kantorovich’s theorem on extensions of positive operators) says that a positive linear operator from a majorizing subspace of a vector lattice to a Kantorovich space admits an extension to the whole lattice with preservation of linearity and positivity (see Theorem 1.32 in [13]). (3) An ordered vector space admitting a dominated extension of linear operators is a Kantorovich space. This is the converse result to the Hahn–Banach–Kantorovich theorem, which is due to Bonnice and Silvermann [17] and, independently, To [18]. An elegant proof of Theorem 3 was proposed by Ioffe [19]. Remark 2. Theorem 3 has a long history; for fundamental steps in the development of this theory, see Ch. 1 in [11]. For more details on various extensions and applications of the Hahn–Banach theorem, see the surveys by Buskes [20], Narici and Beckenstein [21] and Sofi [22]. 3.3. Subdifferential calculusThe Hahn–Banach–Kantorovich theorem is a keystone of operator convex analysis. We present only a few subdifferentiation formulae for convex operators; for a detailed account, see [11]. In the rest of this section we assume that $E$ is a Kantorovich space and $X$ is a vector space. Theorem 4 (subdifferential of a sum). For arbitrary operators $P_1,\dots,P_n$ from $\operatorname{Sbl}(X,E)$ the following representation holds: Formula (7) is known as the subdifferential form of the Hahn–Banach–Kantorovich theorem, because this theorem follows from (7) if $Y=X_0$ is a subspace of $X$ and $T$ is the identity embedding $T:=\iota\colon X_0\hookrightarrow X$:
$$
\begin{equation*}
\partial(P|_{X_0})=\partial(P\circ\iota)= (\partial P)\circ\iota=\{T|_{X_0}\colon T\in\partial P\}= (\partial P)|_{X_0}.
\end{equation*}
\notag
$$
Theorem 6 (subdifferential of a composition). Let $Y$ be an ordered vector space. Then for all $P\in\operatorname{Sbl}(X,Y)$ and $Q\in\operatorname{Sbl}^+(Y,E)$, Remark 3. (1) Formula (6) is also known as the Moreau–Rockafellar formula; formulae (7) and (8) were obtained by Levin [23] and Kutateladze [24], respectively; the proofs of Theorems 4, 5 and 6 can be found in § 1.4.12, (1), § 1.4.14, (4), and § 2.1.6, (3) of [11]; this book also contains references to the original works. Swapping $P$ and $T$ is not always possible in (7), that is, the equality ${\partial(T\circ P)}=T\circ(\partial P)$ holds only for a narrow class of operators; see § 4 below. (2) For an account of operator convex analysis, see [11]; for fundamental steps in the development of this theory, see the surveys by Kutateladze [25], Rubinov [26], and also the book [10] by Akilov and Kutateladze; for classical convex analysis (with $E=\mathbb{R})$ and its applications, see the surveys by Tikhomirov [27], [28] and the books by Borwein and Lewis [29], Brinkhuis and Tikhomirov [30], Hiriart-Urruty and Lemaréchal [31], Magaril-Il’yaev and Tikhomirov [32], Polovinkin and Balashov [33] and Rockafellar [34]. 3.4. Intrinsic characterization of support setsWhat are necessary and sufficient conditions for the set of linear operators from a vector space $X$ to a Kantorovich space $E$ to agree with the subdifferential $\partial P$ of some sublinear operator $P\colon X\to E$? The following result is well known in the scalar case $E=\mathbb{R}$. Theorem 7. A weakly bounded set of linear functionals is a subdifferential if and only if it is convex and weakly closed. Definition 7. Given $\xi\in\Xi$, consider an operator $T_\xi\in L(X,E)$ and an order projection $\pi_\xi$ in $E$. An operator $T\in L(X,E)$ is called a mixing of the family $(T_{\xi})_{\xi\in\Xi}$ relative to $(\pi_{\xi})_{\xi\in\Xi}$ (written $T=\operatorname{mix}_{\xi\in\Xi}(\pi_{\xi}T_\xi)$) if: (1) $\pi_\xi\circ\pi_\eta=0\ (\xi\ne\eta)$ and $\sum_{\xi\in\Xi}\pi_\xi=I_E$; (2) $Tx=o\text{-}\!\sum_{\xi\in\Xi}\pi_{\xi}T_{\xi}x$ for all $x\in X$. Given $\mathcal{D}\subset L(X,E)$, we denote by $\operatorname{mix}(\mathcal{D})$ the set of all possible mixings $\operatorname{mix}_{\xi\in\Xi}(\pi_{\xi}T_\xi)$, where $(T_\xi)\subset\mathcal{D}$. A set $\mathcal{D}\subset L(X,E)$ is said to be cyclic if $\mathcal{D}=\operatorname{mix}(\mathcal{D})$. Definition 8. An operator $S\in L(X,F)$ is a pointwise $o$-limit of a net $(S_i)$ from $L(X,F)$ if the net $(S_ix)$ $o$-converges to $Sx$ in $F$ for each $x\in X$. We let $o\text{-}\mathrm{cl}(\mathcal{D})$ denote the set of all pointwise $o$-limits of nets from $\mathcal D$. We say that $\mathcal{D}$ is pointwise $o$-closed if $\mathcal{D}=o\text{-}\mathrm{cl}(D)$. Theorem 8. A weakly order-bounded set of operators is a subdifferential if and only if it is convex, cyclic and pointwise $o$-closed. Remark 4. Theorem 8 was established in [35] by the method of Boolean valued realization. A standard proof (without recourse to a Boolean valued model) of an intrinsic characterization of subdifferentials can be found in the book [36]. In the case of functionals Theorem 8 is the classical Alaoglu–Bourbaki theorem, which can equivalently be stated as follows: the subdifferential of a continuous sublinear functional $p\colon X\to\mathbb{R}$ defined on a real locally convex space $X$ is weakly compact. At the same time an operator subdifferential is rarely compact in some natural topology and cannot adequately be characterized in terms of scalar separation theorems; see, for example, Theorems 1 and 2 in [23]. This difficulty can be circumvented by invoking the $\operatorname{mix}$-operation and the cyclic compactness property (see § 1.3 in [36]; for details and related results, see § 2.4 in [11]). § 4. Disintegration in Kantorovich spacesIn this section we present a version of Strassen’s disintegration theorem for operators with values in a Kantorovich space. One of the key auxiliary results here is the Radon–Nikodým theorem for positive operators. 4.1. Maharam operatorsIn a series of papers published in the 1950s Maharam developed an original theory of positive operators in spaces of measurable functions. In those papers a special class of operators was singled out, for which an analogue of the classical Radon–Nikodým theorem holds. These operators were called full-valued operators by Maharam, but nowadays the terms Maharam property [37] and Maharam operator [38] are more common (see below). For a brief overview of the method developed by Maharam and the main results, see [39]. Definition 9. Let $E$ be a vector lattice, $F$ be a Kantorovich space and $T$ be a positive operator from $E$ to $F$. The operator $T$ has the Maharam property or is interval preserving if $T([0,\,x])=[0,\, Tx]$ for each $x\in E_+$. A positive order-continuous operator with the Maharam property is called a Maharam operator. Remark 5. Luxemburg and Schep [37] extended a part of Maharam’s theory to positive operators in Kantorovich spaces. For various aspects of the theory of Maharam operators and an extension of this theory to sublinear and convex operators, see § 4.5 of [11] and §§ 3.4 and 3.5 of [12]. For Maharam operators in Boolean valued analysis, see §§ 5.2–5.12 of [40]. Operators with Maharam property appear in various contexts and feature interesting properties: see [12], [13] and [40]. Here, we recall the dual relation with lattice homomorphisms, which we require below. Theorem 9. Let $E$ and $F$ be vector lattices. Assume that the order dual space $E^\sim$ separates points in $E$. Then a positive operator $T\colon E\to F$ is a lattice homomorphism if and only if $T'\colon F^\sim\to E^\sim$ is a Maharam operator. The proof is immediate from Theorems 1.73 and 2.20 in [13]. 