DEMONSTRATIO MATHEMATICA Vol. XXI Mo 3 198» Andrzej Nowakowski NOTE ON CONVEX INTEGRAL FUNCTIONALS Some more refined results on convex integral functional are investigated. This note is also a supplement to paper [3]. The necessity of investigations of convex integral functio- nals is broadly explained by Rockefeller in [4], [5]. Problems which we shall study here arose by investigations of variatio- nal problems and differential equations in Hilbert spaces in [3]. However, some results obtained here are interesting by themselves and generalize several statements from [5]« 2* In what follows, we assume X to be a separable refle- xive Banach space with norm 1 -] and Y its dual, paired by (»). will denote the space X supplied with its weak to- pology. Let (E,M,dt) be a measure space such that M is a 6-algebra of subsets of E and dt is a positive 6-finite mea- sure on M which is complete. We shall refer to a function f : E *X — (-00,+ » ] as an integrand and, for each t in E, we denote by f t the function x f(t,x). f w i l l be called a convex integrand if f t is convex for each t in E. The con- jugate of the integrand f is the integrand g on E*Y defined by g(t,y) = sup{(x,y) - f ( t , x ) | x e x } . The general theory of conjugate functions asserts that in this case g t is convex and lower semicontinuous. Let B be the 6-algebra of Borel subsets of X. M ® B will denote the 6-alge- - 827 -