  In this paper, first-order Sobolev-type spaces on abstract metric measure
spaces are defined using the notion of (weak) upper gradients, where the
summability of a function and its upper gradient is measured by the "norm" of a
quasi-Banach function lattice. This approach gives rise to so-called Newtonian
spaces. Tools such as moduli of curve families and Sobolev capacity are
developed, which allows us to study basic properties of these spaces. The
absolute continuity of Newtonian functions along curves and the completeness of
Newtonian spaces in this general setting are established.
