UC Irvine

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 nonnegative lower semicontinuous functions on the real line are exactly those functions f which can be written as (Formula Presented) with h nonnegative and continuous on R × R and h(s, ·) integrable. There is a similar representation for functions of Baire class 0 or 1 but the integral involved is the (conditional) improper Riemann integral. Generalization leads to a concept of conditional integrals in a more general setting. © 1976, University of California, Berkeley. All Rights Reserved.