Convergence in Measure for Semigroup-valued Integrals

The semigroup-valued integral of M. Sion [S] is reformulated for a general notion of approximation by sums of values taken by a set function integrand. A convergence in measure theorem is established, which yields both his pointwise dominated convergence theorem as well as an integrability criterion which specializes to his existence theorem. In [S] M. Sion introduced and developed an "integral process" for set functions with values in a uniform semigroup, and more particularly for (possibly multi-) uniform space-valued point functions with respect to a topological semigroup-valued finitely additive set function, on the product of whose image with that of the point function a suitable uniform semigroup-valued "multiplication" is available. Among the developments is a convergence theorem for sequences of integrable point functions, patterned after Lebesgue's, thus postulating almost everywhere convergence. Classically this type of convergence theorem can be deduced from one with the weaker hypothesis of convergence in measure; the deduction of this hypothesis from almost everywhere convergence will also be available here—although the usual route via Egoroff's almost uniform convergence seems not be feasible in this generality. This may have been what hindered the development of a convergence in measure theorem in [S]; there may also be some confusion connected with the asserted equivalence of the formulations on pages 40 and 41 (whose verification by "straightforward checking" this reader was unable to believe and conjectures to be incorrect). The task of setting up the indicated convergence in measure theorem is facilitated by the availability of the preceding [F], which generalized the Banach space-valued result from [DS] to a non-absolutely integrable setting1 and was liberally sprinkled with hints as to the modifications which would make it applicable to uniform spaces. The only essentially new ingredient which has to be provided is the displacement of the directed order, which governed convergence to the integral, from the set of finite disjoint subfamilies in the domain of the measure, to an auxiliary set: i.e. the replacement of the order on these subfamilies by a net on them. The result achieved Received by the editors March 10, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 28B10. ' This had actually been done already in [B].