Now, for any a>0, there exists such a partition P such that
VarR(f;P) > a, i.e., E A [(s ),(s)(f) > a.
051O m-1 i'j i+1' j+1
OjSn-1
However, the rectangles ((si,t ),(si+ ,tj+)], 0 S i S m-1,
0 5 j n-1 are disjoint, and their union is contained in R,
so we have
a < A[(s t ),(s ~ t ]
OSi m-1 ) '(Si+1 'tj+ )J
OSj n-1
z I'j ((s +,t ) 1' j+ (f)l
SIml (R).
Thus we have Iml(R) > a. a arbitrary -> ImJ(R) = + -, a contradiction
since m has finite variation. Hence VarR(f) < for R bounded, and
the theorem is proved. I
Remarks.
1) We have said nothing about uniqueness of f. In the case of
functions on the line, f is determined within a constant by m (i.e.,
any other associated function g is determined by adding or subtracting
a constant from f), but here this is not the case. In fact, as we
have seen before (Example 2.1.2) that many completely unrelated
functions can have zero as its associated measure.
2) The o-additivity of m implies that f is incrementally right
continuous, but as we have seen in Chapter II this is insufficient to
imply right continuity in the order sense without imposing finite
variation on the one-dimensional paths f(s;-) and f(-,t).