Journal of Convex Analysis Volume 3 (1996), No. 2, 349–360

A New Tool in the Theory of Integral Representation

R. Becker

U.R.A.754 au C.N.R.S., Université Pierre et Marie Curie-Paris 6, Tour 46-0 – Boite 186, 4 Place Jussieu, 75252 Paris cedex 05, France.

Received October 10, 1995 Revised manuscript received February 6, 1996

1. Introduction

Let E be an l.c.s.. (Hausdorff locally convex space), E′ its dual and X ⊂ E a proper convex cone, not necessarily closed. Recall that an open ray δ of X is said to be extreme if (x ∈ δ and x = y + z with y, z ∈ X \ 0) implies (y, z ∈ δ); we denote by E_{g}(X) the union of all the extreme open rays of X. One of the problems of the theory of integral representation is to give conditions on X in order that for each x ∈ X , there is at least one positive Radon measure m on E_{g}(X), such that:

1. Each ℓ ∈ E′ is m-integrable.

2. One has m(ℓ) = ℓ(x), for each ℓ ∈ E′. Then we say that x is the resultant of m and we write x = r(m).

Definition 1.1. When the cone X has the preceding properties, we say that X has the property of integral representation, denoted by I.R..

Recently E. Thomas ([8]) proved a theorem of integral representation for a class of convex cones, called conuclear. The aim of this work is to give a quite different presentation of his results, with the help of other tools, one of which is new (the pseudo-caps, introduced in 3.1), allowing to avoid some of his hypotheses.

The plan is as follows: In part 2 we recall some results showing the progression of the theory of integral representation and how the present work fits with this theory. In part 3 we prove the main results of this work and we introduce the pseudo-caps. In part 4 we establish the link with the conuclear cones of E. Thomas. In part 5 we discuss some examples. The main results are Theorems 3.11 and 4.8.

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

350 R. Becker / A new tool in the theory of integral representation

2. The theory of Integral Representation

When the cone X has a weakly compact metrizable base then X has I.R.: This is the classical theorem of integral representation for convex compact metrizable sets, proved by G. Choquet. ([7], page 19) The notion of cap of a convex cone, introduced by G. Choquet, permits to cope with more general cones:

Definition 2.1. ([7], page 87) A subset K of X is called a cap of X, when it is non-empty, convex, weakly compact and such that X \ K is also convex. If X is the union of its caps then X is said to be well-capped.

For what follows, it is useful to recall the following characterization of caps of G. Choquet:

Lemma 2.2. ([7], page 89) Let K be a convex subset of X, containing 0; the two following properties are equivalent:

1. K is a cap of X.

2. The gauge j of K is a lower semi-continuous function, defined on X, which is additive and positive-homogeneous, with values in [0, +∞] such that K = j^{−1}([0, 1]) is weakly compact.

The following result of G. Choquet is fundamental:

Theorem 2.3. ([7], page 90) For each x ∈ X, contained in a metrizable cap of X, there is a Radon measure m ≥ 0 on E_{g}(X) such that m(ℓ) = ℓ(x), for each ℓ ∈ E′, or in other words x = r(m).

For example, the cone l_{+}^{1}(N) does not admit a compact base for the duality with c_{o}(N), but it is generated by a cap, namely K = (x : x = (x_{n}) ∈ l_{+}^{1}(N) and ∑_{0}^{∞} x_{n} ≤ 1), which is metrizable. By Theorem 2.3 this cone has property I.R..

Now here is a theorem concerning a class of cones, studied in [6] and in [3], which are not necessarily well-capped.

Theorem 2.4. Let X be a convex cone, generated by a compact metrizable set K ⊂ X, containing 0, for which there exists an affine function ϕ, defined on X, such that for some positive constant M: j_{K}(x) ≤ ϕ(x) ≤ M.j_{K}(x), for each x ∈ X, where j_{K} is the gauge of K. Then the cone X has property I.R..

In fact, this work can be seen as an extension of the preceding theorem. The results that we have recalled yield positive Radon measures; but they can be formulated in an apparently more abstract way which will lead us to a very useful tool.

