An integration theory for vector
functions and operator-valued measures is outlined, and it is shown that in the
setting of locally convex topological vector spaces, the dominated and bounded
convergence theorems are almost equivalent to the countable additivity of the
integrating measure. The measures studied are those representing the continuous
linear operators on a space of continuous functions. When certain restrictions are
imposed on the space involved, actual equivalence of countable additivity and the
above theorems obtains, as well as equivalence of certain compactness properties of
the operator being represented. An example is given which shows that, in general
spaces, convergence in measure no longer implies the almost everywhere convergence
of a subsequence.
