  We revisit the theory of stochastic integral convergence on Skorokhod space,
providing a new comprehensive treatment under Skorokhod's J1 and M1 topologies
based on a simple and tractable notion of good decompositions for the
integrators. Our analysis yields new insights even in the classical J1 setting
and addresses the sharpness of various key results. As regards convergence of
stochastic integrals in the M1 setting, our contribution is twofold. Firstly,
we derive useful sufficient conditions. Secondly, we demonstrate the
limitations of general weak continuity properties for strictly M1 convergent
integrators. Moreover, we provide an important structural result for local
martingales on Skorokhod space, showing that their M1 tightness generally
implies their J1 tightness. The practicality of our results is illustrated
through novel contributions to applications from econometric theory,
statistical mechanics, mathematical finance, and insurance mathematics.
