A well-known result by S. Banach and S. Saks states that every bounded sequence in $L^p([0,1])$, $1 < p < \infty$, has a Cesàro convergent subsequence. By taking the average of the subsequence multiple times, and through a recursive procedure, one can define a transfinite sequence of properties, thus giving rise to an ordinal rank or index for certain subsets of Banach spaces. We explore a different approach to this rank in the separable case using uniform families of finite subsets of integers. This allows us to characterize the existence of such a rank and study some of its properties from the point of view of Descriptive Set Theory.