Proof of 1. For [pt:ideal:explicit], any ideal containing \(A\) must contain \(\mathop{\mathrm{Span}}(A)\) and hence also the right hand side. For the reverse inclusion, it suffices to observe that the right hand side is an ideal; it follows that it contains \(V_A\).

We now assume \(A\subseteq V^+\). For [pt:ideal:sums], let \(v\in V_A\) and consider \(x,y\in\mathop{\mathrm{Span}}(A)\) as in [pt:ideal:explicit]. Drop from the linear combination \(y\) all terms afflicted by a negative coefficient; we still have \(v\leq y\) for this new \(y\). Likewise, drop positive coefficient terms from \(x\). With these new \(x,y\), we obtain \(a_1 + \cdots + a_n\) from \(y-x\).

For [pt:ideal:generate], let \(v\in V_A\). By [pt:ideal:sums], we can choose \(a_1, \ldots, a_n\in A\) with \(- v \leq a\), where \(a=a_1 + \cdots + a_n\). Now \(a\in V_A^+\) and \(v+a\geq 0\) is also in \(V_A^+\), witnessing \(V_A = V_A^+ - V_A^+\). The additional statement holds by definition of the dual cone. ◻

Proof. [pt:VP:monotone] follows from [pt:VP:add] since \(r(v, w) = r(u, w) + r(v-u, w) \geq r(u, w)\). [pt:VP:inv] follows from [pt:VP:cocycle] upon considering separately the values \(0, +\infty\); then [pt:VP:0] follows too, noting that [pt:VP:add] grants \(r(0, v) = 0\).

For the remaining two points [pt:VP:extends] and [pt:VP:hom], fix \(v\neq 0\). Consider the sets \[\begin{align} \mathop{\mathrm{Fin}}^+(v) &= \{ u\in V^+ : r(u, v) \neq +\infty \}\kern3mm\text{and}\\ \mathop{\mathrm{Fin}}(v) &= \mathop{\mathrm{Fin}}^+(v) - \mathop{\mathrm{Fin}}^+(v) \subseteq V. \end{align}\] By [pt:VP:add], \(\mathop{\mathrm{Fin}}^+(v)\) is closed under addition and by [pt:VP:monotone] it is closed under multiplication by \(0\leq t \leq 1\). Therefore \(\mathop{\mathrm{Fin}}^+(v)\) is a cone and \(\mathop{\mathrm{Fin}}(v)\) is a vector subspace. Moreover, \((\mathop{\mathrm{Fin}}(v))^+= \mathop{\mathrm{Fin}}^+(v)\); indeed, if \(u-u'\geq 0\) with \(u, u'\in \mathop{\mathrm{Fin}}^+(v)\) then [pt:VP:monotone] implies \(r(u-u', v) \leq r(u,v) < +\infty\) so that \(u-u'\in \mathop{\mathrm{Fin}}^+(v)\). We note in passing that \(\mathop{\mathrm{Fin}}(v)\) is an ideal because \(\mathop{\mathrm{Fin}}^+(v)\) is closed for order-intervals.

Now that we know that \(\mathop{\mathrm{Fin}}^+(v)\) is the positive cone of the space \(\mathop{\mathrm{Fin}}(v)\) that it spans, it follows that the (finite-valued) positive additive map \(r(\cdot, v)\) on \(\mathop{\mathrm{Fin}}^+(v)\) has a unique extension to \(\mathop{\mathrm{Fin}}(v)\) as a positive linear functional, see e.g. [15] or [16]. Thus [pt:VP:extends] holds because \(V_v \subseteq\mathop{\mathrm{Fin}}(v)\) by [pt:VP:monotone]. At this point we have also obtained [pt:VP:hom] since the latter is trivial for \(u\notin \mathop{\mathrm{Fin}}^+(v)\). ◻

Proof. The weak-* topology is already weak and it is complete because the algebraic dual is isomorphic, as topological vector space, to the product of copies of \(\mathbf{R}\) indexed by a basis of the initial space. See e.g. [18]. The second statement follows since positivity is a weak-* closed condition in the dual. ◻

Proof. We only consider \(r_P(u,v)\) when \(v\neq 0\), see 4. Since \(P(\cdot| u+v)\) is additive with \(P(u+v| u+v)=1\), it cannot happen that both numerator and denominator vanish. Thus, \(r_P(u,v)\) is well-defined in \([0, +\infty]\).

Observe that whenever \(x\geq u+v\) is such that \(P(u+v|x) \neq 0\), [pt:CM:trans] implies \[\frac{P(u|x)}{P(v|x)} = \frac{P(u|u+v) P(u+v|x)}{P(v|u+v) P(u+v|x)} = \frac{P(u|u+v)}{P(v|u+v)} =r_P(u,v).\] In order to check [pt:VP:add], consider \(u,v,w\in V^+\) with \(w\neq 0\). If both \[P(u+w|u+v+w) \neq 0 \kern3mm \text{and} \kern3mm P(v+w|u+v+w)\neq 0,\] then the above observation with \(x=u+v+w\) allows us to verify \[\begin{gather} r_P(u, w) + r_P(v, w) = \frac{P(u|u+v+w)}{P(w|u+v+w)} + \frac{P(v|u+v+w)}{P(w|u+v+w)} = \\ = \frac{P(u+v|u+v+w)}{P(w|u+v+w)} = r_P(u+v, w). \end{gather}\] Otherwise, without loss of generality \(P(u+w|u+v+w)=0\) which implies \(P(w|u+v+w)=0\) and thus \(r_P(u+v, w)=+\infty\). On the other hand, \(P(u+w|u+v+w)=0\) also implies \(P(v|u+v+w)=1\) and hence also \(P(v+w|u+v+w)=1\). Combining this with [pt:CM:trans] applied as \[P(w|v+w) \, P(v+w|u+v+w) = P(w|u+v+w)=0\] implies \(P(w|v+w)=0\) and hence \(r_P(v, w)=+\infty\), so that [pt:VP:add] holds.

