When a liquid stream is injected into a gaseous atmosphere, it destabilizes and continuously passes through different states characterized by different scales and topologies. Throughout this process, the flow and physical variables (turbulence characteristics, surface tension, etc) may act differently depending on the region of the flow and the scales of the involved liquid structures. Exploring this multi-scale and high-dimensional phenomenon requires some new theoretical tools, some of which need yet to be elaborated. In the present study, an innovative general framework is established by transposing the machinery of two-point statistical equations to a relevant metric of liquid-gas flows (the liquid volume fraction). This allows distinguishing the transport of liquid which occurs in geometrical space (i.e. from one position in the flow to the other) and the one occurring in scale space (e.g. from large to small scales). These equations are exact and do not rely on any particular assumptions. The notion of scale is explicit and unambiguously defined. They further apply to the entire flow field, from the injection to the spray dispersion zone and irrespectively of the flow configuration or regime. This new set of equations is here invoked to characterize the air-assisted atomization of a planar liquid layer simulated by means of Direct Numerical Simulation using the ARCHER code.