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...