4.2. Examples(a) We have $T[0,x]\subset[0,Tx]$, $x\in E_+$, for each positive operator $T$. The reverse inclusion holds for positive functionals (that is, for $F=\mathbb{R}$), but it can fail to hold for $F=\mathbb{R}^2$. Consider a vector lattice $E=L^1([0,1])$, the constant function $x_0\equiv1$ and the positive functional $\displaystyle f\colon x\mapsto\int_0^1x(t)\,dt$. For the positive linear operator $T\colon E\ni x\mapsto(f(x),f(x))\in\mathbb{R}^2$ we have $T([0,x_0])=\{(t,t)\colon t\in [0,1]\}\nsubseteq[0,1]^2=[0,Tx_0]$, that is, $T$ is not interval preserving. (b) Let $F$ be a Kantorovich space and $J$ be an arbitrary nonempty set. Let
$$
\begin{equation*}
l^1\,(J,F):=\biggl\{(e_j)_{j\in J}\in F^J\colon o\text{-}\!\sum_{j\in J}|e_j|\in F\biggr\}
\end{equation*}
\notag
$$
be the set of all $o$-integrable families in $F$, indexed by elements of $J$. Consider the operator $T$ from $l^1(J,F)$ to $F$ defined by $Tz:=o\text{-}\!\sum_{j\in J}e_j$, $z:=(e_j)_{j\in J}$. Proposition 1. The set $l^1(J,F)$ with the natural structure of an ordered vector space is a Kantorovich space, and $T$ is a Maharam operator. (c) Consider a finite measure space $(\Omega,\Sigma,\mu)$ and a $\sigma$-subalgebra $\Sigma_0$ of the algebra $\Sigma$. Let $\mu_0$ be the restriction of the measure $\mu$ to $\Sigma_0$. Proposition 2. The conditional expectation operator $\mathcal{E}(\,{\cdot}\,,\Sigma_0)$ is a Maharam operator from $L^1(\Omega,\Sigma,\mu)$ to $L^1(\Omega,\Sigma_0,\mu_0)$. The restriction of $\mathcal{E}(\,{\cdot}\,,\Sigma_0)$ to $L^p(\Omega,\Sigma,\mu)$ is also a Maharam operator in $L^p(\Omega,\Sigma,\mu)$. (d) Let $(\Omega,\Sigma,\mu)$ be as above and $F$ be a Banach lattice. Let $E:=L^1(\mu,F)$ denote the space of all Bochner integrable functions $f\colon \Omega\to F$. Let $I_{\mu}\colon E\to F$ be the integration operator defined by $\displaystyle I_\mu\colon f\,{\mapsto}\int_\Omega f(\omega)\,d\mu(\omega)$. Proposition 3. If $F$ is a Banach lattice with order-continuous norm, then $E$ is also a Banach lattice with order-continuous norm and $I_\mu$ is a Maharam operator. Proof. Under the above assumptions $E$ is also a Banach lattice with order-continuous norm and the operator $I_\mu$ is order continuous. In addition, for $0\leqslant x\in F$ and $0\leqslant y\leqslant I_\mu(x)$, there exists an orthomorphism $\pi\in\mathcal{Z}(F)$ such that $y=\pi(I_\mu(x))= I_\mu(\pi x)$. Therefore, $T$ is a Maharam operator, which proves the proposition. 4.3. The Radon–Nikodým theoremFor Maharam operators an analogue of the Radon–Nikodým theorem [37] holds. To formulate this result recall that $L^\sim(E,F)$ is the space of all order-bounded operators from $E$ to $F$, $\{T\}^{\perp\perp}$ is the band in $L^\sim(E,F)$ generated by $T$, and $S\ll T$ means that $Sx\in\{Tx\}^{\perp\perp}$ for each $x\in E_+$. Theorem 10. Let $E$ and $F$ be Kantorovich spaces and $T\colon E\to F$ be a Maharam operator. Then for a positive operator $S\colon E\to F$ the following assertions are equivalent: Remark 6. (1) If $|S|\leqslant T$, then the orthomorphism $\rho$ in Theorem 10 satisfies ${|\rho|\leqslant I_F}$. In addition, if $T$ is essentially positive $(T(|x|)= 0\Rightarrow x=0)$, then the correspondence $\rho\mapsto T\circ\rho$ is a lattice homomorphism from $\mathcal{Z}$ onto the order ideal in $L^\sim(E,F)$ generated by the operator $T$. (2) Let $T\colon E\to F$ be a positive linear operator, and let $P\colon E\to E$ be the sublinear operator defined by $e\mapsto e^+$. A direct calculation shows that ${\partial(T\circ P)=[0,T]}$ and $\partial P=[0,I_E]$. Hence, in this particular case the ‘chain rule’ $\partial(T\circ P)=T\circ\partial P$ is equivalent to the equality $[0,T]=T\circ[0,I_E]$, which follows from the above restricted version of the Radon–Nikodým theorem. So a Maharam operator is a likely candidate for an operator satisfying the ‘chain rule’. (3) The examples from § 4.2 can be found in § 3.4 of [12]. Theorem 10 is due to Luxemburg and Schep [37]. 4.4. Disintegration formulaeHere we find which operators $P\in\operatorname{Sbl}(X,E)$ and $Q\in\operatorname{Sbl}^+(E,F)$ satisfy the ‘chain rule’ $\partial(Q\circ P)=\partial(Q)\circ\partial(P)$. Definition 10. Let $E$ and $F$ be $K$-spaces and $P$ be an increasing sublinear operator from $E$ to $F$. We say that $P$ satisfies the Maharam condition if for all $e\in E^+$ and $f_1,f_2\in F^+$ the equality $P(e)=f_1+f_2$ implies that there exist $e_1,e_2\in E^+$ such that $e=e_1+e_2$ and $P(e_l)=f_l$, $l:=1,2$. An increasing order-continuous sublinear operator satisfying the Maharam condition is called a sublinear Maharam operator. This definition agrees with Definition 9. The class of sublinear Maharam operators was introduced and studied in [38] and [41], where the following result was also obtained. Theorem 11. Let $E$ and $F$ be Kantorovich spaces and $Q$ be a sublinear Maharam operator from $E$ to $F$. Then for each vector space $X$ and each sublinear operator $P$ from $X$ to $E$, Corollary 3. Let $X, E, F$ and $ P$ be as in Theorem 11. Then for each linear Maharam operator $T\colon X\to E$, Theorem 11 and Corollary 3 are equivalent because for a sublinear Maharam operator $Q$ the set $\partial(Q)$ consists of Maharam operators (see Theorem 4.4.7 in [11]). Corollary 4. Let $X$, $E$ and $F$ be as in Theorem 11. Assume, in addition, that $E^\sim$ separates points in $E$ and that some order ideal $J\subset F^\sim$ separates points in $F$. Also assume that $T\colon E\to F$ is a positive operator whose restriction $S:=T^\sim|_{J}$ is a lattice homomorphism from $J$ to $E^\sim_n$. Then $\partial(T\circ P)=T\circ\partial(P)$ for each sublinear operator $P\colon X\to E$. Proof. By Theorem 9, $S^\sim\colon E^{\sim\sim}\to J^\sim$ is a Maharam operator. Therefore, by Theorem 11 we have $\partial(S^\sim\circ \varkappa\circ P)=S^\sim\circ\partial(\varkappa\circ P)$, where $\varkappa=\varkappa_F$ is the canonical embedding of $F$ in $J^\sim$. By Nakano’s theorem (see Theorem 1.70 in [13]) $\varkappa_F$ and $\varkappa_E$ are lattice isomorphisms from $F$ and $E$ to order ideals in $J^\sim$ and $E^{\sim\sim}$, respectively. If $R\in\partial(\varkappa\circ P)$, then $-\varkappa\circ P(x)\leqslant R(x)\leqslant \varkappa\circ P(x)$ for all $x\in X$, and so the range of $R$ is contained in the same ideal $\varkappa(F)$ as the range of $\varkappa\circ P$. It remains to note that $S^\sim|_{\varkappa(E)}=\varkappa\circ T$ and then return to $T$ and $P$ by means of $\varkappa^{-1}$.