The verification of [pt:VP:cocycle], namely \[\frac{P(u|u+v)}{P(v|u+v)} \, \frac{P(v|v+w)}{P(w|v+w)} =\frac{P(u|u+w)}{P(w|u+w)},\] is entirely similar, considering again \(x=u+v+w\). Namely, if none of the pairwise sums of \(u\), \(v\) and \(w\) are zeros of \(P(\cdot|u+v+w)\), then the identity follows from the above observation. Otherwise, two among \(u\), \(v\) and \(w\) are zeros of \(P(\cdot|u+v+w)\). If this is the case of \(u\) and \(v\), then \(P(w|u+v+w)=1\), which implies \(P(u+w|u+v+w)=1\) and hence using [pt:CM:trans] for \[P(u|u+w) \, P(u+w|u+v+w) = P(u|u+v+w)=0\] we deduce \(P(u|u+w)=0\). Likewise, \(P(v|v+w)=0\); now the desired relation reads \(0=0\).

In case \(v\) and \(w\) are zeros of \(P(\cdot|u+v+w)\), we deduce in the same way that the relation becomes \(+\infty=+\infty\) and finally in case \(u\) and \(w\) are zeros of \(P(\cdot|u+v+w)\), the left hand side is the undefined form \(0\cdot+\infty\).

[pt:VP:one] holds trivially and the remaining statement about \(r\mapsto P_r\) and \(P\mapsto r_P\) is a direct verification. ◻

Proof of 9. We first claim that \(V\) admits a conditional mean \(P\), and only one, such that \[\forall (U, J) \in \mathscr{C}\;\forall 0\leq u\leq v\in U^+\;\text{with } J(v) \neq 0 :\kern3mm P(u|v) = J(u) / J(v).\] To begin with, observe that each \((U,J)\in\mathscr{C}\) is uniquely determined by \(J\). Indeed, if \(\mathscr{C}\) also contained \((U', J)\) with \(U'\neq U\) then without loss of generality \((U, J) \prec (U', J)\), implying \(J(U)=0\), which we excluded in the definition of positive partial functionals.

Therefore we can simplify notation and denote any \((U,J)\in \mathscr{C}\) just by \(J\); accordingly, we write \(J_1\prec J_2\) etc.

Let \(v\in V^+\) be non-zero. By fullness, there is \(J \in \mathscr{C}\) with \(J(v) \neq 0\) (and \(v\) in the domain of \(J\)). Since \(\mathscr{C}\) is a chain, the definition of the order \(\prec\) shows that there can only be one such \(J \in \mathscr{C}\). We thus denote the corresponding pair by \((U^v, J^v)\). (Beware of the misleading similarity with the notation \(V_v\).) We shall use repeatedly that \(u\in U^v\) for all \(0\leq u \leq v\) since \(U^v\) is an ideal.

At this point, the initial existence and uniqueness claim is reduced to stating that the expression \[P(u|v) = J^v(u) / J^v(v) \text{ for } 0\leq u\leq v \neq 0\] defines indeed a conditional mean on \(V\). Only [pt:CM:trans] needs a justification. Pick thus any \(0\leq u \leq v \leq w\) with \(v\neq 0\). We need to justify \[\frac{J^v(u)}{J^v(v)} \;\frac{J^w(v)}{J^w(w)} =\frac{J^w(u)}{J^w(w)}.\] This is tautological if \(J^v=J^w\). Otherwise, recalling \(v\in U^w\), the chain condition on \(\mathscr{C}\) forces \(J^w(v)=0\) and therefore also \(J^w(u)=0\) since \(u\leq v\). In that case, the above relation is of the form \(0=0\). The first claim stands established.

By 6, there is one and only one vector pricing \(r\) corresponding to \(P\). To justify that \(r\) satisfies the statement of the theorem, let \((U, J) \in \mathscr{C}\) and let \(u,v\in U^+\) with \(J(v) \neq 0\). We know \(J=J^v\) and need to show \(r(u,v) = J^v(u) / J^v(v)\), which so far we only established under the additional assumption \(u\leq v\). On the other hand, 6 together with the initial claim grants us \[r(u,v) = \frac{P(u|u+v)}{P(v|u+v)} = \frac{J^{u+v}(u) / J^{u+v}(u+v)}{J^{u+v}(v) / J^{u+v}(u+v)} = \frac{J^{u+v}(u)}{J^{u+v}(v)}.\] In conclusion, to finish the proof of the theorem, it suffices to show that under the current assumptions on \(u,v\) and on \(J=J^v\), we have \(J^v=J^{u+v}\). This holds by uniqueness of \(J^{u+v}\) because \(J^v(u+v) \geq J^v(v)>0\); we used here \(u+v\in U=U^v\). ◻

