In Krylov's book, he asserts that "the generalized derivative $u_x$ of a function $u$ continuous in $\Omega\subset \mathbb{R}^n$ does not exceed a constant $N_1$ almost everywhere if and only if the function $u$ satisfies the Lipschitz condition in $\Omega$ with the same constant". Here he requires a generalized derivative must be Borel measurable as follows:
I knew that a Lipschitz function on $\mathbb{R}^n$ is in $W^{1,\infty}(\Omega)$ and is differentiable Lebesgue almost everywhere. But usually $W^{1,\infty}$ means Lebesuge measurable weak derivatives.
Could you suggest any reference which shows a Lipschitz function has a Borel measurable weak derivative?
I guess one can show it by using a weak convergence argument and the fact that $L^2(\Omega, \mathcal{B}(\Omega), \mathcal{L}_n)$ is a Hilbert space, where $\mathcal{L}_n$ denotes the Lebesgue measure on $\mathbb{R}^n$. But I felt that this should be a fairly standard result and must have been shown somewhere.