right tends to zero as (s',t') (s,t), hence A zzf 0,
which is (b).
Unfortunately, we do not have the converse implication in
general. To see this, we give the following example, which we shall
refer to later in pointing out further weaknesses of using increments
alone.
Example 2.1.2. Let g be any E-valued function defined on [0,=).
For (s,t) E R2, we then put f(s,t) = g(t). Then, for any
(s,t) $ (s',t'), we have
If(s',t')-f(s',t)-f(s(s,t )+f(s,t) = g(t')-g(t)-g(t')+g(t)A
= 0.
The function f is then evidently incrementally right continuous for
any f so defined. If we take, however, g to be a function which is
not right continuous, we have
lim jf(s',t')-f(s,t)[ = lim jg(t')-g(t)j 0,
(s',t')-(s,t) t'4t
(s',t')a(s,t)
hence f is not right continuous. Later on, we shall establish some
additional conditions on f sufficient to have (b) => (a).
Another basic notion for one-parameter functions with regard to
Stieltjes measures is that of increasing function, as we reduce
functions of finite variation to this case via the Jordan
decomposition. Again, we have two definitions, the first the natural
extension of the one-variable definition (order sense), the second