Absolutely Continuous Functions with Values in Metric Spaces

(see e.g. [1, Lemma 1.1]), we could without any loss of generality work with Banach spaces only. The main obstacle in dealing with metric spaces (or arbitrary Banach spaces) is the absence of the Radon-Nikodým property and the resulting non-existence of derivatives. Thus, instead of the “usual” derivative, we have to employ the notion of a “metric derivative” (which was introduced by Kirchheim in [6]). We will need some results about this notion from [2]. Let (M,ρ) be a metric space, and let f : [a, b] → (M,ρ). We say that f is absolutely continuous, provided for each ε > 0 there exists a δ > 0 such that whenever [a1, b1], . . . , [ak, bk] is a sequence of non-overlapping intervals in [a, b] with ∑k i=1(bi − ai) < δ, then k