This proves the corollary. Remark 7. (1) The general result on disintegration (Theorem 11) was proved by Kusraev [38]. Corollary 4 is due to Meyer-Nieberg (Theorem 1 in [42]); that paper also contains a short survey of some operator versions of Strassen’s theorem: see Corollaries 2–5 in [42]. All of these results are particular cases of Theorem 11; also see [11]. (2) From (9) and (10) we can derive some formulae for the Young–Fenchel transform and the $\varepsilon$-subdifferentials of convex operators; see Ch. 4 of [11]. These results are usually called disintegration formulae. By means of general disintegration tricks various facts in the theory of Kantorovich spaces which are based on the Hahn–Banach–Kantorovich and Radon–Nikodým theorems can be unified in the routine form of calculus rules. § 5. Disintegration in measurable Banach bundlesIn this section we present a new version of Strassen’s theorem, where sublinear functionals are replaced by sublinear operators defined on stalks of a Banach bundle and taking values in a Banach lattice. For requisite facts from the theory of lattice normed spaces, see [12]. 5.1. Measurable bundlesFirst recall some necessary facts from the theory of measurable Banach bundles developed by Gutman [43]. In what follows, $(\Omega,\Sigma,\mu )$ is a measure space. Definition 11. A Banach bundle over $\Omega$ is an a map ${\mathcal X}$ of $\Omega$ that associates with each point $\omega\in \Omega$ some Banach space $({\mathcal X}_\omega,\|\cdot\|_\omega):= ({\mathcal X}(\omega),\|\cdot\|_{\mathcal{X}(\omega)})$, the stalk at the point $\omega$. The function $u$ on $\operatorname{dom}(u)\subset\Omega$ is called a section of the bundle ${\mathcal X}$ over $\operatorname{dom}(u)$ if $u(\omega)\in {\mathcal X}(\omega)$ for all $\omega\in\operatorname{dom}(u)$. A section $u$ is called scalarly measurable if the function $ |\kern-1pt|\kern-1pt| u |\kern-1pt|\kern-1pt| \colon \omega\mapsto\|u(\omega)\|_\omega$, $\omega\in\operatorname{dom}(u)$ is measurable. A linear combination of two sections defined almost everywhere is pointwise defined as a section defined almost everywhere (over the intersection of their domains). Let $S_{\sim }(\Omega ,{\mathcal X})$ be the set of all sections $u$ defined almost everywhere, that is, such that $\mu(\Omega\setminus\operatorname{dom}(u))=0$. Definition 12. A measurable structure in ${\mathcal X}$ is a set of scalarly measurable sections ${\mathcal C}\subset S_{\sim}(\Omega,{\mathcal X})$ satisfying the following conditions: A Banach bundle with fixed measurable structure over $\Omega$ is called a measurable Banach bundle over $\Omega$. Definition 13. A section $s\in S_{\sim }(\Omega ,{\mathcal X})$ is a step section if $s=\sum _{k=1}^{n}[A_{k}]c_{k}$ for some $n\in \mathbb N$, $A_{1},\dots,A_{n}\in \Sigma$ and $c_{1},\dots,c_{n}\in {\mathcal C}$, where $[A]u$ is the section coinciding with $u$ on the measurable set $A\in\Sigma$ and vanishing on the complement of $A$. A section $u\in S_{\sim }(\Omega ,{\mathcal X})$ is measurable if for each $\Omega_0\in \Sigma$, $\mu(\Omega_0)<+\infty$, there exists a sequence $(s_{n})_{n\in \mathbb N}$ of step sections such that $s_{n}(\omega )\to u(\omega )$ in $\mathcal{X}(\omega)$ for almost all $\omega \in\Omega_0$. The set of all (almost everywhere defined) measurable sections of a Banach bundle $\mathcal X$ is denoted by ${\mathcal M}(\Omega ,{\mathcal X})$. Given $u, v\in\mathcal{M}(\Omega,{\mathcal X})$, we set $u\sim v$ if $u(\omega )=v(\omega)$ for almost all $\omega \in \Omega $. The equivalence class containing an element $u\in {\mathcal M}(\Omega,{\mathcal X})$ is denoted by $\widetilde u:=u^{\sim }$. The quotient space $L^0(\Omega,{\mathcal X}):=\mathcal{M}(\Omega,{\mathcal X})/{\sim}$ can naturally be identified with a vector space as follows: $(\alpha u+\beta v)^\sim=\alpha\widetilde{u}+\beta\widetilde{v}$ for $\alpha,\beta\in\mathbb{R}$ and $u,v\in \mathcal{M}(\Omega,{\mathcal X})$. Next, for each $\widetilde u\in L^0(\Omega ,{\mathcal X})$ consider the vector $ \mathopen{|\kern-3pt|\kern-3pt|} \widetilde u \mathclose{|\kern-3pt|\kern-3pt|} := |\kern-1pt|\kern-1pt| u |\kern-1pt|\kern-1pt| ^{\sim }\in L^0(\Omega)$. It is clear that the pair $(L^0(\Omega,{\mathcal X}), \mathopen{|\kern-3pt|\kern-3pt|} \,{\cdot}\, \mathclose{|\kern-3pt|\kern-3pt|} )$ is a lattice normed space over $L^0(\Omega,\Sigma,\mu)$; see § 2.1.1 in [12]. Proposition 4. If $E\subset L^0(\mu)$ is an ideal space and then $E(\mathcal X)$ is a Banach–Kantorovich space and $L^0(\mu,\mathcal X)$ is a maximal extension of this space. For a proof, see Theorem 2.5.3 in [12] and Theorem 3.1.14 in [43]. 5.2. Measurable bundles of operator spacesConsider a fixed lifting $\rho\colon L^{\infty}(\Omega )\to \mathcal{L}^{\infty}(\Omega)$ of the space $L^{\infty}(\Omega)$; see Definitions 1.1.13 and 4.3.1 in [43]. The following Theorems 12 and 13 are due to Gutman (see Theorems 4.4.6 and 5.2.3 in [43]). Definition 14. Let $\mathcal{X}$ be a measurable Banach bundle over $\Omega$. A map $\rho_{\mathcal{X}}\colon L^{\infty}(\Omega,\mathcal{X})\to\mathcal{L}^{\infty}(\Omega,\mathcal{X})$ is called a lifting of $L^{\infty }(\Omega,\mathcal{X})$ (associated with $\rho $) if the following conditions are satisfied for all $u, v\in L^{\infty }(\Omega,\mathcal{X})$ and $e\in L^{\infty}(\Omega )$: If there exist a lifting of the space $L^{\infty}(\Omega)$ and an associated lifting of $L^{\infty }(\Omega,\mathcal{X})$, then $\mathcal{X}$ is said to be a measurable Banach bundle with lifting. Let $\mathcal{L}(X,Y)$ be the space of all bounded linear operators from $X$ to $Y$. If $u\in S_\sim(\Omega,X)$ and $H(\omega)\in (\mathcal{X}(\omega), \mathcal{Y}(\omega))$ for each $\omega\in\Omega$, then the section $H\otimes u\in S_\sim(\Omega,\mathcal{Y})$ is defined by $H\otimes u\colon \omega\mapsto H(\omega)u(\omega)$, $\omega\in\Omega$. Theorem 12. Let $\mathcal{X}$ and $\mathcal{Y}$ be measurable Banach bundles over $\Omega$ with liftings associated with the same lifting of $L^\infty(\Omega)$. Then there exists a unique measurable Banach bundle with lifting $\mathcal{L}(\mathcal{X},\mathcal{Y})$ such that: Given a Banach space $X$, consider a constant Banach bundle $\mathcal{X}$ such that ${\mathcal{X}(\omega)=X}$ for all $\omega\in\Omega$, and take the set of all constant functions $c\colon \Omega\to X$ as a measurable structure. Then $\mathcal{M}(\Omega,\mathcal{X})$ is the set of all Bochner measurable $X$-valued vector functions defined almost everywhere on $\Omega$; we denote it by $\mathcal{M}(\Omega,X)$. Theorem 13. For any Banach space $X$, there exists a unique measurable Banach bundle $\mathcal{X}$ with lifting $\rho_{\mathcal{X}}$ over $\Omega$ such that: 5.3. Auxiliary factsLet $(\Omega,\Sigma,\mu)$ be a measurable space and $\mathcal X$ be a measurable Banach bundle over $(\Omega,\Sigma,\mu)$. Consider an ideal Banach space $E\subset L^0(\Omega,\Sigma,\mu)$ and the dual ideal space
$$
\begin{equation*}
E':=\{e'\in L^0(\Omega,\Sigma,\mu)\colon \forall\, e\in E\ \, ee'\in L^1(\Omega,\Sigma,\mu)\}.
