On Relative Content and Green's Lemma*

in which P(xy) =x; and it is to be noted that here C does not need to be rectifiable. In the present paper a definition of relative content is given which makes it possible to prove that if P and Px are subject to certain conditions, the content of K, relative to a certain non-additive function of rectangles derived from P, exists equal to the double integral on the left of (1) and also equal to the line integral on the right of (1) whenever that integral exists. This result includes as a special case the form of Green's lemma for rectifiable C obtained by Gross,t except that in our result Px is deliberately restricted to be properly Riemann integrable instead of summable. In the last section sufficient conditions for the existence of the line integral are given which yield Green's lemma for an important case in which C does not need to be rectifiable. 1. Definitions and elementary theorems. Let *ß denote a class of partitions El of the rectangle R0: a^x^b, c^y^d, such that (1) each partition JJ is formed by dividing R0 into vertical and horizontal strips; and (2) the (greatest) lower bound of the norms of the partitions II of ^? is zero ; here by the norm of a partition n of $ is meant the (least) upper bound of the lengths of the diagonals of the rectangles of which n consists. Moreover let f(R) be a function (not necessarily single-valued) defined for every rectangle R: x'_*_*", y'úyúy" lying in R0. Also, if K\ and K2 are any two sets in R0, let t(Ku K2) = 1 if Ki and K2 have at least one point