Morera Type Theorems

Abstract

Throughout the following each point (x, y) ∊ ℝ² is identified with the complex number $z = x + iy = \rho e^{i\phi } \left( {\rho = \left| z \right|, - \pi < \phi \leqslant \pi } \right)$. Then the group of Euclidean motions of the complex plane can be identified with M(2). Let Ω be a domain in ℂ and let Hol(Ω) be the following set of functions from Ω into ℂ: f ∊ Hol(Ω) if and only if there exists a holomorphic function in Ω coinciding with f almost everywhere (with respect to the Lebesgue measure).