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If $u\in W_{\mathrm{loc}}^{1,2}$ and $\Phi$ is continuously differentiable with bounded derivative on the range in use, then $D(\Phi (u))={\Phi }^{\prime }(u)Du$ almost everywhere. One proves this first for smooth functions and then uses density of smooth functions in a Sobolev space and boundedness of ${\Phi }^{\prime }$. On compact sets one may truncate $\Phi$ away from the range of a bounded $u$.

For a Sobolev space function, almost every line parallel to a coordinate axis gives an absolutely continuous restriction. An absolutely continuous function has derivative zero almost everywhere on each of its level sets. Slicing and Fubini's theorem give the assertion in higher dimension. If $w\in W^{2,p}$, apply it to $w$ and then to each component of its gradient to get $D^{2}w=0$ almost everywhere on $\{w=0\}$. This supplies the contact differentiation used in noncontact of ROF level boundaries.

Suppose $f_k\to f$ almost everywhere and ${\sup }_k|f_k|\in L^1$ in a probability measure-preserving system. Thenalmost everywhere and in $L^1$, where $\mathcal{I}$ is the invariant sigma-algebra. For the almost-everywhere assertion, bound the terms with index at least $M$ by ${\sup }_{k\ge M}|f_k-f|$, apply the Birkhoff ergodic theorem, and then let $M\to \infty$. The finitely many remaining end terms vanish by the linear growth bound for integrable observables.