Proof of 11. Suppose first that \(\mathscr{C}\) is maximal. Let \(v\in V^+\) be a non-zero positive vector; we need to show that there is \((U, J)\in\mathscr{C}\) with \(v\in U\) and \(J(v)\neq 0\). Let \(W_0\subseteq V\) be the union of the ideals \(U\) with \(v\notin U\) when \((U, J)\) ranges over \(\mathscr{C}\); by convention, \(W_0=0\) if there is no such ideal \(U\). Since \(\mathscr{C}\) is a chain, \(W_0\) is an ideal in \(V\) and \(v\notin W_0\). We consider the ideal \(W\subseteq V\) generated by \(W_0\) and \(v\); we further form the quotient \(\overline{W} = W/W_0\). The image \(\overline{v}\) of \(v\) in \(\overline{W}\) is non-zero and positive for the quotient order; we recall that the quotient order is well-defined because \(W_0\) is an ideal (in \(V\) and hence in \(W\)). Moreover \(\mathbf{R}\overline{v}\) is majorizing in \(\overline{W}\) by construction. Therefore, we can apply the Kantorovich¹ theorem ([26] or [16]) to obtain a positive linear functional \(\overline{K} \colon \overline{W} \to\mathbf{R}\) with \(K(\overline{v}) >0\). This gives us by pull-back a positive partial functional \((W, K)\).

We now suppose for a contradiction that there is no \((U, J) \in \mathscr{C}\) with \(v\in U\) and \(J(v) \neq 0\) and we argue that \((W, K)\) can be added to \(\mathscr{C}\), contradicting maximality. Consider any \((U, J)\in \mathscr{C}\). By construction, we have \((U, J) \prec (W, K)\) whenever \(v\notin U\). It suffices therefore to show \((W, K) \prec (U, J)\) when \(v\in U\). The current assumption implies that \(J(v)=0\) holds in that case. On the other hand, if \((U', J')\in \mathscr{C}\) satisfies \(v\notin U'\) then \((U', J') \prec (U, J)\) because the reverse order is impossible. Therefore \(J\) vanishes on any such \(U'\) and hence on \(W_0\). It follows that \(J\) vanishes on \(W\), as was to be shown.

The converse is elementary. If a full chain \(\mathscr{C}\) were not maximal, then there would be some positive partial functional \((U', J')\) not in \(\mathscr{C}\) that is comparable to every element of \(\mathscr{C}\). Choose some \(v\in {U'}^+\) with \(J'(v)\neq 0\). By assumption, there is \((U, J) \in \mathscr{C}\) with \(v\in U\) and \(J(v) \neq 0\). Then both \((U', J') \prec (U, J)\) and \((U, J) \prec (U', J')\) are impossible. ◻

Proof. Suppose for a contradiction that \(\mathscr{C}\) is not full. Thus there is \(v\gneqq 0\) such that for every \((\mathop{\mathrm{Fin}}(u), r(\cdot, u)) \in \mathscr{C}\) we have either \(v\notin \mathop{\mathrm{Fin}}(u)\) or \(r(v,u)=0\). We claim that we can then add \((\mathop{\mathrm{Fin}}(v), r(\cdot, v))\) to \(\mathscr{C}\), contradicting maximality. It suffices to check that for all \((\mathop{\mathrm{Fin}}(u), r(\cdot, u)) \in \mathscr{C}\), that we have \[\begin{align} \text{either} \kern3mm (\mathop{\mathrm{Fin}}(v), r(\cdot, v)) &\prec (\mathop{\mathrm{Fin}}(u), r(\cdot, u))\\ \text{or} \kern3mm (\mathop{\mathrm{Fin}}(u), r(\cdot, u)) &\prec (\mathop{\mathrm{Fin}}(v), r(\cdot, v)). \end{align}\] If \(v\in \mathop{\mathrm{Fin}}(u)\), then our apagogical assumption forces \(r(v,u)=0\). Now [pt:VP:cocycle] implies \(r(w,u)=r(w,v) r(v,u)= 0\) for all \(w\in \mathop{\mathrm{Fin}}(v)\), which shows that the first alternative holds. Otherwise, \(v\notin \mathop{\mathrm{Fin}}(u)\) implies \(r(v, u)=+\infty\), which means \(r(u,v)=0\) and by symmetry the second alternative holds.

This shows that \(\mathscr{C}\) is full and the maximality statement now follows from the easy direction in 11. ◻

Proof of 12. Let \(V\) be a subspace of an ordered vector space \(W\) and let \(r\) be a vector pricing on \(V\). Consider a chain \(\mathscr{C}\subseteq\mathscr{F}_r\) that is maximal within \(\mathscr{F}_r\). Given a positive partial functional \((U,J)\) on \(V\), it can be extended to a positive partial functional \((W_U, J)\) on \(W\), where as usual \(W_U\) denotes the ideal in \(W\) generated by \(U\). Indeed, this follows from the Kantorovich theorem ([26] or [16]) because \(U\) majorizes \(W_U\). There is no ambiguity in the notation \(J\) because \(W_U \cap V = U\) since \(U\) is an ideal in \(V\).

Having chosen such extensions for each \((U, J) \in \mathscr{C}\), we consider \(\mathscr{C}\) as a chain of positive partial functionals on \(W\). We note at this occasion that the definition of the order \(\prec\) does not depend on the ambient vector space, but only on the partial functionals being compared. Indeed, if \((U,J)\) and \((U', J')\) are elements of \(\mathscr{C}\) for \(V\), then \(U \subseteq\ker(J') \Leftrightarrow W_U \subseteq\ker(J')\) under any extension of \(J'\) to \(W_{U'}\).

