Inserted: 8 jul 2016
Last Updated: 30 oct 2017
To appear in ''Measure Theory in Non-Smooth Spaces" (Nicola Gigli, ed), De Gruyter,
This note details how a recent structure theorem for normal $1$-currents proved by the first and third author allows to prove a conjecture of Cheeger concerning the structure of Lipschitz differentiability spaces. More precisely, we show that the push-forward of the measure from a Lipschitz differentiability space under a chart is absolutely continuous with respect to Lebesgue measure.