LX annos nato SPACES OF MEASURES ON COMPLETETY REGULAR SPACES

Let X be a regular topological space. If (Fo)new is a sequence of Radon (i.e., inner regular by compact) measures on X such that (p"(f)r€N converges for every regular open set T of X (i.e., for which 7:Z), then (p,(A)),r * converges for every Borel set,4 of X. This result was proved by P. Gänssler ([4] Theorem 3.1) for real measures and by S. S. Khurana ([fl Theorem 4) for group valued measures. It will be shown in this paper (Theorem 3) that, if X is completely regular, this result can be improved by assuming only that (p,(Z)),.r converges for those regular open sets T of X for which there exists a continuous real function f on X such that {"f =0} c. T c { / =o} (or equivalently f:f7=q1; we denote the set of these sets Z by 3. If the vector lattice of continuous real functions on X is order o-complete, then 3 is exaclly the set ofclosed open sets ofXand so the above formulation contains the corresponding result of Z. Semadeni ([8] Theorem (i)=+(iv)). Let 6 be the vector space of conti-nuous bounded real funclions on X endowed with the strict topology and E be a quasicomplete G-space ([2] Definition 5.9.11). We show (Theorem 12) that a conti-nuous linear map u:6*E is boundedly weakly compact (or equivalently possesses an integral representalion) if and onty if the sels of 3 are sent into E by the biadjoint map of u. The special case of,E equal to the vector space of continuous real functions on a metrizable topological space endowed with the topology of compact conver-gence is discussed in greater detail (Theorems 15 and 16). We use the notations and the terminology of lll and [2]. The expression locally conuex space will mean Hausdorffreal locally convex space. For every locally con-vex space.Ewe denote its dual and bidual by E ' and E", respectively, and identify,E with a subspace of E " Åa the evaluation map E * E", x-(x,.), For every continuous linear map u of locally convex spac€s, u ' and u " will denote the adjoint and the biadjoint of z, respeclively. N, Q, R denote the sets of natural numbers, rational numbers, and real numbers respectively. Throughout this paper we dmote by E a locally convex space, by I a completely regular space, by X a subspace of Y, by G lhe vector space of continuous bounde