By Hausdorff’s maximality principle, there exists a maximal chain \(\mathscr{C}'\) containing \(\mathscr{C}\), maximal among all chains of positive partial functionals on \(W\). We apply 11 and 9 to the chain \(\mathscr{C}'\) for \(W\) and obtain a vector pricing \(r'\) on \(W\). It remains to show that \(r'\) coincides with \(r\) on \(V\). This follows from the uniqueness statement of 9 applied to the chain \(\mathscr{C}\) for \(V\) because 13 validates the required hypothesis. ◻

Proof of 14. We can restrict to \(u,v\neq 0\) because of 4. The equivalence of the first four points follows readily from axiom [pt:VP:cocycle]. It remains to discuss [pt:def-inv:2] for \(u\gneqq 0\). By [pt:VP:inv], that statement is equivalent to \(r(u + g u, u)=2\), which is equivalent to [pt:def-inv:1] since \(r(u + g u, u)=1+ r(g u, u)\) and \(r(g u, u)=1\Leftrightarrow r(u, g u)=1\). ◻

Proof of 16. We suppose that \(P\) is invariant and proceed to verify the criterion [pt:def-inv:2] in 14. Our assumption applied to \(v=u + g u\) gives \(r(g u, u+g u) = r(u, g u + u)\). Therefore \(r(g u+u, g u + u)=1\) yields the desired conclusion. The converse implication is tautological. ◻

Proof of 18. We fix a family of elementary cylinders witnessing that \(\mathscr{C}\) satisfies \(\pi\). Our definition of conditional means is given by an intersection of closed properties; therefore, it suffices to prove the following statement.

\((*)\) Given any finite set \(0\in E\subseteq V^+\), there is a map \(P\colon \mathscr{O}\to [0, 1]\) satisfying [pt:CM:add], [pt:CM:trans] and [pt:CM:one] when \(u,v,w\in E\) and such that \(P(u|v)\in C\) for those of the given elementary cylinders determined by some \((u, v)\in \mathscr{O}\) and \(C\subseteq[0, 1]\) for which \(u, v\in E\).

Since [pt:CM:add] involves sums, we consider the finite set \(E'= E+E \supseteq E\). In proving \((*)\), it suffices to define \(P\) on \(\mathscr{O}\cap (E'\times E)\) because we can take \(P\) arbitrary outside this set. We apply the hypothesis of the theorem to the finite set \(F = E' \smallsetminus\{0\}\), obtaining a suitable chain \(\mathscr{C}_F \subseteq\mathscr{C}\).

Since \(\mathscr{C}_F\) is a chain, it follows (as in the proof of 9) that, given \(v\in F\), there is a unique \((U, J)\) in \(\mathscr{C}_F\) with \(v\in U\) and \(J(v) \neq 0\); moreover, \(J\) determines \(U\) within \(\mathscr{C}_F\). Thus, we again denote this pair by \((U^v, J^v)\) or simply by \(J^v\).

We now define \(P\) on \(\mathscr{O}\cap (E'\times E)\) by \[P(u|v) = J^v(u) / J^v(v).\] This is well-defined because \(u\in U^v\) and it ranges in \([0, 1]\) since \(J^v(u) \leq J^v(v)\neq 0\). By construction, \(P(u|v)\in C\) in case one of the elementary cylinders is given by this pair \((u,v)\).

Therefore, to complete the proof, it suffices to verify that \(P\) satisfies [pt:CM:add], [pt:CM:trans] and [pt:CM:one] for all \(u,v,w\in E\). For [pt:CM:add] and [pt:CM:one], this is by construction. For [pt:CM:trans], the argument given in the proof of 9 applies without any change. ◻

Proof of 19. The strategy is to apply 18 for the elementary property \(\pi\) of \(G\)-invariance, recalling from 17 that it is indeed elementary. This will establish the existence of a \(G\)-invariant conditional mean. The corresponding vector pricing constructed in 6 will then be invariant by 16.

We define a collection \(\mathscr{C}\) of partial positive functionals as follows. Given any non-zero \(v\geq 0\), let \(V_{G v}\) be the ideal generated by the orbit \(G v\); this is a \(G\)-invariant ideal. By assumption, there exists a non-zero \(G\)-invariant positive linear map \(J\colon V_{G v} \to \mathbf{R}\). We note that \(J(v)\neq 0\); indeed otherwise by invariance \(J(G v) =\{0\}\) so that \(J\) would vanish on the entire ideal \(V_{G v}\).

We define \(\mathscr{C}\) to be the collection of all such pairs \((V_{G v}, J)\) as \(v\) ranges over all non-zero elements of \(V^+\) and \(J\) is as above. By construction, \(\mathscr{C}\) satisfies property \(\pi\).

It now remains only to verify the hypothesis of 18. Let thus \(F\subseteq V^+\) be a finite set of non-zero positive vectors. We denote by \(\Sigma F\subseteq V^+\) the finite set of all sums of elements of \(F\) (without repetitions). We shall prove by induction on the number of elements of \(F\) that \(\mathscr{C}\) contains a chain \(\mathscr{C}_F\) such that:

1. \(\forall v\in F \;\exists (U, J)\in \mathscr{C}_F : \kern2mm v\in U\) and \(J(v) \neq 0\),

2. \(\forall (U, J)\in \mathscr{C}_F \;\exists u\in \Sigma F : \kern2mm U= V_{G u}\).

