Notes on measure theory

Our aim is to provide the reader with a clear and simple (but nevertheless complete) analysis of the main topics of measure theory needed to fully understand the theory behind necessary optimality conditions for optimal control problems with state constraints. In so doing we deliberately left out many subjects of interest in measure theory; we refer the reader to bibliography in the end of this notes. Another important restriction of these notes is that they mainly focus on the real line (which is enough for the usual formulations of state constrained optimal control problems). Once again, generalization of measures in Rn, with n 2, can be found in the bibliography. These notes were written as a support for a short course on Measure Theory given by Maria do Rosário de Pinho at Lab I2002, Department of Electrical Engineering Department, Faculdade de Engenharia, University of Porto in June-July 2011