A C^k Lusin approximation theorem for real-valued functions on Carnot groups

We study the Lusin approximation problem for real-valued measurable functions on Carnot groups. We prove that k-approximate differentiability almost everywhere is equivalent to admitting a Lusin approximation by CᵏG maps. We also prove that existence of an approximate (k-1)-Taylor polynomial almost everywhere is equivalent to admitting a Lusin approximation by maps in a suitable Lipschitz function space.