This will complete the proof since point (a) is the desired hypothesis. The base case where \(F\) is empty holds with \(\mathscr{C}_F\) empty. For \(F\) non-empty, consider the element \(w=\sum_{v\in F} v\) of \(\Sigma F\) and select some \((U, J)\in \mathscr{C}\) with \(U=V_{G w}\), \(J(w)\neq 0\). We note that every \(v\in F\) lies in \(U\) and hence \(J(v)\) makes sense; we define \[F' = \big\{ v\in F : J(v)=0\big\} \subseteq F.\] Since \(J(w)\neq 0\), there is at least one \(v\in F\) not in \(F'\). Therefore, we can apply the induction hypothesis to \(F'\) and obtain a corresponding chain \(\mathscr{C}_{F'}\). We now define \(\mathscr{C}_F = \mathscr{C}_{F'} \cup \{(U,J)\}\).

By construction, both properties (a) and (b) hold for \(\mathscr{C}_F\). Thus we only need to justify that \(\mathscr{C}_F\) is a chain; we claim that \((U', J') \prec (U,J)\) holds for all \((U', J') \in \mathscr{C}_{F'}\). Indeed, by (b) there is \(u'\in \Sigma F'\) with \(U'= V_{G u'}\). The definition of \(F'\) implies \(J(u')=0\) and by invariance \(J(g u')=0\) follows for all \(g\in G\). Therefore, \(J(U')=0\), confirming the claim. ◻

Proof of 20. The main implication is [pt:FPC-a:FPC]\(\Longrightarrow\)[pt:FPC-a:VP-a]. Let \(V\) be a non-singular ordered vector space and endowed with an order-bounded representation of \(G\). We can assume that the order is generating upon possibly replacing \(V\) by its subspace \(V^+ - V^+\). Indeed, none of the assumptions is affected by this change and neither is the conclusion. We fix a non-zero positive vector \(v \in V\) and proceed to establish the hypothesis of 19.

We let \(E=(V_{G v})^*\) be the algebraic dual of \(V_{G v}\), endowed with the weak-* topology. The dual positive cone \(E^+\) is a proper cone by 1[pt:ideal:generate]. We claim that the dual \(G\)-action on \(E^+\) has all the properties required to apply the fixed-point property for cones, which will then finish the proof.

First, the cone \(E^+\) is weakly complete, see 5. Secondly, the “coboundedness” condition states that the topological dual \(E'\) of the locally convex space \(E\) should contain a continuous linear functional \(\varphi\colon E \to \mathbf{R}\) such that any other \(\lambda\in E'\) can be bounded by some sum of \(G\)-translates of \(\varphi\). We recall that the topological dual of \(E\) is canonically identified with \(V_{G v}\) endowed with its finest locally convex topology ([27] and [18]). Thus we can take \(\varphi=v\) and the boundedness condition for \(\lambda\in E'\) now holds by definition of the ideal \(V_{G v}\) as expressed in 1[pt:ideal:sums].

Finally, the “local boundedness” condition postulates the existence of a non-zero element \(x\in E^+\) whose orbit is bounded in the sense of topological vector spaces. Since the representation is order-bounded, there is \(w\in V\) with \(g v\leq w\) for all \(g\in G\). Since the order is non-singular, we can find a positive linear functional \(K\colon V\to\mathbf{R}\) with \(K(v)\neq 0\). Now we claim that the restriction of \(K\) to \(V_{G v}\) is the desired element \(x\in E^+\). Indeed, weak-* boundedness of \(G x\) means that any basic neighbourhood of \(0\) of the form \[U=\big\{ y\in E : |y(v_i)|<\epsilon \;\forall i \big\} \kern3mm (v_1, \ldots, v_n\in V_{G v}, \epsilon >0)\] we must absorb \(Gx\), i.e. \(G x \subseteq t U\) must hold for all sufficiently large \(t>0\). Given such \(v_i\) and \(\epsilon\), we can find for each \(i\) elements \(g_{i,j}\in G\) with \(j=1, \ldots, k_i\) and \(\pm v_i \leq \sum_{j=1}^{k_i} g_{i,j}v\) because \(v_i\in V_{G v}\) (again 1[pt:ideal:sums]). Now \[\pm (gx)(v_i) =\pm (g K) (v_i) = K(\pm g^{-1}v_i) \leq K\left(\sum_{j=1}^{k_i} g^{-1}g_{i,j}v\right) \leq k_i \, K(w).\] This shows that any \(t > \max(k_1, \ldots , k_n) \, K(w) / \epsilon\) witnesses absorption and completes the proof of this implication.

[pt:FPC-a:VP-a]\(\Longrightarrow\)[pt:FPC-a:CM-a] and [pt:FPC-a:VP]\(\Longrightarrow\)[pt:FPC-a:CM] hold since any vector pricing gives in particular a conditional mean. Both implications admit a converse: 6 constructs a vector pricing from a conditional mean and 16 shows that invariance is preserved. Moreover, [pt:FPC-a:VP] is a special case of [pt:FPC-a:VP-a].

