CHAPTER III
STIELTJES MEASURES ON THE PLANE
In this chapter we extend the classical correspondence between
functions of finite variation on the real line and Stieltjes measures
on the real line to the case of functions and Stieltjes measures on
2
R.
3.1 Measures Associated With Functions
2
Given a function f: R2 E with finite variation on bounded
rectangles, right continuous (in the order sense!) on R we can
associate a unique measure with finite variation. The statement and
proof we give are due essentially to Radu [16]. The statement is a
little more general than we really need, but no further difficulties
are encountered by this; we also use right-limits instead of Radus's
left-limits, but this is just a matter of choice. The term "bounded
variation" in the statement refers to the variation of f on rectangles
as in Definition 2.2.4; as we have seen, this is weaker than the
requirement that Ifl be bounded.
Theorem 3.1.1 (Radu). If the function f: R + E is of bounded
variation on R2 and if the right limit f(s+,t+) (cf. Theorem .3.3 for
definition) exists at each point (s,t) of R then there exists a
2
Stieltjes measure m on R with values in E, uniquely determined,
with finite variation, and such that for all rectangles