During the last years representation theory of additive operators on spaces of measurable functions has been of interest for many writers. L. DREWNOWSKI, W. ORLICZ ([4], [5]), and V. MIZEL ([19], [20]) considered scalar valued operators (functionalu) on various function spaces, then in a subsequent paper V. MIZEL and K. SUNDARESAN ([21]) extended these results and characterjzed those transformations from LEBESGUE-BOCHNER function spaces L p ( S ) into a BANACH space, which admit a representation by integral kernels with BANACH space valued CARATHI~ODORY functions, integrated with respect t o a positive measure. J. BATT ([l]) finally got a weak integral representation for a very general class of additive operators from L i ( 0 ) into a BANACH space.
In this paper we will characterize those additive transformations, which admit a representation by integral kernels with real valued CARATH~ODORY functions, integrated with respect to ~1 a-additive vector measure. Of course, not every additive operator can be characterized this way. On the other hand our representation is rather general, because we suppose nothing on the variation of the vector meaaure and moreover, it may take values in an arbitrary quasi-complete locally convex vector space. Finally the operator may be defined on a general (not necessarily locally convex) P-normed function space.
The present study is motivated by the fact that one gets very good information on the range of such an operator. This leads to conditions for weak