CHAPTER IV
 VECTOR-VALUED PROCESSES
 WITH FINITE VARIATION



 An important part of the general theory of processes in one

 parameter is the correspondence between processes of finite variation

 and measures on R+xn (see for example Dellacherie and Meyer [5, VI.

 64-89] and also Kussmaul [10]). This correspondence finds

 applications in the notion of dual projections of processes, which are

 used in the theory of potentials and in decomposition of

 supermartingales (see for example Dellacherle and Meyer [5, nos. VI.

 71-113], also Rao [17] and Metivier [11]).

 In the one-parameter case, the extension of the correspondence to

Banach-valued processes is done in Dellacherie and Meyer [5]. In two

parameters, the correspondence for real-valued processes is stated

(more or less) in Meyer [12]; we shall presently give a more directly

applicable (for our purposes) version, along with a proof, as the case

of finite variation on R2 is more delicate (as we have seen). In

fact, many times, in the literature results are given for increasing

processes, and then extended by defining a process of finite variation

as a difference of two increasing processes. The method we use here

is a little more constructive.