Distinguishing Between Symmetric Spaces and $L^\infty$ by a Differential Basis

One of the fundamental problems in the theory of differentiation of integrals is the following. Let $X$ and $Y$ be two spaces which are different in some sense. Does there exist a differential basis that differentiates the space $X$, i.e., all integrals of functions from $X$, but not integrals of functions from $Y$, i.e., there exists a function from $Y$ whose integral cannot be differentiated by this basis. In this paper we construct a basis which differentiates the space $L^\infty$ but does not differentiate any other symmetric space $X\ne L^\infty$.