In conclusion, it remains only to justify [pt:FPC-a:VP]\(\Longrightarrow\)[pt:FPC-a:FPC]. To that end, we recall that by Theorem 7 in [5], the fixed point-property for cones is equivalent to the following statement. For every non-zero \(f\geq 0\) in \(\ell^\infty(G)\), there is an invariant positive linear functional \(J\) on the space \[\left\{ h \in \ell^\infty(G) : \exists g_1, \ldots, g_n\in G \;\text{ with } \;\pm h \leq \sum_{i=1}^n g_i\, f \right\}\] normalized by \(J(f)=1\). The above space is precisely the ideal generated by the orbit of \(f\), see 1[pt:ideal:sums]. Therefore, given an invariant vector pricing \(r\) on \(\ell^\infty(G)\), we obtain \(J=r(\cdot, f)\). ◻

Proof. Let \(r\) be an equivariant vector pricing on \(\ell^\infty(G)\). Since \(\boldsymbol{1}_G\) is an invariant vector, the map \(u\mapsto r(u, \boldsymbol{1}_G)\) is invariant. On the other hand, the ideal generated by \(\boldsymbol{1}_G\) is all of \(\ell^\infty(G)\), so that in view of [pt:VP:extends] we have obtained an invariant mean. In other words, \(G\) is amenable.

We now fix an arbitrary \(u\gneqq 0\) in \(\ell^\infty(G)\) and proceed to show that \(r(u, g u)=1\) holds for all \(g\in G\), which is the invariance criterion [pt:def-inv:1] from 14. Consider \[S = \big\{ \, g\in G : r(u, g u) \in[1, +\infty]\, \big\} \subseteq G.\] Given \(g,h\in S\) we have \[r(u, gh u) = r(u, g u)\, r(g u, gh u) = r(u, g u)\, r(u, h u) \geq 1\] and thus \(S\) is a monoid. We need to show that the subgroup \(S \cap S^{-1}\) coincides with \(G\). If not, then the normal subgroup \[N = \bigcap_{g\in G} g \big( S \cap S^{-1}\big) g^{-1}\] is a proper normal subgroup of \(G\). The monoid \(S\) defines a left-invariant pre-order \(\sqsubseteq\) on \(G\) by setting \(x\sqsubseteq y\Leftrightarrow x^{-1}y\in S\). This pre-order is total because [pt:VP:inv] implies \(S \cup S^{-1}=G\). It follows that the quotient \(G/N\) admits a (total, left-invariant) order, see for instance [28]. Thus \(G/N\) is a non-trivial amenable left-orderable group. By Theorem B of [29], it follows that \(G/N\) is locally indicable. This means by definition that every non-trivial finitely generated subgroup of \(G/N\) admits a non-trivial homomorphism to \(\mathbf{Z}\). We apply this to the image \(G_1/(G_1\cap N)\) in \(G/N\) of the subgroup \(G_1<G\) generated by \(F\cup\{g_1\}\), where \(g_1\in G\) is any element outside \(N\). It follows that there is a non-trivial homomorphism \(G_1\to\mathbf{Z}\), which is absurd.

If instead we start with an equivariant conditional mean \(P\), it suffices to observe that the formula in 6 defines an equivariant vector pricing. ◻

Proof. Start with the finitely supported case for general \(V\). Then 22 gives everything except the bound \(t\geq \varrho(\mu)\) for \(\mu\) symmetric. In that case, we can suppose \(\mu\) non-degenerate upon replacing \(G\) by the subgroup generated by the support (which does not change the statement). Consider the non-negative function \[h\colon G \longrightarrow\mathbf{R}^+, \kern3mm h(g) = J(g v) \kern3mm (g\in G)\] which is well-defined since now \(g v\in V_{\langle \mu \rangle v}\). Since \(\mu\) is symmetric, we obtain for all \(g\in G\) \[\begin{gather} (\mu * h)(g) = \sum_{s\in G} \mu(s) \, h(s^{-1}g) =\sum_{s\in G} \mu(s) \, h(s g) =\\ =\sum_{s\in G} \mu(s) J(s g v) =J\left(\sum_{s\in G} \mu(s) s g v \right) = J(\mu *(g v)) = t \, h(g). \end{gather}\] (All the above sums are finite.) Thus \(\mu * h = t h\) and it follows \(\mu^{* n}*h = t^n h\) for all \(n\in \mathbf{N}\). In particular, using \(h\geq 0\) and \(h(e)=1\), \[\mu^{* n} (e) = \mu^{* n} (e) \, h(e) \leq (\mu^{* n} *h)(e) = t^n \, h(e) = t^n.\] We deduce \(t\geq \varrho(\mu)\) as desired.

When \(V=\ell^\infty(G)\) and \(\mu\) is not necessarily finitely supported, the point needing additional justification is \(\mu * h = t h\), including the fact that the left-hand side is a convergent sum (a priori \(h\) is unbounded). In fact we only used \(\mu * h \leq t h\) and therefore we obtain everything we need if we show that for every finite set \(F\subseteq G\) we can bound the partial sum by \[\sum_{s\in F} \mu(s) \, h(s^{-1}g) \leq t \, h(g).\] This, in turn, follows from the positivity of \(J\) and of \(v\): \[\begin{gather} \sum_{s\in F} \mu(s) \, h(s^{-1}g) =\sum_{s\in F^{-1}} \mu(s) \, h(s g) =\sum_{s\in F^{-1}} \mu(s) J(s g v) =\\ =J\left(\sum_{s\in F^{-1}} \mu(s) s g v \right) \leq J\left(\sum_{s\in G} \mu(s) s g v \right) = J(\mu *(g v)) = t \, h(g). \end{gather}\] ◻

Proof of 9. We are given a finitely supported symmetric probability distribution \(\mu\) on an amenable group \(G\) and an order-bounded \(G\)-representation on some non-singular ordered vector space \(V\). The statement is that \(V\) admits a \(\mu\)-stationary vector pricing.