Definition 2.5. (Weakly complete cones and conical measures) ([4], par.30) Let S be the class of weakly complete convex proper cones contained in an l.c.s. E. And let X ∈ S. We denote by h(X) the Riesz vector space of functions on X generated by E′ |_{X} and by s(X) the subcone of h(X) consisting of all finite suprema of elements of E′ |_{X}. The space h(X) is ordered by the set of its positive elements. A positive linear form on h(X) is called a positive conical measure on X. We denote by M^{+}(X) their set.

Each μ ∈ M^{+}(X) has a resultant r(μ) ∈ X such that: μ(ℓ) = ℓ(r(μ)) for each ℓ ∈ E′. The cone M^{+}(X) is ordered by: (λ ≺ μ) iff ( λ(f) ≤ μ(f) for each f ∈ s(X)).

The following theorem is fundamental:

Theorem 2.6. ([4], 30.13) For each x ∈ X there is a μ ∈ M^{+}(X), which is maximal for the order ≺, and we have r(μ) = x.

The following characterization of maximal conical measures will be very useful:

Theorem 2.7. ([4], 30.pb1) If μ ∈ M^{+}(X) the three following properties are equivalent:

1. μ is maximal for the order ≺ .

2. For each f ∈ h(X) then: μ(f) = inf(μ(g) : g ∈ −s(X) and g ≥ f on X ).

3. Same condition as in 2. but only for each f ∈ s(X).

Finally we recall a theorem of H. Fakhoury ([5]) which will be useful for us:

Theorem 2.8. Let Z ⊂ E be a proper convex cone, covered by an increasing sequence of weakly compact convex sets (Z_{n}), which are hereditary for the order of Z. Then Z ∈ S for the duality with the space of all affine and homogeneous functions on Z, whose restriction to each Z_{n} is continuous.

3. Pseudo-caps and main results

We introduce now the notion of pseudo-cap of a cone, which will show that this work is in some sense parallel to the theory of caps.

Definition 3.1. (The pseudo-caps) Let X be a proper convex cone, contained in a vector space V, not necessarily equipped with a topology. We call pseudo-cap of X any non-empty convex set K ⊂ X, such that X \ K is also convex and that each ray of X intersects K along a compact interval.

We shall give in 5.2 an example of a pseudo-cap K of a cone contained in an l.c.s., such that \overline{K} is compact, but is not a cap. Here is an analytical characterization of pseudo-caps, analoguous to that of caps:

Proposition 3.2. Let K be a pseudo-cap of a convex cone X; the gauge of K is additive and homogeneous on X, with values in [0, ∞], and it is > 0 on X \ 0. Reciprocally, for each function f, defined on X, with values in [0, ∞], which is additive and homogeneous, and which is > 0 on X \ 0, the set K = f^{−1}([0, 1]) is a pseudo-cap of X and f is the gauge of K.

Proof. In order to show that the gauge j of K is additive on X, it is sufficient to consider the two dimensional sub-cones of X: Let Y be such a sub-cone, generated by x, y ∈ X; it is sufficient to prove that: K ∩ Y = conv((K ∩ R^{+}.x) ∪ (K ∩ R^{+}.y)), which is easy to prove since each ray of X intersects K along a segment [0, a] and hence K ∩ (R^{+}.x) and K ∩ (R^{+}.y) are compact intervals.

One has j(x) > 0 for each x ≠ 0, since each ray of X intersects K along a segment [0, a]. The remainder of the proposition is easy.

Remark 3.3. Here is an example proving that, in the preceding proposition, the condition that the intersection of K with each ray of X is a compact interval cannot be removed: In R^{2}, let X be the cone ((x, y) : x, y ≥ 0). If K = ((x, y):(x, y) ∈ X and y ≥ x − 1) obviously K is convex, contains 0 and X \ K is also convex. But the gauge j of K is not additive: Indeed, one has j((1, 0)) = 1 and j((0, 1)) = 0, but j((1, 1)) = 0.

Recall that a face of a proper convex cone X is a convex sub-cone of X which is hereditary for the order of X.

Proposition 3.4. For each pseudo-cap K of X the cone X_{K} generated by K is a face admitting (x : j_{K}(x) = 1) as a base.

Proof. One has X_{K} = (x : x ∈ X and j_{K}(x) < ∞) and j_{K} is additive on X.

The two following theorems are fundamental; in particular the following theorem is analogous to Theorem 2.3.

