On integrals over sets of points
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The developments of the last twenty years in the theory of sets of points and in the applications of this theory to the theory of functions of real variables, besides leading to a tremendous extension of the ordinary theory of integration, have given rise to a number of definitions of the integral for the case of a function defined for a set of points. These may be classified as (a) geometrical and (b) analytical. We may characterize the geometrical definitions by saying that they define the integral as the volume (possibly n-dimensional) or area, as the case may be, of a set of points. As it is not the purpose of this paper to discuss the various ways in which this may be done, we shall content ourselves with saying here that for any geometrical definition the equivalent analytical definition may be found, and conversely. Of the types of analytical definition of the integral, two are fundamental in the theory; (i) those definitions which regard the integral as the inverse of the derivative, and (ii) those which define the integral as the limit of a sum, or set of sums. A third type of definition makes the extension to sets of points by the aid of an auxiliary function, but these definitions are not independent, since they imply a definition of one of the preceding types. By far the greater number of the definitions which we have are given from the standpoint of the integral as the limit of a sum, or set of sums. The fact that a difference in the way two sums are defined sometimes means a difference in the resulting integrals leads us to the considerations of this paper. It is our purpose to consider all the different kinds of sums which one might reasonably expect to lead to desirable results, to find out which of the resulting integrals are equivalent to integrals already defined, and which of them lead to new integrals of interest. In order to bring the discussion within reasonable limits we shall confine ourselves to what are ordinarily termed proper integrals, that is, integrals of limited functions defi
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