A Green's Theorem in Terms of Lebesgue Integrals

The method of proof, that of approximating -polynomials, was suggested by Professor G. C. Evans, who uses it in the proof of a similar theorem,* where, however, the functions u, aujax, j, etc., have to obey certain restrictions as to continuity owing to the fact that the integrals in the equation are of the ordinary kind. Here, however, Lebesgue integrals are used; consequently, as might be expected, the properties of u, aujax, j, etc., are less restricted. The region, R, here considered is a rectangle. The works cited in footnotes are the Cours d' analyse of de la ValleePoussin and the article Sur l'integrale de Lebesgue by the same author in the Transactions of the American Mathematical Society, Volume XVI} 1916. They are referred to, briefly, as Cours d'analyse and Transactions respectively. § 1. The following theorems and definitions will be used in the course of this discussion. THEOREM A.t If the transformation