In geometric measure theory an approximate tangent space is a measure theoretic generalization of the concept of a tangent space for a differentiable manifold.
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Definition
In differential geometry the defining characteristic of a tangent space is that it approximates the smooth manifold to first order near the point of tangency. Equivalently, if we zoom in more and more at the point of tangency the manifold appears to become more and more straight, asymptotically tending to approach the tangent space. This turns out to be the correct point of view in geometric measure theory.
Definition for sets
Definition. Let
in the sense of Radon measures. Here for any measure
Certainly any classical tangent space to a smooth submanifold is an approximate tangent space, but the converse is not necessarily true.
Multiplicities
The parabola
is a smooth 1-dimensional submanifold. Its tangent space at the origin
then
Definition for measures
One can generalize the previous definition and proceed to define approximate tangent spaces for certain Radon measures, allowing for multiplicities as explained in the section above.
Definition. Let
in the sense of Radon measures. The right-hand side is a constant multiple of m-dimensional Hausdorff measure restricted to
This definition generalizes the one for sets as one can see by taking
Relation to rectifiable sets
The notion of approximate tangent spaces is very closely related to that of rectifiable sets. Loosely speaking, rectifiable sets are precisely those for which approximate tangent spaces exist almost everywhere. The following lemma encapsulates this relationship:
Lemma. Let
has approximate tangent spaces