Theorem 3.5. Let X be a convex proper cone, not necessary closed, contained in an l.c.s. E, equipped with its weak topology. Let K be a pseudo-cap of X, contained in a compact convex set S of X, which is metrizable and hereditary for the order associated to X. Then, for each x ∈ X_{K}, there is a positive Radon measure m on E _{g}(X), integrating all the elements of E′, such that r(m) = x.

Proof. There are three steps. We suppose that x ≠ 0.

1. Let Y be the cone generated by S. Using Theorem 2.8, if we denote by F the space Y − Y, there is F′ ⊂ F^{∗}, containing E′ |_{F}, such that Y ∈ S for the duality with F′. Moreover, S is compact and metrizable for this topology by Theorem 2.8. Since S is hereditary in X, the cone Y is a face of X and hence E_{g}(Y) ⊂ E_{g}(X). Hence we can use the cone Y to prove the theorem.

2. Since Y ∈ S, there exists, by Theorem 2.6, a conical measure μ ∈ M^{+}(Y), for the duality with F′, which is maximal and such that x = r(m). Since Y ∈ S, there exists an ultrafilter U on the set of all conical measures ν on Y such that: ν = ∑_{1}^{n} ε_{x_{i}}, with ∑_{1}^{n} x_{i} = x, and x_{i} ≠ 0, for each i, and this ultrafilter converges to μ for the duality with h(F). This follows from ([4], 30.9). Let j be the gauge of K; since one has j > 0 on X \ 0 and since j(x) < ∞, one can write: ∑_{1}^{n} ε_{x_{i}} = ∑_{1}^{n} j(x_{i}).ε_{y_{i}}, with y_{i} = x_{i}/j(x_{i}). The Radon measure ∑_{1}^{n} j(x_{i}).ε_{y_{i}}, on S, converges, with respect to U, to a Radon measure m on S, since ∑_{1}^{n} j(x_{i}) = j(x), so that one has μ = m on h(F).

3. Let us show that m is carried by E_{g}(X): For each f ∈ h(Y), let f̂ be the function on Y defined by: f̂(y) = inf(g(y) : g ∈ −s(Y) and g ≥ f on Y ), for each y ∈ Y.

Since μ is maximal on Y, one has m(f̂) = m(f) for each f ∈ h(F), by Theorem 2.7. Since S is metrizable for the duality with F′, it is easy to find a sequence (f_{n}) of h(F), such that (X \ 0) ∩ (f̂_{n} = f_{n}) = E_{g}(Y). Since Y is a face of X, one has E_{g}(Y) ⊂ E_{g}(X), hence the result.

Theorem 3.6. (With the hypotheses and notations of the preceding theorem) Suppose that for each x ∈ X_{K} all the positive Radon measures on E_{g}(X) having x as resultant, induce the same conical measure on X. Then X_{K} is a Riesz cone for its own order.

Proof. By the hypotheses, for each x ∈ X_{K}, there is only one maximal conical measure μ ∈ M^{+}(Y) having x as resultant. Since X_{K} is a face of Y, for each λ ∈ M^{+}(Y), such that λ ≤ μ , one has r(λ) ∈ X_{K}. Hence X_{K} is isomorphic to a face of M^{+}(Y) and so X_{K} is a Riesz cone.

Remark 3.7. The converse of the preceding theorem is false, even for a well-capped closed cone. We shall give an example in 5.1.

The following elementary lemma will permit us to use Theorem 3.5:

Lemma 3.8. Let X be a convex proper cone, not necessarily closed, contained in an l.c.s. E, equipped with its weak topology. Let K_{1} and K_{2} be two pseudo-caps of X such that \overline{K_{1}} and \overline{K_{2}} are two compact subsets of X, satisfying \overline{K_{1}} ⊂ K_{2}. Then, the hereditary envelope of \overline{K_{1}} in X, denoted by s(\overline{K_{1}}), is a convex compact subset of X, contained in \overline{K_{2}}.

Proof. Suppose x ∈ s(\overline{K_{1}}); there are y ∈ \overline{K_{1}} and z ∈ X such that x + z = y; one has also y ∈ K_{2} since \overline{K_{1}} ⊂ K_{2}, hence x, z ∈ K_{2}, since K_{2} is a pseudo-cap. Hence the lemma, since \overline{K_{1}} and \overline{K_{2}} are compact convex subsets of X: Using an ultrafilter U on s(\overline{K_{1}}), it is immediate that y converges to an element of \overline{K_{1}} and x, z to elements of \overline{K_{2}}.