\end{equation*}
\notag
$$
Lemma 1. Given a family $(P_\omega)_{\omega\in\Omega}$ of continuous sublinear operators $P_\omega\colon {\mathcal X}_\omega\,{\to}\, E$, assume that the map $\omega\mapsto P(\omega,x(\omega)):=P_\omega(u(\omega))$ is $\mu$-measurable for each ${u\in E(\mathcal{X})}$ and the function $\omega\mapsto\|P_\omega\|:=\sup\{\|P_\omega(x)\|\colon \|x\|_\omega\leqslant 1\}$ is majorized by some measurable function $e'$ from $E'$. Then defines a sublinear operator $Q$ from $E(\mathcal X)$ to $E$. Proof. The operator $Q$ is well defined in view of the estimates That $Q$ is sublinear follows from elementary properties of the Bochner integral.
This proves the lemma. Let $\displaystyle\int_\Omega\partial(P(\omega,(\,{\cdot}\,)(\omega))\,d\mu(\omega)$ be the set of linear operators $\overline{S}\colon E(\mathcal{X})\to E$ representable in the form
$$
\begin{equation*}
\overline{S}(u)=\int_\Omega S_\omega(u(\omega))\,d\mu(\omega),
\end{equation*}
\notag
$$
where $(S_\omega)_{\omega\in\Omega}$ is the family of linear operators $S_\omega\in\mathcal{L}(\mathcal{X}(\omega),E)$ such that ${S_\omega\,{\in}\,\partial P_\omega}$ for all $\omega\in\Omega\,$ and the map $S\otimes u\colon \omega\mapsto S_{\omega}(u(\omega))$ is Bochner-integrable, that is, $S\mathbin{\widetilde{\otimes}}u:=(S\otimes u)^\sim\in L^1(\Omega,\Sigma,\mu,E)$ for each $u\in E(\mathcal{X})$. Definition 15. Let $\mathcal{X}$ and $\mathcal{Y}$ be measurable Banach bundles with liftings over $\Omega$, and $E$ and $F$ be order dense ideals in $L^0(\Omega,\mu)$. A linear operator $T\colon E(\mathcal{X})\to F(\mathcal{Y})$ is said to be bounded if there exists a function $g\in L^0(\Omega,\mu)$ such that $g\cdot E\subset F$ and $ \mathopen{|\kern-3pt|\kern-3pt|} Tu \mathclose{|\kern-3pt|\kern-3pt|} \leqslant g \mathopen{|\kern-3pt|\kern-3pt|} u \mathclose{|\kern-3pt|\kern-3pt|} $ for each $u\in E(\mathcal{X})$. Lemma 2. A map $S$ from $E(\mathcal{X})$ to $F(\mathcal{X})$ is a bounded linear operator if and only if there exists a measurable section $\mathbb{S}\in\mathcal{L}^0(\Omega,\mathcal{L}(\mathcal{X},\mathcal{E}))$ such that $S\widetilde{u}=\mathbb{S}\mathbin{\widetilde{\otimes}} u$ for all $\widetilde{u}\in E(\mathcal{X})$. An analogous fact was established by Gutman (see Proposition 6.5.13 in [43]) for operators acting on continuous Banach bundles. The claim of the lemma can easily be reduced to this fact using the machinery of the Stone transform of a measurable Banach bundle with lifting, which was also developed by Gutman; see Theorems 4.3.4 and 4.3.5 in [43]. 5.4. Disintegration in Banach bundlesNow we can formulate the main result in this section. Theorem 14. Let $(\Omega,\Sigma,\mu)$ be a measure space with the direct sum property, $E$ be an ideal Banach space with order-continuous norm and $\mathcal X$ be a Banach bundle over $(\Omega,\Sigma,\mu)$ with lifting associated with a fixed lifting of $L^\infty(\Omega)$. Assume that a family $(P_\omega)_{\omega\in\Omega}$ satisfies the above conditions. Then Proof. By Lemma 1 a sublinear operator $Q$ from $E(\mathcal{X})$ to $L^1(\Omega,E)$ is well defined by the following rule: if $u\in E(\mathcal{X})$, then $Q(u)$ is the equivalence class of the integrable vector function $\Omega\ni\omega\mapsto P(\omega,u(\omega))$. In addition, $P=I_\mu\circ Q$, where $I_\mu$ is the Bochner integral. Now, $I_\mu$ is a Maharam operator by Proposition 3, and by the disintegration formula from Corollary 3 we have $\partial(P)=I_\mu\circ\partial(Q)$. It remains only to describe the set of operators $\partial(Q)\subset L(E(\mathcal{X}), L^1(\Omega,E))$.
By Theorem 13 there exists a measurable Banach bundle $\mathcal{E}$ with lifting $\rho_{\mathcal{E}}$ over $\Omega$ (associated with the same lifting as $\rho_{\mathcal{X}}$) such that the map $J\colon \widetilde{u}\mapsto \widetilde{u}\cap L^0(\Omega,\mathcal{E})$ is a lattice isomorphism from $L^1(\Omega,\mathcal{E})$ to $L^1(\Omega,E)$ preserving the $E$-valued norm. Hence $\partial(J\circ Q)=J\circ\partial(Q)$, and so we can assume without loss of generality that $Q(E(\mathcal{X}))\subset L^1(\Omega,\mathcal{E})$. By Theorem 12 there exists a measurable Banach bundle $\mathcal{L}(\mathcal{X},\mathcal{E})$ with lifting $\rho_\mathcal{L}$ whose arbitrary measurable section $S$ satisfies $S(\omega)\in \mathcal{L}(\mathcal{X}(\omega),\mathcal{E}(\omega))$ for all $\omega\in\Omega$. In addition, $\rho_{\mathcal{E}}((S\mathbin{\overline{\otimes}} u)^\sim)= \rho_\mathcal{L}(\widetilde{S})\otimes \rho_{\mathcal{X}}(\widetilde{u})$ for all $u\in\mathcal{L}^\infty(\Omega,\mathcal{X})$ and $S\in\mathcal{L}^\infty(\Omega,\mathcal{L}(\mathcal{X},\mathcal{E})$. If $S_0\in\partial(Q)$, then $ \mathopen{|\kern-3pt|\kern-3pt|} S_0u \mathclose{|\kern-3pt|\kern-3pt|} \leqslant e' \mathopen{|\kern-3pt|\kern-3pt|} u \mathclose{|\kern-3pt|\kern-3pt|} $ for all $u\in E(\mathcal{X})$, and so $S_0$ is a bounded operator in the sense of Definition 15. Note that $ \mathopen{|\kern-3pt|\kern-3pt|} Q(u) \mathclose{|\kern-3pt|\kern-3pt|} \leqslant e' \mathopen{|\kern-3pt|\kern-3pt|} u \mathclose{|\kern-3pt|\kern-3pt|} $, and therefore $Q(u)\in L^1(\Omega,\mathcal{X})$ whenever $\widetilde{u}\in E(\mathcal{X})$. By Lemma 2 there exists a measurable section $\mathbb{S}\in \mathcal{L}^0(\Omega,\mathcal{L}(\mathcal{X},\mathcal{E}))$ such that $S_0\widetilde{u}=(\mathbb{S}\mathbin{\overline{\otimes}} u)^\sim\leqslant Q(\widetilde{u})$ for all $\widetilde{u}\in E(\mathcal{X})$. Using the direct sum property and expanding $L^0(\Omega)$ into disjoint bands, we can assume without loss of generality that $\mathbb{S}\in L^\infty(\Omega,\mathcal{L}(\mathcal{X},\mathcal{E}))$ and $\mathbb{S}(L^\infty(\Omega,\mathcal{X}))\subset L^\infty(F(\mathcal{E}))$. We set $S_\omega:=\rho_\mathcal{L}(\widetilde{\mathbb{S}})(\omega)$ for all $(\omega\in\Omega)$. Using Theorem 12 again, we find that $\rho_\mathcal{E}(Q(\widetilde{u}))\geqslant \rho_\mathcal{E}(S_0\widetilde{u})= \rho_\mathcal{L}(\widetilde{\mathbb{S}})\mathbin{\overline{\otimes}}\rho_{\mathcal{X}}(\widetilde{u})$. Therefore, for each $u\in L^\infty(\Omega,\mathcal{X})$,
$$
\begin{equation*}
P_\omega(u(\omega))=\rho_\mathcal{E}(Q(\widetilde{u}))(\omega)\geqslant \bigl(\rho_\mathcal{L}(\widetilde{\mathbb{S}})(\rho_{\mathcal{X}}(\widetilde{u})\bigr)(\omega) =S_\omega(u(\omega))
\end{equation*}
\notag
$$
for almost all $\omega\in\Omega$. It remains to put $S_\omega:=\rho_\mathcal{L}(\mathbb{S})(\omega)$ for all $\omega\in\Omega$ and note that $S_\omega u=(\rho_\mathcal{L}(\mathbb{S})u)(\omega)\leqslant P(\omega(\omega))$.
