In the theory of vector valued functions there is a theorem which states
that if a function from a compact interval I into a normed
linear space X is of weak bounded variation, then it is of
bounded variation. The proof uses in a straightforward way the Uniform
Boundedness Principle (see [2, p. 60]). The present paper grew
from the question of whether an analogous theorem holds for absolutely
continuous functions. The answer is in the negative, and an example will be
given (Theorem 7). But it will also be shown that if X is
weakly sequentially complete (e.g. an Lp space, 1 ≦ p < ∞ ), then a weakly absolutely
continuous point function from / into X is absolutely
continuous. The method of proof involves the construction of a countably
additive set function in the standard Lebesgue-Stieltjes fashion.The paper is divided into three parts. In Section 1 extensions of finitely
additive, absolutely continuous set functions are carried out in an abstract
setting. Section 2 applies this to vector valued (point) functions on the
real line.
Vector valued absolutely continuous functions on idempotent semigroups