Stone-Cech compactification with applications to evolution equations on Banach spaces

In this note we study the question of Stone-Cech compactification of Banach spaces. This is used to construct regular countably additive measure valued functions as solutions for evolution equations on Banach spaces where the standard notions of solutions fail. We show that if one admits finitely additive measure valued solutions one can define the notion of measure solutions in terms of the original state space.