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.