Statistical modelling of complex dependencies in extreme events requires meaningful sparsity structures in multivariate extremes. In this context two perspectives on conditional independence and graphical models have recently emerged: One that focuses on threshold exceedances and multivariate pareto distributions, and another that focuses on max-linear models and directed acyclic graphs. What connects these notions is the exponent measure that lies at the heart of each approach. In this work we develop a notion of conditional independence defined directly on the exponent measure (and even more generally on measures that explode at the origin) that extends recent work of Engelke and Hitz (2019), who had been confined to homogeneous measures with density. We prove easier checkable equivalent conditions to verify this new conditional independence in terms of a reduction to simple test classes, probability kernels and density factorizations. This provides a pathsway to graphical modelling among general multivariate (max-)infinitely distributions. Structural max-linear models turn out to form a Bayesian network with respect to our new form of conditional independence.[-]