ABSTRACT
This chapter is devoted to constructions of measures on Cartesian products of measurable spaces, whose marginals agree with the measures given a priori on the individual spaces. It focuses on the finite products and the infinite products. The chapter presents a measure theoretical characterization of positive definite functions. It provides basic definitions and properties of the product measures and integrals along with theorems and proofs. The chapter establishes the fundamentals of the Fubini-Stone and Tonelli theorems. It considers the Cartesian product of measures, and indicates how other (non-Cartesian) products can be obtained. The latter also find applications in areas such as probability (under the name “transition probabilities”), game theory, integral equations with different types of kernels, and statistical decision functions. The chapter includes multiple exercises that help students try themselves and perform product measures and integrals.