This proves the theorem. Remark 8. For $E=\mathbb{R}$ this result was obtained in [44]. Using the lifting machinery König ([45], pp. 506–507) proved a version of the original Strasse theorem (${\mathcal{X}(\omega)=X}$ for all $\omega$) without the separability assumption, but under a condition on the family of sublinear functionals $(p_\omega)$: the measurable function $\omega\mapsto p_\omega(x)$ had to be bounded for each $x$. Neumann ([46], p. 419) obtained a generalization in another direction — namely, he showed that Strassen’s theorem remains true if the sublinear functionals $p_\omega$ are replaced by sublinear operators $\vartheta_\omega$ from a separable Banach space $X$ to a Banach space $E$ with the Radon–Nikodým property which is simultaneously an order complete vector lattice. Meyer-Nieberg ([42], p. 311) showed that in this result the Radon–Nikodým property can be relaxed by assuming that the norm is order continuous on $E$. In this case the lifting machinery cannot be used to get rid of the separability assumption because $L^\infty(\mu,E)$ has a lifting if and only if either $E$ is finite-dimensional or $\mu$ is atomic (Gutman [43], Theorem 5.2.2). § 6. Positive operators with given marginalsIn this section we consider the operator version of Strassen’s marginal theorem. As a technical tool, we use the tensor product of vector lattices in the sense of Fremlin [47]. 6.1. Tensor product of vector latticesIn what follows $E_1,\dots,E_n$ and $F$ are vector lattices. Definition 16. An $n$-linear operator $T\colon E_1\times\dots\times E_n\to F$ is said to be: We let $L^r(E_1,\dots,E_n;F)$ denote the vector space of all regular $n$-linear operators from $E_1\times\dots\times E_n$ to $F$ ordered by the cone of positive $n$-linear operators, that is, $T_1\geqslant T_2$ means that the operator $T_1-T_2$ is positive. It is clear that the set of regular $n$-linear operators is an ordered vector space. In the following two theorems we collect the most important properties of the tensor product of vector lattices; see Fremlin [47] and Schep [48]. Theorem 15. There exist a (unique up to an isomorphism) vector lattice $E_1\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}} E_n$ and a lattice $n$-morphism $\otimes$ from $E_1\times\dots\times E_n$ to $E_1\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}} E_n$ such that the following assertions hold: Definition 17. The vector lattice $\mathop{\overline{\bigotimes}}_{k=1}^{\,n}E_k:=E_1\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}} E_n$ is called the (Fremlin) tensor product of the vector lattices $E_1,\dots,E_n$. The universal property of the Fremlin tensor product (see assertion (1) of Theorem 15) holds also for a wider class of operators if the vector lattice $F$ has an additional feature. Theorem 16. Let $E_1,\dots, E_n$, $F$ be vector lattices, where $F$ is uniformly complete. Then there exists an order-preserving bijection $T\mapsto T^{\otimes}$ between the regular $n$-linear operators $T\colon E_1\times\dots\times E_n\to F$ and the regular linear operators ${T^{\otimes}\colon E_1\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}} E_n\to F}$ such that $T=T^{\otimes}\circ\otimes$. In addition, if $F$ is order complete, then $L^r(E_1,\dots,E_n;F)$ is an order complete vector lattice and the mapping $T\mapsto T^{\otimes}$ is a lattice isomorphism from $L^r(E_1,\dots,E_n;F)$ to $L^r(E_1\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}} E_n,F)$. 6.2. The existence of an operator with prescribed marginalsGiven a set of elements $\mathbf{e}=(e_1,\dots,e_n)\in E_1\times\dots\times E_n$, consider the lattice homomorphism $[\mathbf{e}]_k$ from $E_k$ to $\mathop{\overline{\bigotimes}}_{k=1}^{\,n}{E_k}$ defined by $[\mathbf{e}]_k\colon x\mapsto e_1\otimes\dots\otimes x\otimes\dots\otimes e_n$, where $x$ is at the $k$th position. Theorem 17. Let $E_1,\dots,E_n$, $F$ and $G$ be vector lattices, where $F$ and $G$ are order complete. Also let $0\leqslant e_i\in E_+$, $\mathbf{e}=(e_1,\dots,e_n)$ and $S_i\in L^+(E_i,F)$, $i=1,\dots,n$. Assume that $\mathcal{T}$ is a convex cyclic weakly closed weakly bounded subset of $L^+\bigl(\mathop{\overline{\bigotimes}}_{k=1}^{\,n}E_k,F\bigr)$, and $Q\colon F\to G$ is a sublinear Maharam operator. Then the following assertions are equivalent: (1) there exist $T\in\mathcal{T}$ and $R\in\partial Q$ such that $S_i=R\circ T\circ[\mathbf{e}]_i$, $i=1,\dots,n$; (2) for all $x_1\in E_1,\dots,x_n\in E_n$, Proof. By Theorem 8 and Corollary 2 there exists an increasing sublinear operator $\overline{P}\colon \mathop{\overline{\bigotimes}}_{k=1}^{\,n}E_k\to F$ such that $\partial(\overline{P})=\mathcal{T}$. Consider the linear operator $\Phi_{\mathbf{e}}$ from $E_1\times\dots\times E_n$ to $E_1\otimes\dots\otimes E_n$ and the linear operator $\mathbb{S}$ from $E_1\times\dots\times E_n$ to $F$ defined, respectively, by
$$
\begin{equation*}
\Phi_{\mathbf{e}}\colon (x_1,\dots,x_n)\mapsto \sum_{k=1}^n[\mathbf{e}_k]x_k\quad\text{and}\quad \mathbb{S}\colon (x_1,\dots,x_n)\mapsto\sum_{k=1}^nS_kx_k.
\end{equation*}
\notag
$$
Also consider the sublinear operator $P$ from $E_1\times\dots\times E_n$ to $F$ defined by
$$
\begin{equation*}
P\colon (x_1,\dots,x_n)\mapsto\sup_{T\in\mathcal{T}} \biggl\{Q\circ T\biggl(\sum_{i=1}^n[\mathbf{e}]_ix_i\biggr)\biggr\}.
