Let Lm,p(Rn) be the homogeneous Sobolev space for p∈(n,∞), μ be a Borel regular measure on Rn, and Lm,p(Rn)+Lp(dμ) be the space of Borel measurable functions with finite seminorm ‖f‖Lm,p(Rn)+Lp(dμ):=inff1+f2=f{‖f1‖Lm,p(Rn)p+∫Rn|f2|pdμ}1/p. We construct a linear operator T:Lm,p(Rn)+Lp(dμ)→Lm,p(Rn), that nearly optimally decomposes every function in the sum space: ‖Tf‖Lm,p(Rn)p+∫Rn|Tf−f|pdμ≤C‖f‖Lm,p(Rn)+Lp(dμ)p with C dependent on m, n, and p only. For E⊂Rn, let Lm,p(E) denote the space of all restrictions to E of functions F∈Lm,p(Rn), equipped with the standard trace seminorm. For p∈(n,∞), we construct a linear extension operator T:Lm,p(E)→Lm,p(Rn) satisfying Tf|E=f|E and ‖Tf‖Lm,p(Rn)≤C‖f‖Lm,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.