Integrals of continuous functions

Abstract

PACIFIC JOURNAL OF MATHEMATICS Vol. 67, No. 2, 1976 INTEGRALS OF CONTINUOUS FUNCTIONS MARK FINKELSTEIN A N D ROBERT WHITLEY Semicontinuous and related functions are characterized as integrals of continuous functions in several variables. For example: a new result of classical type is that the non- negative lower semicontinuous functions on the real line are exactly those functions / which can be written as /(s) = Γ h(s, t)dt, with h nonnegative and continuous on R x R and h{s, •) in- tegrable. There is a similar representation for functions of Baire class 0 or 1 but the integral involved is the (con- ditional) improper Riemann integral. Generalization leads to a concept of conditional integrals in a more general setting. We will consider a locally compact but not compact metric space All functions on S will be real valued, not extended real valued. Recall that [1, 2, 5, 6]: a function / is lower semicontinuous iff / ^ α , oo) is open for each a, and a function / is l.s.c. iff there is a monotone increasing sequence of continuous functions converging pointwise to / . S. THEOREM 1. A nonnegative function f on S is lower semicon- tinuous iff there is a nonnegative function h, continuous on S x R, with h(s, •) integrable for each s, and f(s)= h(s,x)dx. J-oo Proof. Suppose that (1) holds as described. The function h(s, x)dx -n is continuous. The sequence {f n } converges monotonically to / which is therefore l.s.c. Conversely, suppose that / is l.s.c. and let {/»}, with f ί = 0, be a sequence of continuous functions increasing pointwise to / . By truncating each function f n at ±n (redefine f n to be n when f n (s) > n and redefine f n to be — n when f n (s) < — n) we have \f n \^n. There is obviously a sequence of continuous functions h n on R satisfying 0 ^ h n ^ l/(2n + 1)2* and h n {x)dx = 1. Consider