Here is a first consequence of Theorem 3.5 and of the preceding lemma:

Theorem 3.9. Let X be a proper convex cone, not necessarily closed, contained in an l.c.s. E, equipped with its weak topology. Let K_{1} and K_{2} be two pseudo-caps of X, such that \overline{K_{1}} and \overline{K_{2}} are compact metrizable subsets of X, satisfying \overline{K_{1}} ⊂ K_{2}. Then, for each x ∈ X_{K_{1}}, there is a Radon measure m ≥ 0 on E_{g}(X), integrating all the elements of E′, and such that r(m) = x.

Proof. We apply Theorem 3.5, with K = K_{1} and S = s(\overline{K_{1}}), which is a metrizable compact convex subset of X, by Lemma 3.8.

The preceding theorem suggests the following definition, which can be compared with Definition 2.1 of well-capped cones.

Definition 3.10. (Pseudo-well-capped cones) Let X be a proper convex cone, not necessarily closed, contained in an l.c.s. E, equipped with its weak topology. We say that X is pseudo-well-capped by K, if there is a family K of pseudo-caps of X such that:

1. ∪K = X, where K ∈ K.

2. For each K ∈ K, then \overline{K} is a compact subset of X.

3. For each K_{1} ∈ K there is K_{2} ∈ K, such that \overline{K_{1}} ⊂ K_{2}. Of course this definition remains meaningfull if E is an l.c.s. not necessarily equipped with its weak topology. In what follows we shall never use such an extension of the definition.

Now we are able to prove the following theorem of integral representation which is the main result of this work.

Theorem 3.11. (With the notations of Definition 3.10) Suppose E is an l.c.s. equipped with its weak topology. Let X ⊂ E be a cone which is pseudo-well-capped by K, and such that each K ∈ K is metrizable. Then the cone X has I.R.

Proof. Apply Theorem 3.9.

Remark 3.12. We shall give in 5.2 an example, due to A. Goullet de Rugy, of a pseudo-cap K of a cone X, such that \overline{K} is a compact subset of X, but not a cap of X.

The following proposition concerns the stability of the class of pseudo-well-capped cones under some operations:

Proposition 3.13.

1. Any (relatively) closed convex subcone of a pseudo-well-capped cone is itself pseudowell-capped.

2. Any denumerable product of pseudo-well-capped cones is also pseudo-well-capped.

Proof.

1. Obvious.

2. Let X_{n} ⊂ E_{n} be a sequence of pseudo-well-capped cones and let K_{n} be a family of pseudo-caps satisfying the three conditions of Definition 3.10. Let E = ∏ E_{n} and let X = ∏ X_{n}. Let K be the family of all pseudo-caps K of X such that, for each x = (x_{n}), one has: j_{K}(x) = ∑_{1}^{∞} a_{n}.j_{n}(x_{n}), where j_{n} is the gauge of K_{n} ∈ K_{n} and each a_{n} is > 0. It is immediate that K is a pseudo-cap of X; let us prove that the family K satisfies the three conditions of Definition 3.10:

1. ∪K = X, where K ∈ K : It is immediate.

2. If K ∈ K, then K ⊂ ∏ K_{n}/a_{n}, hence \overline{K} is a compact subset of X, since \overline{K_{n}} is a compact subset of X_{n}.

3. If j_{K}(x) = ∑_{1}^{∞} a_{n}.j_{n}(x_{n}), using the preceding notations, for each integer n, choose K′_{n} ∈ K_{n}, such that \overline{K_{n}} ⊂ K′_{n}; we denote by j′_{n} the gauge of K′_{n}.

For each x = (x_{n}) we set: j′(x) = ∑_{1}^{∞} 2^{−n}.a_{n}.j′_{n}(x_{n}); it is immediate that j′ is the gauge of some K′ ∈ K; then we have:

\overline{K} ⊂ ∏ \overline{K_{n}}/a_{n} ⊂ ∏ K′_{n}/a_{n} ⊂ K′. The proof is now complete.

Recall the following definition which is useful for Theorem 3.15.

