Introduction to Measure Theory and Integration Measure spaces Integration Spaces of measurable functions Hilbert spaces Fourier series Operations on measures The fundamental theorem of integral calculus Operations on measures Appendix: Riesz representation theorem of the dual of C(K) and integrals depending on a parameter Solutions to the exercises. This textbook collects the notes for an introductory course in measure theory and integration. The course was taught by the authors to undergraduate students of the Scuola Normale Superiore in the years 2000 2011. The goal of the course was to present in a quick but rigorous way the modern point of view on measure theory and integration putting Lebesgues Euclidean space theory into a more general context and presenting the basic applications to Fourier series calculus and real analysis. The text can also pave the way to more advanced courses in probability stochastic processes or geometric measure theory. Prerequisites for the book are a basic knowledge of calculus in one and several variables metric spaces and linear algebra. All results presented here as well as their proofs are classical. The authors claim some originality only in the presentation and in the choice of the exercises. Detailed solutions to the exercises are provided in the final part of the book.