\end{equation*}
\notag
$$
Using Corollary 1 and the Hahn–Banach–Kantorovich theorem, we find that for all $x_i\in E_i$, $i=1,\dots,n$,
$$
\begin{equation*}
\begin{aligned} \, P(x_1,\dots,x_n) &=\sup_{T\in\mathcal{T}} \biggl\{Q\biggl(\sum_{i=1}^n T\circ [\mathbf{e}]_ix_i\biggr)\biggr\} \\ &=\sup_{T\in\mathcal{T}} Q\circ{T}\circ\Phi_{\mathbf{e}}(x_1,\dots,x_n) =Q\circ\widehat{P}\circ\Phi_{\mathbf{e}}(x_1,\dots,x_n), \end{aligned}
\end{equation*}
\notag
$$
that is, $P=Q\circ\overline{P}\circ\Phi_{\mathbf{e}}$. By Theorems 5 and 11 In addition, $T\circ\Phi_{\mathbf{e}}(x_1,\dots,x_n)=\sum_{k=1}^nT\circ[\mathbf{e}_k]x_k$ for each $T\in L(\mathop{\overline{\bigotimes}}_{k=1}^{\,n}E_k,F)$. It remains to note that assertions (1) and (2) of the theorem are equivalent to the inclusions $\mathbb{S}\in(\partial Q)\circ(\partial\overline{P})\circ\Phi_{\mathbf{e}}$ and $\mathbb{S}\in\partial(P)$, respectively. Now the required result follows from the equality $\partial(P)=\partial(Q)\circ\partial(\overline{P})\circ\Phi_{e,f}$ (see above).
This proves the theorem. Remark 9. There is an analogue of Theorem 17 for multilinear operators — one only needs to replace there $\mathcal{D}$ by the set of multilinear operators from $E_1\times\dots\times E_n$ to $F$ and define the operator $[\mathbf{e}]_k$ from $E_k$ to $E_1\times\dots\times E_n$ by $[\mathbf{e}]_k\colon x\mapsto (e_1,\dots,x,\dots,e_n)$, where $x$ is at the $k$th position. Indeed, in view of Theorem 16 the linearization by means of the Fremlin tensor product $T\mapsto\overline{T}:=T^{\otimes}$ is an affine bijection between the sets $\mathcal{D}\subset L^{\rm r}(E_1,\dots,E_n;G)$ and $\overline{\mathcal{D}}:=\{\overline{D}\colon D\in\mathcal{D}\}\subset L^{\rm r}\bigl(\mathop{\overline{\bigotimes}}_{k=1}^{\,n} E_k,G\bigr)$. Note that $\overline{\mathcal{D}}$ is a convex cyclic weakly closed weakly bounded subset of $L^\sim\bigl(\mathop{\overline{\bigotimes}}_{k=1}^{\,n} E_k,G\bigr)$, and so by Proposition 2 and Theorem 8 there exists $\overline{P}\in{\rm Sbl}^+\,\bigl(\mathop{\overline{\bigotimes}}_{k=1}^{\,n} E_k, G\bigr)$ such that $\overline{\mathcal{D}}=\partial\overline{P}$. § 7. Multidimensional Monge–Kantorovich problemIn this section we consider the operator version of the multidimensional Monge–Kantorovich problem. For the history of this problem, foundations and the current state of this theory, see the books by Bogachev, Kolesnikov and Shaposhnikov [49], Rachev and Rüschendorf [50], and Villani [51] and [52]. 7.1. The direct and dual problemsGiven probability spaces $(\Omega_i,\Sigma_i,\mu_i)$, $i=1,\dots,n$, set $\Omega:=\Omega_1\times\dots\times\Omega_n$ and $\Sigma:=\Sigma_1\otimes\dots\otimes\Sigma_n$. For a probability measure $\mu$ on $\Sigma$, we denote by $\pi_i\mu$ the probability measure on $\Sigma_i$ defined by $\pi_i\mu(A):=\mu(\pi_i^{-1}(A))$, where $\pi_k\colon \Omega\to \Omega_k$ is the canonical projection. Also let $\mu_i\colon \Sigma_i\to\mathbb{R}_+$ be probability measures and $c$ be a nonnegative measurable function on $\Omega$ (a cost function). The multidimensional Monge–Kantorovich problem consists in finding the minimum of the functional
$$
\begin{equation*}
\int_{\Omega}c(\omega_1,\dots,\omega_n)\,d\mu(d\omega_1,\dots,d\omega_n)
\end{equation*}
\notag
$$
over all probability measures $\mu\colon \Sigma\to\mathbb{R}_+$ such that $\pi_k\mu=\mu_k$, $k=1,\dots,n$. In other words, the problem is to minimize a linear functional on the space of bounded measures over a convex set of probability measures with fixed marginals. The dual problem calls for the maximization of the functional
$$
\begin{equation*}
\sum_{k=1}^n\int_{\Omega_k}u_k(\omega_k)\,d\mu_k(\omega_k)
\end{equation*}
\notag
$$
over the set of tuples $(u_1,u_2,\dots,u_n)$, where we assume that $u_k\in L^1(\mu_k)$ and $\sum_{k=1}^nu_k(x_k)\leqslant c(x_1,\dots,x_n)$. The duality theorem asserts that, under certain conditions, the direct and dual problems have the same values; see Theorem 2.1.1 in [50]. 7.2. The operator statement of the problemLet $E_1,\dots,E_n$ and $F$ be vector lattices, where $F$ is order complete. Consider the Cartesian product $E_1\times\dots\times E_n$ and the Fremlin tensor product $\mathop{\overline{\bigotimes}}_{k=1}^{\,n} E_k$ of the vector lattices $E_1,\dots,E_n$; we also consider an arbitrary vector lattice $\overline{E}$ containing $\mathop{\overline{\bigotimes}}_{k=1}^{\,n} E_k$ as a majorizing sublattice. The last condition means that for each $\overline{u}\in \overline{E}$ there exists $u\in\mathop{\overline{\bigotimes}}_{k=1}^{\,n} E_k$ such that $\overline{u}\leqslant u$. Note that if $e_1,\dots,e_n$ are positive, then $[\mathbf{e}]_k\colon E_k\to\overline{E}$ is a lattice homomorphism and $[\mathbf{e}]_k(E_k)$ is a sublattice of $\overline{E}$ (see the notation at the beginning of § 6.2). In what follows the $e_k\in E_k$ are fixed positive elements. Now we state an operator version of the problem in § 7.1. Consider positive operators $S_k\colon E_k\to F$, $k=1,\dots,n$, and an element $u\in\overline{E}$. The operator statement of the multidimensional Monge–Kantorovich problem is as follows:
$$
\begin{equation*}
T\in L^+(\overline{E},F), \qquad T\circ[\mathbf{e}]_k=S_k, \quad k=1,\dots,n, \qquad Tu\to\inf.
\end{equation*}
\notag
$$
To be more precise, one is to calculate $\inf Tu$ in $F$ over all positive operators $T\colon \overline{E}\to F$ under the condition that $T\circ[\mathbf{e}]_k=S_k$ for all $k=1,\dots,n$. The dual problem has the form
$$
\begin{equation*}
\sum_{k=1}^n [\mathbf{e}]_kx_k\leqslant u, \quad k=1,\dots,n, \qquad \sum_{k=1}^n S_kx_k\to\sup.
\end{equation*}
\notag
$$
Thus, we need to find the supremum in $F$ of the sum $\sum_{k=1}^n S_kx_k$ over all $(x_1,\dots,x_n)\in E_1\times\dots\times E_n$ satisfying $\sum_{k=1}^n [\mathbf{e}]_kx_k\leqslant u$. We let $L(u)$ and $U(S_1,\dots,S_n)$ denote the values of the direct and dual Monge–Kantorovich problems, respectively, that is,
$$
\begin{equation*}
L(u):=\inf\bigl\{Tu\colon T\in L^+(\overline{E},F),\, S_k=T\circ[\mathbf{e}]_k,\,k=1,\dots, n\bigr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
U(S_1,\dots,S_n):=\sup\biggl\{\sum_{k=1}^n S_kx_k\colon x_k\in E_k\, 1\leqslant k\leqslant n,\, \sum_{k=1}^n [\mathbf{e}]_kx_k\leqslant u\biggr\}.
