This book has grown out of my research interests in the theory of oper ator algebras, orthomodular structures and mathematical foundations of quantum theory. It is based on a series of lectures on measure theory on nonboolean operator structures which I prepared for Ph. D. students in Workshops on Measure Theory and Real Analysis in Italy (Gorizia 1999, Grado 2001) and which I have delivered at the conferences of the Interna tional Quantum Structures Association in Berlin 1996, Cesena 2001, and Vienna 2002. I have worked on these subjects in the framework of Prague's Semi nar on Mathematical...