Characterization of AC and Sobolev curves via Lipschitz post-compositions

Let X:=(X,d) be an arbitrary metric space. For each p ∈ [1,∞], we prove that a map γ:[a,b] → X is p-absolutely continuous if and only if, for every Lipschitz function h:X → R, the post-composition h ∘ γ is a p-absolutely continuous function. Furthermore, if X is complete and separable, then, for each p ∈ (1,∞), we show that the equivalence class (up to L¹-a.e. equality) of a Borel map γ:[a,b] → X belongs to the Sobolev Wₚ¹([a,b],X)-space if and only if, for every Lipschitz function h:X → R, the equivalence class (up to L¹-a.e. equality) of the post-composition h ∘ γ belongs to the Sobolev Wₚ¹([a,b],R)-space.