By Kesten’s criterion, \(\varrho(\mu)=1\). Therefore, 23 states that every orbit ideal \(V_{\langle \mu \rangle v}\) admits a non-zero \(\mu\)-invariant positive linear functional. This is the needed assumption to apply 19, not to the group \(G\), but rather to the monoid generated by \(\mu\). The conclusion follows. ◻

Proof of 3. Let \(\mu\) be a symmetric non-degenerate measure on the group \(G\). If \(G\) is amenable, then we are done by applying 9 to \(V=\ell^\infty(G)\), except that we have not supposed \(\mu\) finitely supported. However this restriction was only used in 23, which we proved without finite support restriction in the case of \(\ell^\infty(G)\).

It remains to consider the case where \(G\) is non-amenable. By Kesten’s criterion, we now have \(\varrho(\mu)<1\). We can therefore choose a number \(z\) with \[1 < z < 1/\varrho(\mu).\] We have \(\mu^{*n} (e) \leq \varrho(\mu)^n\) for all \(n\), see e.g.Lemma 1.9 in [35]. Let \(x\in G\); since \(\mu\) is symmetric, we have \[\big(\mu^{*n} (x)\big)^2 = \mu^{*n} (x)\, \mu^{*n} (x^{-1}) \leq \mu^{*2n} (e) \leq \varrho(\mu)^{2n},\] and therefore \[\mu^{*n} (x) \leq \varrho(\mu)^n \kern3mm \forall x\in G, \forall n\in \mathbf{N}.\] This estimate shows at once that the generalized Green function \[\mathscr{G}_z(x) = \sum_{n=0}^{+\infty} z^n \mu^{*n}(x) \kern3mm (\text{where } x\in G)\] converges and that it is a (non-zero, positive) bounded function on \(G\). Indeed we have the geometric series bound \(\mathscr{G}_z(x) \leq 1/(1-z \varrho(\mu) )\) thanks to our choice of \(z\). The definition of \(\mathscr{G}_z\) implies \[z \,\mu * \mathscr{G}_z = \mathscr{G}_z - \mu^{*0} =\mathscr{G}_z - \delta_e \leq \mathscr{G}_z.\] Suppose now for a contradiction that \(r\) is a \(\mu\)-stationary vector pricing on \(\ell^\infty(G)\). We then have \[1 = r(\mathscr{G}_z, \mathscr{G}_z) = r(\mu* \mathscr{G}_z, \mathscr{G}_z) \leq \frac{1}{z} r(\mathscr{G}_z, \mathscr{G}_z) = \frac{1}{z} < 1,\] quod est absurdum. ◻

Proof of 2. This time \(\mu\) is a finitely supported non-degenerate probability distribution on an arbitrary group \(G\) (which is necessarily finitely generated). Let \(V\) be the ordered vector space of step-functions on \(G\). Given any non-zero \(v\in V^+\), let \(c_\mathrm{max}, c_\mathrm{min}>0\) denote the maximal, respectively minimal non-zero value in the (finite!) range of \(v\). We then have \[c_\mathrm{min} \, \boldsymbol{1}_A \leq v \leq c_\mathrm{max}\,\boldsymbol{1}_A, \kern3mm\text{where}\kern3mm A=\mathrm{supp}(v) \neq \varnothing.\] This implies that the orbit ideals \(V_{G v}\) and \(V_{G \boldsymbol{1}_A}\) coincide.

The main result of [3] is that for any non-empty set \(A\subseteq G\), there is a finitely additive \(\mu\)-stationary measure \(m\geq 0\) on \(G\) with \(m(A)=1\). It is understood that \(m\) can take infinite values; but \(m\) defines a bona fide positive linear functional on \(V_{\langle \mu \rangle v} = V_{G v} = V_{G \boldsymbol{1}_A}\), which is non-zero and \(\mu\)-stationary.

In other words, we are in a position to apply 19 to the monoid generated by \(\mu\) as in the proof of 9 and deduce that \(V\) admits a \(\mu\)-stationary vector pricing. This vector pricing gives in particular a \(\mu\)-stationary conditional mean on \(V\), as claimed in 4, and hence also a \(\mu\)-stationary conditional probability. ◻

Proof. Given a non-empty subset \(A\subseteq X\), we consider \(P(\cdot|A)\) as a finitely additive probability measure on \(A\), which has therefore a unique extension to a mean \(P'(\cdot| A)\) on the space \(\ell^\infty(A)\). Thus in particular \(P'(\boldsymbol{1}_B|A) = P(B|A)\) when \(B\subseteq A\).