Definition 3.14. (Localizable conical measures) Let X ∈ S; a conical measure μ ∈ M^{+}(X) is said to be representable or localizable on X \ 0 if there is a positive Radon measure on X \ 0 which is equal to μ on h(X).

Theorem 3.15. (With the notations of Definition 3.10) Suppose E is an l.c.s. equipped with its weak topology. Let X ⊂ E be a pseudo-well-capped cone. Let (K_{n}) be a sequence of elements of K such that \overline{K_{n}} ⊂ K_{n+1} for each n. Then the cone Z, generated by the union of the sets K_{n}, has the three following Properties:

1. Z is a face of X.

2. Z ∈ S, for the duality with the space A_{0} consisting of all affine homogeneous functions on Z, whose restrictions to each Z_{n} are continuous.

3. Each positive conical measure on Z, for the duality with A_{0}, is localizable on Z \ 0.

Proof.

1. Obvious by Proposition 3.4.

2. Obvious by Theorem 2.8 and Lemma 3.8.

3. The proof of part 2 of Theorem 3.5 works also for a conical measure which is not maximal: Indeed, if r(μ) ∈ K_{n}, then μ is localizable on \overline{K_{n}}.

4. Pseudo-well-capped cones and conuclear cones

In this part, we will compare the class of pseudo-well-capped cones and that of conuclear cones of E. Thomas ([8]). Indeed, in Theorem 4.8 we shall prove that these two classes of cones are identical when the space is equipped with its weak topology. First we need the following definition:

Definition 4.1. Let X be a proper convex cone, contained in a vector space V, not necessarily equipped with a topology. Let A and B be two convex subsets of X, containing 0; we say that A is capped by B, and we note A ≺ B, if, for each finite sequence (x_{1}, …, x_{n}) of X, such that ∑_{1}^{n} x_{i} ∈ A, one has: ∑_{1}^{n} j_{B}(x_{i}) ≤ 1. It is immediate that any pseudo-cap of X is capped by itself.

The following elementary lemma enables us to consider only hereditary subsets.

Lemma 4.2. (With the notations of the preceding definition) If A ≺ B then Â ≺ B̂, if we denote by Â and by B̂ the hereditary envelopes of A and B, for the order of X.

Proof. It is sufficient to prove that Â ≺ B, since jB̂ ≤ j_{B}.

If ∑_{1}^{n} x_{i} ∈ Â, there is x_{0} ∈ X such that (x_{0} + ∑_{1}^{n} x_{i}) ∈ A . The lemma follows immediately.

We will give an interpretation of the relation A ≺ B in terms of pseudo-caps. For this purpose we shall use a technique of L. Asimow ([1], prop.2.3. and [2], prop.5.).

Lemma 4.3. Let X be a proper convex cone, not necessarily closed, contained in an l.c.s. E, equipped with its weak topology. Let A and B be two compact convex subsets of X, containing 0. We suppose that A is hereditary for the order of X. Then the two following properties are equivalent :

(1) One has A ≺ B.

(2) For each ε > 0, there is an algebraic hyperplane of the vector subspace E_{A} generated by A, which separates A and E_{A} ∩ j_{B}^{−1}(1 + ε).

Proof.

(1) ⇒ (2) (The heredity of A will not be used here). Let q be the norm on E_{A} such that the associated unit ball is (A − A). Let us show that the two following sets are disjoint: B_{ε} = conv(E_{A} ∩ j_{B}^{−1}(1 + ε)) and A_{ε} = A + ε/2.(A − A). It is sufficient to prove that the following equality is impossible: ∑_{1}^{n} α_{i}.x_{i} = a_{0} + ε/2.(a_{1} − a_{2}), where each x_{i} is an element of X, j_{B}(x_{i}) = 1 + ε, and each α_{i} is ≥ 0, with sum 1 and a_{0}, a_{1}, a_{2} ∈ A: Indeed, suppose that: ε/2.a_{2} + ∑_{1}^{n} α_{i}.x_{i} = a_{0} + ε/2.a_{1}. Since A ≺ B we have (1 + ε) ≤ 1 + ε/2, which is a contradiction. Now it is sufficient to apply the separation theorem of Hahn-Banach to B_{ε} and A_{ε}.

