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