ABSTRACT
The concept of measurability of sets and functions plays an important role in various fields of classical and modern mathematical analysis. This chapter introduces and examines the three classes of functions: the class of sup-continuous mappings; the class of sup-measurable mappings; and the class of weakly sup-measurable mappings. From the view-point of the theory of first-order ordinary differential equations, the notion of a weakly sup-measurable mapping is more preferable than the notion of a sup-measurable mapping, because any solution of a first-order ordinary differential equation must be continuous everywhere and differentiable almost everywhere. The chapter also shows that all functions of two variables which satisfy the Carathéodory conditions are good from the point of view of sup-measurability.
