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
Highlights
Semicontinuous and related functions are characterized as integrals of continuous functions in several variables
Consider h(s, X) = Σ (Λ+i(β) - fM)K(x). This nonnegative function is continuous on S x R because of the uniform convergence of the series
What properties of Lebesgue measure and of the real line support these theorems? That is, for a measure space (Γ, μ), when can we find h continuous on S x T and represent f(s) by \ h(s, t)dμ in place h{s, x)dxtί We are not asking for minimal hypotheses, that is an interesting problem, but for reasonably general sufficient conditions
Summary
Semicontinuous and related functions are characterized as integrals of continuous functions in several variables. The sequence {fn} converges monotonically to / which is l.s.c. suppose that / is l.s.c. and let {/»}, with fί = 0, be a sequence of continuous functions increasing pointwise to / . A function f belongs to Baire class 0 or 1 iff there is a function h, continuous on S x R, with h{s, ) improperly Riemann integrable on R, and
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