more closely related to measure theory: namely, a condition
sufficient to generate a positive measure.
2
Definition 2.1.3. Let f: R R be a (real-valued) function.
a) We say f is increasing (in the order sense) if
z z' => f(z) S f(z').
b) We say f is incrementally increasing if A ,(f) 1 0 for
zz
all z z'.
The scalar-valued functions we shall typically consider are defined
using the variation of vector-valued functions: these (as we shall
see later) are increasing in both senses. In general, however, the
two notions are distinct--neither implies the other, as the following
two examples show. In these, we focus our attention on the unit
square [(0,0),(1,1)] for simplicity, but we can extend them (by
constants, say) to give a perfectly good counterexample defined on all
of R2
Examples 2.1.4.
i) We define here a function satisfying definition 2.1.3(a) but
not (b). The particular function we shall give is defined on the unit
square; we could extend it arbitrarily outside [(0,0),(1,1)], but we
shall not give an explicit extension--the square is sufficient to
indicate how things can go wrong.
The idea consists of writing A ,f = f(s',t')-f(s',t)-f(s,t')+f(s,t)
as f(s',t')-f(s',t)-[f(s,t')-f(s,t)], so if the second difference is
larger than the first, the increment will be negative even if f is
increasing in the sense of 2.1.3(a). Accordingly, for (s,t) in the