CHAPTER V
 CONCLUSION


 We have seen that the usual definition of the variation on a

 rectangle of a function of two variables is insufficient to yield all

 the properties necessary to extend the theory of Stieltjes measures to

 functions of finite variation on the plane. We have given some

 additional conditions sufficient to establish a proper definition of

 the variation of a function, and although these were not shown to be

 minimal, it would seem to be difficult to weaken them further.

 We have shown that, starting with a two-parameter stochastic

 function X with values in L(E,F), we can associate a measure p with

 values in L(E,Z') and that under certain conditions px has values in

 L(E,F) as well. We have also established a similar correspondence,

 starting with a measure and obtaining a stochastic function. We have

 also shown that, if the spaces E and F are not "too large," we have

 the equality



 IUXI = "IXI.


We hope that this lays the groundwork for exploring the question of

existence of optional and predictable projections of vector-valued

multlparameter processes.