Indag. Mathem., N.S., 3 (3), 337-351 September 21, 1992 Singular linear functionals on non-locally convex Orlicz spaces by Marian Nowak Institute of Mathematics, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznan’, Poland Communicated by Prof. W.A.J. Luxemburg at the meeting of February 24, 1992 ABSTRACT Singular linear functionals on an Orlicz space Lv defined by an arbitrary Orlicz function cp over a a-finite atomless measure space are described. 1. INTRODUCTION AND PRELIMINEARIES In the theory of duality of function spaces an investigation of the space of all singular linear functionals is of importance. It is well known that the topo- logical dual X* of a complete F-normed function space X can be represented in the form: X* = Xi +X;, where Xl is the order continuous dual of X and X; (= the disjoint complement of XL in X*) is the space of all singular linear functionals on X. For an Orlicz space zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC L9 defined by a continuous convex Orlicz function (p (an N-function) the space (L9), was examined by T. Ando ([2], 1960). He proved that the space (L9); is an AL-space, and using this fact he established the general form of singular linear functionals on Lq in terms of some special class of purely finitely additive set functions. The Ando’s results were extended by M.M. Rao ([19], 1968) to the case of discontinuous convex Orlicz functions. His definition of singular linear functionals, however, did not cover all possible cases. The general situation for Orlicz spaces (defined by convex Orlicz func- tions) was discussed by E. de Jonge ([4], 1975). For normal function spaces (= Kothe normed spaces) E. de Jonge ([6], 1977) obtained some sufficient and necessary conditions for the spaces XL to be an AL-space. Moreover, he proved (151, 1976) that if Xi is an AL-space, then there exists a space of purely finitely 337