This chapter presents the precise construction of expected value (or expectation) for random variables and distributions within the framework of integrals with respect to the (probability) measure. The approach enables the calculation of expectations not only for discrete, absolutely continuous (and convex linear combination of discrete and absolutely continuous) distributions, but also for singular distributions. The chapter includes proofs of Lebesgue’s Monotone Convergence Theorem, Fatou’s Lemma, and Lebesgue’s Dominated Convergence Theorem for expected values. Finally, the expectation is expressed as the Lebesgue-Stieltjes integral with respect to increments of the distribution function. This sets the stage for the (subsequent) definition of the path integral for stochastic processes.