\end{equation*}
\notag
$$
7.3. The Young–Fenchel transformFor a further analysis of the duality under consideration we need an auxiliary fact from convex analysis. Let $X$ be a vector space, $Y$ be an ordered vector space, $C\subset X$ be a nonempty convex set, $f\colon X\to Y$ be a convex operator, and let $\delta_C\colon X\to E\cup\{+\infty\}$ be the ($E$-valued) indicator function defined by $\delta_C(x)=0\in E$ for $x\in C$ and $\delta_C(x)=+\infty$ otherwise. Definition 18. The Young–Fenchel transform of an operator $g\colon X\to E$ is the operator $g^\ast\colon L(X,E)\to E\cup\{\pm\infty\}$ defined by In place of $(\delta_C)^\ast$ one writes $C^\ast$, and so $C^\ast(T)=\sup\{Tx\colon x\in C\}$, $T\in L(X,E)$. Lemma 3. If the set $\{f\leqslant0\}:=\{x\in X\colon f(x)\leqslant0\}$ is nonempty and $f(X)\subset E$, then for each linear operator $S\colon X\to E$, In addition, $\{f\leqslant0\}^*(S)=(T\circ f)^*(S)$ for some $T\in L^+(Y,E)$. For the arguments underling the proof, see [11], Proposition 4.1.11, (4); the required condition of algebraic general position (see § 5.5.4 in [11]) is met because $f(X)\subset E$. For general change-of-variable formulae for the Young–Fenchel transform; see [11], Ch. 4, and, in particular, Theorems 4.1.4, 4.1.8, 4.1.13, 4.5.3 and 4.5.4 there. 7.4. The duality theoremNow we formulate and prove the main result in this section; here $E_1,\dots,E_n$ and $F$ are as in § 7.2. Theorem 18. Assume that $\overline{x}_k\in E_k$ and $S_k\in L(E_k,F)$, $k=1,\dots,n$, are such that $\sum_{k=1}^n [\mathbf{e}]_k\overline{x}_k\leqslant u$ and $\sum_{k=1}^n S_kx_k\leqslant T\bigl(\sum_{k=1}^n [\mathbf{e}]_k\overline{x}_k\bigr)$ for appropriate $u\in E_+$ and $T\in L^+\bigl(\mathop{\overline{\bigotimes}}_{k=1}^{\,n} E_k, F\bigr)$. Then $L(u)=U(S_1,\dots,S_n)$, and the infimum on the left-hand side is attained. Proof. To apply Lemma 3 we set $X:=E_1\times\dots\times E_n$, $Y:=\overline{E}$, and define maps $S\colon X\to F$ and $f\colon X\to Y$ by
$$
\begin{equation*}
S(x_1,\dots,x_n)=\sum_{i=1}^n S_ix_i\quad\text{and} \quad f(x_1,\dots,x_n)=\sum_{k=1}^n [\mathbf{e}]_kx_k -u.
\end{equation*}
\notag
$$
By the assumptions of the theorem the set $\{f\leqslant0\}$ is nonempty. By Theorem 17 and Kantorovich’s theorem on an extension of a positive operator (see Remark 1, (2)) there exists a positive extension $\overline{T}\colon \overline{E}\to F$ of the operator $T$ such that $S_k=\overline{T}_k\circ[\mathbf{e}]_k$, $k=1,\dots,n$. Hence $\sum_{k=1}^n S_k\leqslant \overline{T}(u)$, and therefore we have $L(u), U(S_1,\dots,S_n)<+\infty$. The operators $S$ and $f(\,{\cdot}\,)+u$ are linear, hence for each linear operator $T\colon Y\to F$ the quantity is different from $+\infty$ if and only if $S=T\circ f$, and in this case $(T\circ f)^\ast(S)=Tu$. Note also that the equality $S=T\circ f$ is equivalent to the system of equalities $S_i=T\circ[\mathbf{e}]_i$, $i=1,\dots,n$. Now, recalling Definition 18 of the Young–Fenchel transform and employing Lemma 3, we find that
$$
\begin{equation*}
\begin{aligned} \, U(S_1,\dots,S_n) &:=\sup\biggl\{\sum_{k=1}^n S_kx_k\colon x_k\in E_k,\, \sum_{k=1}^n [\mathbf{e}]_kx_k\leqslant u\biggr\} \\ &=\sup\bigl\{S(x)\colon x\in X,\ f(x)\leqslant0\bigr\}=\{f\leqslant0\}^\ast(S) \\ &=\inf\{(T\circ f)^*(S)\colon T\in L^+(Y,F)\} \\ &=\inf\bigl\{Tu\colon T\in L^+(Y,F),T\circ[e]_i=S_i,\,1\leqslant i\leqslant n\bigr\}=:L(u). \end{aligned}
\end{equation*}
\notag
$$
In addition, by Lemma 3, $\{f\leqslant 0\}^*(S)=(T_0\circ f)^*(S)$ for some $T_0\in L^+(Y,F)$, and therefore $L(u)=T_0(u)$ and $T_0\circ[\mathbf{e}]_i=S_i$ for all $i=1,\dots,n$, as required.
Theorem 18 is proved. Remark 10. The Monge–Kantorovich problem has for decades attracted the attention of specialists from various fields of mathematics: probability theory, functional analysis, differential geometry, convex analysis, dynamical systems, and so on; see the books [49]–[52] and the surveys [53]–[56]. The duality theorem (Theorem 18) dates back to Kantorovich [57], who obtained it as an application of linear programming, which he had discovered; Kantorovich became aware of Monge’s paper [58] only later (see [59]). In §§ 7.2 and 7.4 we presented a version of the Monge–Kantorovich problem in the context of operator convex analysis [11], which, as we think, deserves a special study. § 8. On Choquet’s theory in Kantorovich spacesCiosmak [55] pioneered a new approach to the mass transfer problem which is based on Strassen’s disintegration theorem and Choquet’s theory. In this section we show that the operator Choquet theory [10] has similar capabilities. The necessary machinery here is the Maharam extension of a positive operator and a description of the set of extreme points for a special class of sublinear operators (submorphisms). 8.1. Maharam extensionWe require one useful result to the effect that each positive operator from a vector lattice to a Kantorovich space can be extended to a Maharam operator. This construction was proposed in [60] and, independently, [61]. By $\mathcal{Z}(F)$ we denote the ideal centre of a vector lattice $F$, that is, the order ideal in the space of order-bounded operators generated by the identity operator $I_F$; see [12], § 3.3.2, (5)–(8). The family of pairwise disjoint order projections $(\pi_\xi)$ in $F$ is called a partition of the projection $\pi$ if $\pi=\sup_\xi\pi_\xi$. For details on the following result and historical notes, see § 3.5 in [12]. Theorem 19. Let $T$ be a positive operator from a vector lattice $X$ to a Kantorovich space $F$. Then there exist a (unique up to a lattice isomorphism) vector lattice $E$, a lattice homomorphism $\iota_X\colon X\to E$, a lattice ring isomorphism $h\colon \mathcal{Z}(F)\to\mathcal{Z}(E)$, and an essentially positive Maharam operator $\widehat{T}\colon E\to F$ such that: In what follows we set by definition $L^1(T):=E$ and $ \mathopen{|\kern-3pt|\kern-3pt|} u \mathclose{|\kern-3pt|\kern-3pt|} :=\widehat{T}(|u|)$, and denote by $\overline{X}$ the order ideal in $L^1(T)$ generated by the sublattice $\iota_X(X)$; see § 3.5.2 in [12]. The following result, which describes the band $\{T\}^{\perp\perp}$, extends Theorem 10 (the Radon–Nikodým theorem) to arbitrary positive operators; see Theorem 3.5.4 in [12]. Theorem 20. For each operator $S\in\{T^{\perp\perp}\}$ there exists a unique operator ${\overline{S}\in\{\overline{T}\}^{\perp\perp}}$ such that $S=\overline{S}\circ\iota_X$. The correspondence $S\mapsto\overline{S}$ is an isomorphism of the vector lattices $\{T\}^{\perp\perp}$ and $\{\overline{T}\}^{\perp\perp}$. 