  We develop a general convergence theory for adaptive discontinuous Galerkin
methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as
well as all practically relevant marking strategies. Another key feature of the
presented result is, that it holds for penalty parameters only necessary for
the standard analysis of the respective scheme. The analysis is based on a
quasi interpolation into a newly developed limit space of the adaptively
created non-conforming discrete spaces, which enables to generalise the basic
convergence result for conforming adaptive finite element methods by Morin,
Siebert, and Veeser [A basic convergence result for conforming adaptive finite
elements, Math. Models Methods Appl. Sci., 2008, 18(5), 707--737].
