Definition
A subset of Euclidean space is said to be -rectifiable set if there exist a countable collection of continuously differentiable maps
such that the -Hausdorff measure of
is zero. The backslash here denotes the set difference. Equivalently, the may be taken to be Lipschitz continuous without altering the definition.
A set is said to be purely -unrectifiable if for every (continuous, differentiable), one has
A standard example of a purely-1-unrectifiable set in two dimensions is the cross-product of the Smith-Volterra-Cantor set times itself.
Read more about this topic: Rectifiable Set
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