Differentiability of Lipschitz functions, structure of null sets, and other problems

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On the differentiability of Lipschitz functions with respect to measures in the Euclidean space (2014, background): “Then for every ε > 0 and every r′ > r there exists a smooth function g : Rn → R such that (i) ‖g‖∞ ≤ ε and the support of g is contained in B′ := B(x̄, r′); and for every x ∈ B′, (ii) −ε ≤ Deg(x) ≤ 1, and Deg(x) = 1 if x ∈ K ∩B; (iii) |dW g(x)| ≤ 2/ tanα where W := e⊥.(3) Proof of Lemma 5.”