Quasi-additive set functions and the concept of integral over a variety

Abstract

In the last years a great deal of research has been done on the axiomatization of the concept of linear integral over a variety T, but very little for the nonlinear integrals 1=fTf(P, q), p = (Pi, * * * , PM), q = (qi, * * * , qk), though they have their natural place in Volterra's functional analysis. In the present paper we give the first elements of a very general axiomatization concerning such integrals. This is done by introducing a concept of quasi normality (quasi additivity, quiasi subadditivity, etc.) which extends Banach's concept of normality. As we will see, if 4 (I) is a quasi additive set function of bounded variation, then the integral 3(f, T, 4) of a function f(p, q) over any continuous parametric mapping T with respect to 4 (f positive homogeneous of degree one in q but not necessarily linear) can be defined by a process of limit on the set function d1(I) =f [T(r), p (I) ], rCI, and this set function is proved to be itself quasi additive [?6, Theorem (i) ]. It appears, therefore, that the concept of quasi additivity is a generalization of the usual additivity which is reproduced by the integrand f, though the latter is not linear. The integral 3 is obtained by means of a process of limit involving a mesh a which is defined axiomatically. Such a process is more general than usual since a is not required to decrease nor approach zero by refinements only. Examples of this situation are known (?4). In successive papers we will discuss the problem of extension and other properties of the integral 3 in connection with measure theory, Radon-Nikodym derivatives, and weak convergence. The present integral 3 contains, as a particular case, the concept of integral over a continuous surface T of finite area, as defined by the author and used in surface area theory and the calculus of variations by L. Sigalov, J. M. Danskin, V. E. Bononcini, J. Cecconi, L. H. Turner, and the author. In this particular situation 45(I) is the usual vector signed area of the same mapping T. As is natural, the same integral 3 contains, as a particular case, the classic Weierstrass integral over a curve of finite Jordan length. Finally, the process of limit considered in the present paper can be thought of as extending Burkill's integration process for normal interval functions. Burkill's integral has been used in connection with total variation, Jordan