ABSTRACT
T heorem 1. Let fd v be summable on a cell K where f > 0 and dv > 0. Then there exists a measurable function f > 0 on K such that: (i) fd v is integrable on K , (ii) f K fd v = JKfdv , (in) f > f dv-everywhere, and (iv) g > f dv-everywhere for ev ery measurable function g on K such that g > f dv-everywhere,
P roof. Define F on K = [a, 6] by
(1)
which is finite since fd v is summable. By Theorem 2 (§2.5)
(2)
and (3) dF is the smallest integrable differential such that dF > fdv.
