ADVANCES IN APPLIED MATHEMATICS 3, 430--434 (1982) Integration on Fuzzy Sets* MADAN L. PURI Department of Mathematics, Indiana University, Bloomington, Indiana 47405 AND DAN A. RALESCU Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221 In this paper we define the integral of a function on a fuzzy set. Such an integral generalizes the integral over ordinary (nonfuzzy) sets. Some of the basic properties of this integral are stated. The main result shows that under suitable hypotheses, the integral on a fuzzy set equals the integral over some level set. Some applications are indicated. l. INTRODUCTION The concept of fuzzy set as introduced by Zadeh [7] was proved to be useful in modelling inexact systems [2], where subjective judgments and the nonstatistical nature of inexactness make classical methods of probability theory unsuitable. The concept of fuzzy integral as developed in [5] and [3] has been used in the problems of optimization with inexact constraints [6, 4]. In this paper we define the integral of a (measurable) function over a fuzzy set. The measure with respect to which this integral is defined is a classical measure. Throughout this paper, (X, d~,/~) will denote a measure space consisting of a set X, a o-algebra ~ of subsets of X, and a positive measure/~ on ft.. If f- X---, [0, o¢] is a positive extended real-valued measurable function, we denote the integral of f w.r.t./~ by fxfdtt. Also fafd~ = fxfXAdl ~, where XA is the indicator (characteristic) function of A ~ d~. A fuzzy subset of X (or fuzzy set) is a function u: X ---, [0, 1]. Let F(X) denote the set of all fuzzy subsets of X. It is well known [2] that F(X) is a *Research supported by the National Science Foundation Grant IST-7918468. 430 0196-8858/82/040430-0550500/0 Copyright © 1982 by Academic Press, Inc. All fights of reproduction in any form reserved.