A New Tool in the Theory of Integral Representation

Let $E$ be a Hausdorff locally convex space, $E^\prime$ its dual, and $X\subset E$ a proper convex cone, not necessarily closed. Recall that an open ray $\delta$ of $X$ is said to be extreme if $(x \in \delta$ and $x = y +z$ with $y,z \in X \setminus 0)$ implies $(y,z \in \delta)$. We denote by $\mathcal{E}(X)$ the union of all extreme open rays of $X$. We say that $X$ has the property of integral representation if for each $x \in X$ there is a positive Radon measure $m$ on $\mathcal{E}(X)$ such that each $f \in E^\prime$ is $m$-integrable and $m(f)=f(x)$.
Recently E. Thomas ["Integral representations in conuclear cones", J. Convex Analysis 1/2 (1994) 225–258] 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), allowing to avoid some of his hypotheses.