  Let $L^{m,p}(\mathbb{R}^n)$ be the homogeneous Sobolev space for $p \in
(n,\infty)$, $\mu$ be a Borel regular measure on $\mathbb{R}^n$, and
$L^{m,p}(\mathbb{R}^n) + L^p(d\mu)$ be the space of Borel measurable functions
with finite seminorm $\|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(d\mu)} :=
\text{inf}_{f_1 +f_2 = f} \{ \|f_1\|_{L^{m,p}(\mathbb{R}^n)}^p +
\int_{\mathbb{R}^n} |f_2|^p d\mu \}^{1/p}$. We construct a linear operator
$T:L^{m,p}(\mathbb{R}^n) + L^p(d\mu) \to L^{m,p}(\mathbb{R}^n)$, that nearly
optimally decomposes every function in the sum space:
$\|Tf\|_{L^{m,p}(\mathbb{R}^n)}^p + \int_{\mathbb{R}^n} |Tf-f|^p d\mu \leq C
\|f\|_{L^{m,p}(\mathbb{R}^n) + L^p(d\mu)}^p$ with $C$ dependent on $m$, $n$,
and $p$ only. For $E \subset \mathbb{R}^n$, let $L^{m,p}(E)$ denote the space
of all restrictions to $E$ of functions $F \in L^{m,p}(\mathbb{R}^n)$, equipped
with the standard trace seminorm. For $p \in (n, \infty)$, we construct a
linear extension operator $T:L^{m,p}(E) \to L^{m,p}(\mathbb{R}^n)$ satisfying
$Tf|_E = f|_E$ and $\|Tf\|_{L^{m,p}(\mathbb{R}^n)} \leq C \|f\|_{L^{m,p}(E)}$,
where $C$ depends only on $n$, $m$, and $p$. We show these operators can be
expressed through a collection of linear functionals whose supports have
bounded overlap.