(2) ⇒ (1) Let l_{ε} be a linear form on E_{A} such that (l_{ε} = 1) is the hyperplane mentioned in (2). If ∑_{1}^{n} x_{i} ∈ A, where each x_{i} is an element of X, then x_{i} ∈ E_{A} for each i, since A is hereditary, hence ∑_{1}^{n} l_{ε}(x_{i}) ≤ 1. Consequently one has (∑_{1}^{n} j_{B}(x_{i})/(1 + ε) ≤ ∑_{1}^{n} l_{ε}(x_{i}) ≤ 1. Hence the result when ε → 0.

The following lemma is a translation, in terms of pseudo-caps, of the preceding lemma:

Lemma 4.4. (With the hypotheses of the preceding lemma) The two following properties are equivalent:

(1) One has A ≺ B.

(2) For each ε ≥ 0 there is a pseudo-cap K_{ε} of X, such that A ⊂ K_{ε} ⊂ (1 + ε).B.

Proof.

(1) ⇒ (2) Since A is hereditary in X, the set E_{A} ∩ X is a face of X; consequently the set (x : x ∈ E_{A} ∩ X and l_{ε}(x) ≤ 1) is a pseudo-cap of X, denoted by K_{ε}, with property (2).

(2) ⇒ (1) This follows from (2) ⇒ (1) of the preceding lemma, using the gauge of the pseudo-cap K_{ε}.

We are now able to give the definition of the conuclear cones of E. Thomas ([8]):

Definition 4.5. (Conuclear cones) Let X be a proper convex cone, not necessarily closed, contained in an l.c.s. E, not necessarily equipped with its weak topology. The cone X is called conuclear if there is a family Θ of compact convex subsets of X, containing 0, whose union is X, and such that for each A ∈ Θ, there is B ∈ Θ such that A ≺ B. Of course X remains conuclear if E is equipped with its weak topology. Indeed E. Thomas uses this definition even when E is not equipped with its weak topology. We shall see why in Remarks 4.10.

The following elementary lemma is useful, for the use of hereditary subsets:

Lemma 4.6. For each A ∈ Θ, its hereditary envelope, denoted by Â, is a compact convex subset of X.

Proof. Let x ∈ Â; there are y ∈ A and z ∈ X such that y = x + z. Choose B ∈ Θ such that A ≺ B; one has x, z ∈ B, hence the result, since B is compact.

Lemma 4.7. If X is conuclear for the family Θ, it is also conuclear for the following family: Θ̂ = {Â : A ∈ Θ}.

Proof. Obvious by Lemmas 4.2 and 4.6.

Here now is the identification we were looking for:

Theorem 4.8. Let X be a proper convex cone, not necessarily closed, contained in an l.c.s. E equipped with its weak topology. The two following properties are equivalent:

1. X is pseudo-well-capped.

2. X is conuclear.

Proof.

(1) ⇒ (2) Let K be a family of pseudo-caps of X, satisfying the conditions of Definition 3.10: Let Θ = (\overline{K} : K ∈ K); it is immediate that \overline{K_{1}} ≺ \overline{K_{2}} if \overline{K_{1}} ⊂ K_{2}. Hence X is conuclear for the family Θ.

(2) ⇒ (1) Let Θ be a family of convex compact subsets of X, satisfying the conditions of definition 4.5: By Lemma 4.7, we may assume that the menbers of Θ are hereditary and that Θ is stable by positive scalar multiplications. Let K be the family of all pseudo-caps K of X such that there is A, B ∈ Θ satisfying A ⊂ K ⊂ B. Then, it is immediate that X is pseudo-well-capped, by Lemma 4.4.

As a consequence of the preceding theorem, Theorem 3.11 can be given the following formulation:

Theorem 4.9. (With the notations of Definition 4.5) Suppose E is an l.c.s. not necessarily equipped with its weak topology. Let X ⊂ E be a conuclear cone for a family Θ such that each A ∈ Θ is metrizable. Then X has I.R.

Proof. Obvious, since X is also conuclear when E is equipped with its weak topology.

Remark 4.10. E. Thomas ([8]) proved the preceding theorem using two other hypotheses on X:

1. X is closed.

2. Condition ⟨⟨Convex Envelope⟩⟩, or (C.E.): Each compact subset of X is contained in a convex compact subset of X.

Note that X can satisfy condition (2) for the topology of E but not for its weak topology and compare with Definition 4.5.

