Let
E
E
be a Hausdorff locally convex space,
E^\prime
E
′
its dual, and
X\subset E
X
⊂
E
a proper convex cone, not necessarily closed. Recall that an open ray
\delta
δ
of
X
X
is said to be extreme if
(x \in \delta
(
x
∈
δ
and
x = y +z
x
=
y
+
z
with
y,z \in X \setminus 0)
y
,
z
∈
X
∖
0
)
implies
(y,z \in \delta)
(
y
,
z
∈
δ
)
. We denote by
\mathcal{E}(X)
E
(
X
)
the union of all extreme open rays of
X
X
. We say that
X
X
has the property of integral representation if for each
x \in X
x
∈
X
there is a positive Radon measure
m
m
on
\mathcal{E}(X)
E
(
X
)
such that each
f \in E^\prime
f
∈
E
′
is
m
m
-integrable and
m(f)=f(x)
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.