*K*-Causality between Measures

Received: 31 January 2017 / Revised: 1 March 2017 / Accepted: 11 March 2017 / Published: 20 March 2017

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**Abstract**

Drawing from the optimal transport theory adapted to the Lorentzian setting, we propose and study the extension of the Sorkin–Woolgar causal relation ${K}^{+}$ onto the space of Borel probability measures on a given spacetime. We show that it retains its fundamental properties
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Drawing from the optimal transport theory adapted to the Lorentzian setting, we propose and study the extension of the Sorkin–Woolgar causal 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 “measure-theoretic” analogue of the characterization of ${K}^{+}$ in terms of time functions.
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(This article belongs to the Special Issue Varying Constants and Fundamental Cosmology)