In this article we focus mainly on the class of almost periodic functions in view of the Lebesgue measure (briefly: \(\mu \) -a.p. functions) and on some if its subclasses. We are going to deal with autonomous superposition operators acting in the space of \(\mu \) -a.p. functions. We will indicate necessary and sufficient conditions under which the autonomous superposition operator maps the space under consideration into itself as well as conditions under which it is continuous. As a corollary from these results we indicate when the autonomous superposition operator defined on that space is a bijection. Next, we will analyse in detail the situation when the composition of \(\mu \) -a.p. function with a continuous function or with a homeomorphism gives a Stepanov almost periodic function. As an application of our results we indicate a subclass of \(\mu \) -a.p. functions for which linear differential equations with a non-homogeneous term belonging to this subclass may not have \(\mu \) -a.p. solutions.
