Lorentzian coarea inequality

In this article, we introduce the notion of locally uniformly d-controlling map between Lorentzian pre-length spaces which is preserving the diameters of causal diamonds, and through that we establish the coarea inequality for Lorentzian Hausdorff measure which is introduced by McCann and S\"{a}mann. Besides that we get a covering lemma for subsets in a Lorentzian pre-length space with a new local assumption named the local causal enlargement property, which enables us to enlarge causal diamonds.