Remark 14.5.1.label Past me made the mistake of believing that in Lemma 14.5.5, it is possible to approximate rectifiable curves with piecewise linear curves in total variation distance. However, this is not possible: as every piecewise linear curve is absolutely continuous, and the limit of these curves in total variation distance must also be absolutely continuous. As such, this strong approximation exists if and only if the curve is absolutely continuous.