The course introduces a rigorous treatment of probability theory based on measure and Lebesgue integration theory. We will develop the necessary measure and integration theory together with the probabilistic interpretation so that the two subjects (measure theory and probability) are developed simultaneously. Measure theoretic topics include σ-algebras, measures, Lebesgue integrals, convergence theorems, and the Radon-Nikodym theorem. Probabilistic topics include conditional probability and expectation, modes of convergence of random variables, characteristic functions, laws of large numbers and martingales.