09 - Rectifiable Sets, I

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Rectifiable sets are measure theoretic versions of smooth manifolds: they can be quite rough, but they still have an almost everywhere differentiable structure. Moreover, it is typically easier to verify a set is rectifiable than smooth. They are important since many results that hold for smooth surfaces extend to the rectifiable sets, and in some instances this is the right amount of differentiable structure needed for a result to hold. The focus this week will be on showing how to establish rectifiability in terms of cone points and that almost every point of a rectifiable set has a ?weak tangent plane? in the sense of tangent measures.