8.2. Lattice submorphismsWe need the description of extreme points of the subdifferential for a special class of sublinear operators (submorphism); see § 3.3.9 in [12]. Definition 19. A sublinear operator $P\colon E\to F$ is said to be a lattice submorphism if $P(x\vee y)=P(x)\vee P(y)$ for all $x, y\in E$. Any submorphism is an increasing operator. Given an order complete vector lattice $F$, we define the operator $\varepsilon_{\mathcal{U}}:=\varepsilon_{\mathcal{U},F}$ from $l_\infty (\mathcal{U}, F)$ to $F$ by $\varepsilon_{\mathcal{U}}\colon f\mapsto\sup\{f(u)\colon u\in \mathcal{U}\}$, $f\in l_\infty(\mathcal{U},F)$; cf. Example (b) in § 4.2. It is easily checked that $\varepsilon_{\mathcal{U}}$ is a lattice submorphism, which is usually referred to as the canonical sublinear Kutateladze operator; see § 2.3 in [27]. By $\mathcal{H}(E,F)$ we denote the set of all lattice homomorphisms from $E$ to $F$. Lemma 4. If $\mathfrak{A}\subset L(X,F)$ and for each $x\!\in\! X$ there exists $P(x)\!:=\!\sup\{Tx\colon T\!\in\!\mathfrak{A}\}$, then $P$ is a sublinear operator from $X$ to $F$ and $P=\mathcal{E}_\mathfrak{A}\circ\langle\mathfrak{A}\rangle$. Theorem 21. Let $E$ and $F$ be vector lattices, where $F$ is order complete. For an increasing sublinear operator $P\colon E\to F$ the following assertions are equivalent: (1) $P$ is a lattice submorphism; (2) the extreme points of $\partial(P)$ are lattice homomorphisms; (3) $P(x)=\sup\{Tx\colon T\in\partial(P)\cap\mathcal{H}(E,F)\}$ for all $x\in E$; (4) $P$ factors as $P=\varepsilon_\mathcal{U}\circ\mathcal{T}$, where $\mathcal{T}\in\mathcal{H}(E, l^\infty(\mathcal{U},F))$. The equivalences (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3) are due to Buskes and van Rooij [62]; the implications (3) $\Rightarrow$ (4) $\Rightarrow$ (1) are clear. 8.3. The balayage theoremConsider the convex cone $H$ in a vector lattice $X$, that is, $H+H=H$ and $\lambda H\subset H$ for all $\lambda\in\mathbb{R}_+$. We assume that $H$ is a lower lattice $(x,y\in H\Rightarrow x\wedge y\in H)$ which majorizes $X$. (For each $x\in X$ there exists $h\in H$ such that $ x\leqslant h$). Let $S,T\in L^+(X, F)$. Definition 20. An operator $S$ is called a balayage of an operator $T$ (relative to the cone $H$), written $T\preccurlyeq_H S$, if $Th\geqslant Sh$ for all $h\in H$; the relation $\preccurlyeq_H$ is known as the Choquet ordering. The positive germ of $T$ on the cone $H$ is the set of all balayages of $T$, that is, An operator $T$ is a maximal operator (relative to) $H$ if $\operatorname{Spr}(T,H)=\{T\}$. It is easily seen that the Choquet ordering is a preorder, which is an order if $H-H$ is dense in $X$ relative to some topology in which positive operators are continuous. It is also clear that $\operatorname{Spr}(T,H)$ is a nonempty convex set (which contains $T$). Now we can formulate the main result in this section, where we describe the germ of a positive operator. Theorem 22. Let $T$, $\widehat{T}$, $\iota_X$ and $L^1(T)$ be as in § 8.1. Let the map $P_H:=P_{H,T}$: $X\to L^1(T)$ be defined by $P_H(x):=\inf\{\iota_X(h)\colon h\in H,\, x\leqslant h\}$, where the infimum is taken in $L^1(T)$. Then $P_H$ is a submorphism, and the following assertions hold: (1) $\operatorname{Spr}(T,H)=\widehat{T}\circ\partial(P_H)$; (2) $\operatorname{Spr}(T,H)=\widehat{T}\circ\operatorname{Spr}(\iota_X,H)$; (3) $\operatorname{Spr}(T,H)=o\text{-}\operatorname{cl}\widehat{T}\bigl(\operatorname{co}_\Lambda (\mathcal{H}(X,L^1(\widehat{T}))\cap\partial P_H)\bigr)$; (4) $\operatorname{Spr}(T,H)=\widehat{T}\circ \partial(\mathcal{E}_{\mathfrak{A}})\circ\langle\mathfrak{A}\rangle$ for some set $\mathfrak{A}\subset\mathcal{H}(X,L^1(T))$. Proof. The map $x\mapsto Q (x):=\inf\{T(\iota_X(h))\colon h\in H,\,x\leqslant h\}$ is a sublinear operator from $X$ to $G$. In addition, $Q=\widehat{T}\circ P_H$ since the set $\{\iota_X(h)\colon h\in H,\,x\leqslant h\}$ is downward filtered and the operator $\widehat{T}$ is order continuous. Assertion (1) follows from Corollary 3 (which must be used in the form of Corollary 3 in [63]) because $\operatorname{Spr}(T,H)=\partial(Q)$; to verify (2) we must also recall that $\operatorname{Spr}(\iota_X,H)=\partial(P_H)$. Assertion (3) can be derived from Theorems 8 and 21; assertion (4) is a consequence of (3) and Lemma 4.
This proves the theorem. Corollary 5. An operator $S\in L^+(X,F)$ lies in $\operatorname{Spr}(T,H)$ if and only if $S=\widehat{T}\circ R$ for some $R\in\operatorname{Spr}(\iota_X,H)$. In addition, for an appropriate net $(R_\alpha)$ in the convex hull $\operatorname{mix}\bigl(\mathcal{H} (X,L^1(\widehat{T}))\cap\partial(P_{H})\bigr)$. 8.4. Extreme balayagesLet $Q\colon X\to F$ be a submorphism. We denote by $\operatorname{Blg}(Q,H)$ the set of all pairs $(T,S)$ such that $T,S\in L^+(X,F)$, $T\in \partial Q$ and $T\preccurlyeq_H S$. The following result describes the extreme points of the convex set $\operatorname{Blg}(Q,H)$. Theorem 23. The following assertions hold: Proof. (1) Given a pair $(T,S)\in\operatorname{ext}(\operatorname{Blg}(Q))$, assume that $T=\alpha T_1+(1-\alpha)T_2$ for some $T_1,T_2\in\partial Q$ and $1<\alpha<1$. By the decomposition theorem (see § III.1.2 in [10] or § 2 in [64])
$$
\begin{equation*}
\operatorname{Spr}(\alpha T_1+(1-\alpha)T_2,H)=\alpha\operatorname{Spr}(T_1,H)+(1-\alpha)\operatorname{Spr}(T_2,H),
\end{equation*}
\notag
$$
and so $(T,S)=\alpha(T_1,S_1)+(1-\alpha)(T_2,S_2)$ for appropriate $S_1\in\operatorname{Spr}(T_1,H)$ and $S_2\in\operatorname{Spr}(T_2,H)$. By assumption $T=T_1=T_2$, and so $T$ is an extreme point of the set $\partial Q$. Thus, $T$ is a lattice homomorphism by Theorem 21.
Assertion (2) is proved similarly using Theorems 21, 20 and 22 and also an auxiliary result due to Buskes and van Rooij; see [12], Proposition 3.3.9, (2). This proves the theorem. Remark 11. The classical Choquet theory is based on the duality idea: to a convex compact set $K$ there corresponds the space of all continuous affine functions $A(K)\subset C(K)$ on this compact set, and vice versa, to any closed subspace $A$ of $C(K)$ which contains constant functions there corresponds the space of its states (a compact convex set). This idea also underlies both versions (the commutative and noncommutative ones) of the operator Choquet theory. Remark 12. The first operator version of Choquet theory for vector measures with values is an order complete vector lattice with strong unit was proposed by Vincent-Smith [65]. The theory of maximal operators in Kantorovich spaces was put forward by Kutateladze [64]; see also the book by Akilov and Kutateladze [10], Ch. 3. Roth [66] proposed another version of the operator Choquet theory, where the linear operators are defined on the space of continuous vector functions, and the Choquet ordering is not introduced in terms of a selected cone, but rather in terms of a family of seminorms (see p. 190 in [66]). The noncommutative Choquet theory originates from Arveson (see, for example, [67]); for further advances, see Davidson and Kennedy [68] and the references there.
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