Lemma 8.10.1EB20E121128A

Proof. For open $G_ i$ choose Sobolev functions $0\leq f_ i\leq 1$ equal to one almost everywhere on $G_ i$, with a summably small excess in their squared norms over the capacity infima. The Sobolev chain rule gives

Indeed, almost everywhere the gradient of the maximum is the gradient of an active function, and the squared value is bounded by the sum of the squared values. If the sum is finite, these maxima converge in $L^2$, are bounded in $H^1$, and have a weak $H^1$ limit equal to one almost everywhere on $\bigcup _ iG_ i$. Lower semicontinuity proves subadditivity for open sets. Taking open supersets proves it for arbitrary sets. If the sum is infinite there is nothing to prove. The level-set estimate follows by testing with $\min \{ 1,|f|/a\} $ and the same Sobolev chain rule.

Fix a compact $K\subset G$ and a Sobolev competitor $0\leq f\leq 1$ for the open set $G$. Positive local mollification and a partition of unity give smooth nonnegative $f_\epsilon \to f$ in $H^1$ that satisfy $f_\epsilon \geq 1$ on $K$ for small $\epsilon $: every convolution ball meeting $K$ then lies in $G$, where $f=1$ almost everywhere. The assumed estimate gives $\mu (K)\leq C\lim _\epsilon \| f_\epsilon \| _{H^1}=C\| f\| _{H^1}$. Take the infimum over $f$ and then the supremum over compact $K\subset G$ to obtain the estimate for $G$. Outer regularity and open supersets give it for every Borel $E$. $\square$