If (F) is a finite measure space and I is a continuous linear functional on >1, there exists ge. where (1/p)+(1/4)-1, such that (f)- Sadu for all fe, This is known as the (Riesz) representation theorem Hint: Let be finite. Since 1, F.) for all Ae, a set function on Fis defined by v[4] () It is finitely additive and, moreover, additive by con- tinuity of and the fact that V(0) -0. Further, vis finite since V is bounded (Exercise 8); is absolutely continuous with respect to . By the Radon-Nikodym theorem there is an F-measurable g finite ae, with V(1) - {4}-{