Approximation Theorems for a Generalized Riemann Integrals

This note concerns the generalized Riemann integrals using free tagged subdivisions, and leading to absolute integration theories. We first establish an approximation theorem by step functions. This result will allow us to define a natural notion of equi-integrability in Lebesgue space $L^1$. Some applications, as a new caracterization of compact parts of $L^1$, are presented.