# On the causality and $K$-causality between measures

## TL;DR

This paper extends the concept of causality relations from spacetime points to measures, preserving key properties and providing new characterizations, thereby broadening the understanding of nonlocal causality in spacetime.

## Contribution

It introduces an extension of the $K^+$ relation to measures, maintaining transitivity and closedness, and offers new characterizations including a nonlocal time function analogue.

## Key findings

- $K^+$ relation retains transitivity and closedness for measures
- Provides new characterizations of the $K^+$ relation in the measure space
- Generalizes the causal precedence relation $J^+$ to measures

## Abstract

Drawing from our earlier works on the notion of causality for nonlocal phenomena, we propose and study the extension of the Sorkin--Woolgar relation $K^+$ onto the space of Borel probability measures on a given spacetime. We show that it retains its fundamental properties of transitivity and closedness. Furthermore, we list and prove several characterizations of this relation, including the `nonlocal' analogue of the characterization of $K^+$ in terms of time functions. This generalizes and casts new light on our earlier results concerning the causal precedence relation $J^+$ between measures.