  Conditional independence and graphical models are well studied for
probability distributions on product spaces. We propose a new notion of
conditional independence for any measure $\Lambda$ on the punctured Euclidean
space $\mathbb R^d\setminus \{0\}$ that explodes at the origin. The importance
of such measures stems from their connection to infinitely divisible and
max-infinitely divisible distributions, where they appear as L\'evy measures
and exponent measures, respectively. We characterize independence and
conditional independence for $\Lambda$ in various ways through kernels and
factorization of a modified density, including a Hammersley-Clifford type
theorem for undirected graphical models. As opposed to the classical
conditional independence, our notion is intimately connected to the support of
the measure $\Lambda$. Our general theory unifies and extends recent approaches
to graphical modeling in the fields of extreme value analysis and L\'evy
processes. Our results for the corresponding undirected and directed graphical
models lay the foundation for new statistical methodology in these areas.
