Equivalence of integrals

Abstract

Suppose R R is the set of real numbers and N N is the set of nonnegative real numbers, each of G G and F F is a function from R × R R \times R to N N . All integrals considered are of the subdivision-refinement type. This paper gives necessary and sufficient conditions for ∫ a b F = ∫ a b G \int _a^b {F = } \int _a^b G . A necessary and sufficient condition for ∫ a b G 2 = 0 \int _a^b {{G^2} = 0} is also given.