Vol. 67, No. 2, 1976

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ISSN: 0030-8730
Integrals of continuous functions

Mark Finkelstein and Robert James Whitley

Vol. 67 (1976), No. 2, 365–372
Abstract

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

      ∫
∞
f(s) = −∞ h(s,t)dt,

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.

Mathematical Subject Classification 2000
Primary: 28A25
Secondary: 26A42
Milestones
Received: 4 June 1976
Published: 1 December 1976
Authors
Mark Finkelstein
Robert James Whitley