Unimodular Random Measured Metric Spaces and Palm Theory on Them

In this work, we define the notion of unimodular random measured metric spaces as a common generalization of various other notions. This includes the discrete cases like unimodular graphs and stationary point processes, as well as the non-discrete cases like stationary random measures and the continuum metric spaces arising as scaling limits of graphs. We provide various examples and prove many general results; e.g., on weak limits, re-rooting invariance, random walks, ergodic decomposition, amenability and balancing transport kernels. In addition, we generalize the Palm theory to point processes and random measures on a given unimodular space. This is useful for Palm calculations and also for reducing some problems to the discrete cases.