From measures to Itô integrals Undergraduate mathematics syllabi vary considerably in their coverage of measure theoretic probability theory so beginning graduates often find substantial gaps in their background when attending modules in advanced analysis stochastic processes and applications. This text seeks to fill some of these gaps concisely. The exercises form an integral part of the text. The material arose from my experience of teaching AIMS students between 2004 and 2007 of which I retain many fond memories. The AIMS series format allows few explorations of byways; and the objective of arriving at a reasonably honest but concise account of the Itô integral decided most of the material. With motivation from elementary probability we discuss measures and integrals leading via L2 theory and conditional expectation to discrete martingales and an outline proof of the Radon Nikodym Theorem. The last two chapters introduce Brownian Motion and Itô integrals with a brief look at martingale calculus. Here proofs of several key results are only sketched briefly or omitted. The Black Scholes option pricing model provides the main application. None of the results presented is new; any remaining errors are mine From Measures to Itô Integrals gives a clear account of measure theory leading via L2 theory to Brownian motion Itô integrals and a brief look at martingale calculus. Modern probability theory and the applications of stochastic processes rely heavily on an understanding of basic measure theory. This text is ideal preparation for graduate level courses in mathematical finance and perfect for any reader seeking a basic understanding of the mathematics underpinning the various applications of Itô calculus