This monograh is mainly devoted to the theory of integral representation

in convex cones due to G. Choquet and to several of

its applications to Analysis :

Classical theorems of Bochner-Weil and of Berstein, theorem of

Choquet-Deny, axiomatics of Brelot an Bauer in potential theory,

results of Talagrand concerning invariant measures and

capacities and those of Royer and Yor concerning Quasi-invariant

measures in field theory.

This book is accessible to a student with a Master's degrees

M1 and M2, with emphasis on analysis, and can be considered

as an introduction to the richness and variety of the subjects.

The three tools created by G. Choquet in this framework (weakly

complete cones, caps, conical measures) are also studied for

themselves. Moreover, there are also useful in other fields close

to Analysis : zonoforms and vector measures, statistical decision

spaces. bireticulated cones and normal cones contained in

Banach spaces. Remark that in the postface G. Choquet relates

the thought process which led him to find out the three tools

mentioned above.