Let \(V\) denote the space of step-functions on \(X\). Given any \(v\in V\) we write \(A_v\subseteq X\) for the support of \(v\). The definition of step-functions implies that for any non-zero \(v\in V^+\) we have \(v\geq c_\mathrm{min} \boldsymbol{1}_{A_v}\), where \(c_\mathrm{min}\) is the smallest non-zero value of \(v\) as in the proof of 2. This implies \(P'(v | A_v) \geq c_\mathrm{min} >0\). We now define the conditional mean \(P''\) on \(V\) by \[P''(u|v) = \frac{P'(u|A_v)}{P'(v|A_v)} \in[0, 1]\kern3mm \text{when}\kern3mm 0\leq u \leq v \neq 0.\] [pt:CM:add] and [pt:CM:one] hold by definition. For [pt:CM:trans], consider \(0\leq u \leq v \leq w\) with \(v\neq 0\). In particular, \(\varnothing\neq A_v \subseteq A_w\) and for any \(B\subseteq A_v\) the axioms of conditional probability give \(P(B|A_v) P(A_v|A_w) = P(B|A_w)\). By uniqueness of the extension from (unconditional) probability to mean, it follows \[P'(u|A_v) \, P(A_v|A_w) = P'(u|A_w)\kern3mm \text{and}\kern3mm P'(v|A_v) \, P(A_v|A_w) = P'(v|A_w).\] This implies \[\begin{gather} P''(u|v) \, P''(v|w) =\frac{P'(u|A_v)}{P'(v|A_v)} \, \frac{P'(v|A_w)}{P'(w|A_w)} =\frac{P'(u|A_v)}{P'(v|A_v)} \, \frac{ P'(v|A_v) \, P(A_v|A_w)}{P'(w|A_w)} \\ = \frac{P'(u|A_w)}{P'(v|A_v)} \, \frac{ P'(v|A_v)}{P'(w|A_w)} = P''(u|w), \end{gather}\] confirming [pt:CM:trans]. Now that we can use [pt:CM:trans], the uniqueness of the extension from \(P'\) to \(P''\) follows from the definition of \(P''\). ◻

Proof that 28 implies 27. A result going essentially back to Amemiya (end of § I.6, p. 126 in [42]) is that in a hyper-Archimedean vector lattice every element is self-majorizing, and therefore 28 applies. The expression for \(P(u|v)\) with \(u\in V^+\) determines \(P\) everywhere since in any vector lattice the positive cone is generating.

An alternative and detailed reference instead of [42] is Corollary 7.2 in [44]. ◻

Proof of 28. Let \(v\in V^+\) be a non-zero self-majorizing element. Given \(u\in V^+\), let \(\lambda>0\) be a scalar such that \(u \wedge n v \leq \lambda v\) for all \(n\in\mathbf{N}\). It follows \[\forall n, m\in \mathbf{N}\text{ with } m\geq \lambda : \kern3mm u \wedge n v \leq u \wedge \lambda v \leq u \wedge m v.\] In other words, the sequence \(u \wedge n v\) is eventually constant and in particular admits a maximum.

Now we recall three facts from Proposition 1 in [45] (which are all consequences of the above calculation through manipulations of mins and absolute values). First, the ideal \(V_v\) is a band, which means that every set in \(V_v\) that admits a supremum in \(V\) has its supremum in \(V_v\). Secondly, it is a projection band, which means that it comes with a positive linear projection \(p_v\colon V \to V_v\). Finally, this projection satisfies \[\forall u\in V^+: \kern3mm p_v(u) = \sup\{ u \wedge n v : n\in \mathbf{N}\} \in V_v.\] That last expression is implicit in the above reference but explicit e.g.in [46] or [47].

We now define the desired extension of \(P(\cdot|v)\) by considering \(r_P(p_v(\cdot), v)\), where \(r_P\) is the vector pricing corresponding to \(P\), as exhibited in 6. This is indeed a positive linear functional on \(V\).

In order to justify that we do have the stated formula for this extension, we fix some \(u\in V^+\) and some \(\lambda>0\) such that \(u \wedge n v \leq \lambda v\) for all \(n\in\mathbf{N}\). We can and do take \(\lambda\) to be an integer, so that the above discussion gives \[p_v(u) = u \wedge \lambda v = u \wedge n v \kern3mm \forall n\geq \lambda.\] Therefore, for every \(n\geq \lambda\) we have \[r_P(p_v(u), v) = r_P(u \wedge n v, v) = n \, r_P(u \wedge n v, n v) = n\, P(u \wedge n v| n v),\] confirming the claim. ◻

Proof of 30. We start with the observation that \(r(0,u)=0\) for all \(u\neq 0\). Indeed, the following application of [pt:VP:add] \[1 = r(u,u) = r(u+0, u) = r(u, u) + r(0,u) = 1 + r(0,u)\] cannot be undefined. Using this, we deduce that \(r(-u, u) = - 1\) holds for all \(u\neq 0\) because again \[0 = r(0,u) = r(u-u, u) = r(u, u) + r(-u,u) = 1 + r(-u,u)\] cannot be undefined.

Suppose now first that \(G\) contains an element \(g\) of infinite order. Then we define a step-function \(u\in \ell^\infty(G)\) by \(u(x)=(-1)^n\) if \(x=g^n\) for some \(n\in\mathbf{Z}\) and \(u(x)=0\) otherwise. If \(r\) were an invariant vector pricing, then \[1 = r(u, u) = r (g u, u) = r(-u, u) = -1,\] which is absurd.

Therefore it remains only to consider the case of a non-trivial group \(G\) in which every element has finite order (in fact, even orders can also be treated as above). Let thus \(g\in G\) have finite order \(q>1\). This time we set \(u=\delta_e - \delta_g \neq 0\) and compute \[q = q\, r(u,u) = \sum_{n=0}^{q-1} r(g^n u,u) = r\left(\sum_{n=0}^{q-1} g^n u, u\right) = r(0, u) = 0,\] again absurd. All values of \(r\) appearing in the above rather dim sum are \(1\), then \(0\), so that no hypothetical \(+\infty - \infty\) could be used as a fig leaf. ◻