This chapter gathers various tools and results from measure and probability theory, martingale theory, uniform integrability, essential extrema and stochastic calculus (Itô’s formula)—some with a complete proof, others simply with precise references—which are used throughout the book. We also included the proofs of two specific mathematical results (discrepancy of the Halton sequence and the Pitman–Yor identity), which are not essential in the context of numerical applications but give the mathematical flavor of the underlying theories we use at several places in the book.