Order-invariant measures on fixed causal sets

Abstract

A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers -- we call such a linear extension a {\em natural extension}. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of {\em order-invariance}: if we condition on the set of the bottom $k$ elements of the natural extension, each possible ordering among these $k$ elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.