Vol. 111, No. 1, 1984

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Weak integral convergence theorems and operator measures

William Victor Smith and Don Harrell Tucker

Vol. 111 (1984), No. 1, 243–256

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.

Mathematical Subject Classification 2000
Primary: 46G10
Secondary: 28B05
Received: 31 July 1978
Revised: 1 March 1983
Published: 1 March 1984
William Victor Smith
Don Harrell Tucker