The classical uniform almost periodic (a.p.) functions have been generalized, by omitting the continuity hypothesis, into Stepanov, Weyl and Besicovitch a.p. functions and a comprehensive account of these appears in \textit{A. S. Besicovitch}'s book [``Almost periodic functions'' (1932; Zbl 0004.25303)]. In 1938 Slutsky studied the corresponding a.p. concept for stationary processes for a Besicovitch class. Others have studied the remaining a.p. notions for the same type of processes. Here the author formulates the a.p. concept for the more general (nonstationary) harmonizable class, and studies the problem using the mean square sense instead of continuity. Then sufficient conditions are presented for a strongly harmonizable process to have almost all of its sample paths uniform, Stepanov, or Besicovitch a.p. property.