Removability of zero modular capacity sets

We introduce a notion of modular with a corresponding modular function space in order to build a modular capacity theory. We give two different definitions of capacity, one of them of variational type, the other one through either the modular of the test functions, or the modular of their gradients. We study, in both cases, the removability of sets of zero capacity in fairly general abstract Sobolev spaces with zero boundary values. As a key tool, we establish a modular Poincare inequality. With the notion of modular function space in hands, we find a way to introduce a Banach function space, which allows to compare the zero capacity sets with respect to both notions. Thanks to this comparison, we characterize the compact sets of zero variational type capacity as removable sets. The paper is enriched with several examples, extending and unifying many results already known in literature in the settings of Musielak-Orlicz-Sobolev spaces, Lorentz-Sobolev spaces, variable exponent Sobolev spaces.