Rectifiability - an overview | ScienceDirect Topics

By the rectifiability property we can find countably many oriented C1 hypersurfaces Σi and pairwise disjoint Borel sets Ei ⊂ Σi ∩ Σ such that Hd−1(Σ\∪iEi)=0; From: Handbook of Differential Equations: Evolutionary Equations, 2007 Frank Morgan, in Geometric Measure Theory (Fifth Edition), 2016 Im is F closed in Nm, (“boundary rectifiability”) 𝐼 𝑚 + 1 = { 𝑇 ∈ 𝑅 𝑚 + 1 : 𝑀 ( ∂ 𝑇 ) < ∞ } , 𝑅 𝑚 = { 𝑇 ∈ 𝐹 𝑚 : 𝑀 ( 𝑇 ) < ∞ } . Consequently, 𝑇 = { 𝑇 ∈ 𝐼 𝑚 : spt 𝑇 ⊂ 𝐵 𝑛 ( 0 , 𝑅 ) , 𝑀 ( 𝑇 ) ≤ 𝑐 , and 𝑀 ( ∂ 𝑇 ) ≤ 𝑐 } is 𝐹 complete. Remarks This result, specifically (1), is the difficult part of the compactness theorem. It depends on Lemma 4.12, the characterization of rectifiable sets by rectifiable slices. The original proof of Lemma 4.12 used structure theory. However, in 1986 White [3] found a simpler and more direct argument. It follows directly from the definition of 𝐹 𝑚 that { 𝑇 ∈ 𝐹 𝑚 : spt 𝑇 ⊂ 𝐵 ( 0 , 𝑅 ) } is 𝐹 complete. Consequence (4) now fol

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