The two hypotheses of the preceding remark were used by E. Thomas ([8]) to prove the following lemma, not proved here, which can be compared with part 2 of the proof of Theorem 3.5:

Lemma 4.11. Let X be a closed convex cone, having (C.E), contained in an l.c.s. E. For each positive Radon measure m on X \ 0, integrating all the elements of E′, such that r(m) ∈ X, the conical measure μ_{m} induced by m, is the limit, for the duality with h(X), of a net of conical measures of the following form: ∑_{1}^{n} ε_{x_{i}}, with x_{i} ∈ X for each i, and ∑_{1}^{n} x_{i} = r(m).

E. Thomas ([8]) proved also the following theorem not proved here and which can be compared with Theorem 3.6:

Theorem 4.12. (With the hypotheses of Theorem 4.9) Suppose that X is closed and has (C.E). If X is a Riesz cone then, for each x ∈ X, all the positive Radon measures on E_{g}(X) having x as resultant, induce the same conical measure on X.

Remark 4.13. We shall give in 5.1 an example of a cone X, showing that, in the preceding theorem, the condition (C.E.) cannot be dropped.

5. Examples

Here are a few examples, interesting for various reasons:

Example 5.1. (Showing that condition (C.E.) cannot be dropped in Theorem 4.12) Let T = [0, 1]. Let E be the vector space generated by all the Dirac measures ε_{x}, where x ∈ T, and by the Lebesgue measure λ. We equip E with the weak* topology: Hence E′ is not the dual of E for the norm topology. Let X be the convex subcone of E generated by all the Dirac measures ε_{x} and by λ: It is immediate that X is a Riesz cone, but λ is the resultant of two different conical measures, localizable on E_{g}(X): λ is the resultant of ε_{λ} and of ∫_{T} ε_{x}.λ(dx). The cone X does not enjoy condition (C.E.), although it is the union of its finite dimensional caps and it is closed in E.

Example 5.2. (A. Goullet de Rugy [6]) Let M be the Banach space of all Radon measures on [0, 1], equipped with the duality with the space C = C ([0, 1]); we denote by M_{1} the unit ball of M. Choose a ∈ [0, 1] and let X be the cone generated by the following compact convex set A = {λ − ε_{a} : λ ∈ M_{1}}. We set X_{1} = X ∩ M_{1}.

We will recall the main properties of X, giving only sketchy proofs. Here is an easy key lemma without proof:

Lemma 5.3. One has:

X = ⋃_{k≥0} k((λ − ε_{a}) : ∥λ∥ ≤ 1; λ({a}) = 0)

Theorem 5.4.

1. Let ϕ be the linear form on M defined by ϕ(ν) = −2.ν({a}); then:

∥x∥ ≤ ϕ(x) ≤ 2.∥x∥, for each x ∈ X.

2. One has X ∈ S, for the duality with C.

3. The element -ε_{a} does not belong to any cap of X.

Proof.

1. It is an immediate consequence of the preceding lemma.

2. It is a consequence of Theorem 2.4, since X_{1} contains a ball of M.

3. Let j be the gauge of a cap containing -ε_{a}; since X contains a ball of center -ε_{a} it is immediate that j is finite and continuous, hence X would have a compact base for the duality with C. One can prove that this is impossible, since a is not an isolated point of T.

As an immediate consequence of the preceding theorem we have:

Theorem 5.5. The cone X is pseudo-well-capped and has I.R.

Proof. X is pseudo-well-capped by the family of all the sets: k.{x : x ∈ X and ϕ(x) ≤ 1} where k is a positive scalar. X has I.R. by Theorem 3.11, since X_{1} is compact and metrizable for the duality with C.

Remark 5.6. The construction of example 5.2 can be carried out starting from any compact set T and any non isolated point a of T: Theorem 5.4 remains true for the cone constructed in this way; Theorem 5.5 also, if T is metrizable.

5.7. Other examples Proposition 3.13 shows the existence of a lot of pseudo-well-capped cones: For example, if X^{1}, X^{2}, … denotes a sequence of cones built as in 5.2 and 5.6, starting from T_{1}, T_{2}, … and from a_{1}, a_{2}, …, where for each n, a_{n} ∈ T_{n} and is not isolated, one has: Any closed convex subcone of the product ∏ X^{n} is pseudo-well-capped and it has I.R., if the spaces T_{n} are metrizable.

References

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