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 M \subset \mathbb{R}^n be a set that is
measurable with respect to
m-dimensional
Hausdorff measure \mathcal{H}^m, and such that the restriction measure \mathcal{H}^m \llcorner M is a
Radon measure. We say that an
m-dimensional subspace P \subset \mathbb{R}^n is the
approximate tangent space to M at a certain point x, denoted T_x M = P, if : \left( \mathcal{H}^m \llcorner M \right)_{x,\lambda} \rightharpoonup \mathcal{H}^m \llcorner P as \lambda \downarrow 0 in the sense of
Radon measures. Here for any measure \mu we denote by \mu_{x,\lambda} the rescaled and translated measure: :\mu_{x,\lambda}(A) := \lambda^{-n} \mu(x + \lambda A), \qquad A \subset \mathbb{R}^n Certainly any classical tangent space to a smooth submanifold is an approximate tangent space, but the converse is not necessarily true.
Multiplicities The parabola :M_1 := \{ (x, x^2) : x \in \mathbb{R} \} \subset \mathbb{R}^2 is a smooth 1-dimensional submanifold. Its tangent space at the origin (0,0) \in M_1 is the horizontal line T_{(0,0)} M_1 = \mathbb{R} \times \{0\}. On the other hand, if we incorporate the reflection along the
x-axis: :M_2 := \{ (x, x^2) : x \in \mathbb{R} \} \cup \{ (x, -x^2) : x \in \mathbb{R} \} \subset \mathbb{R}^2 then M_2 is no longer a smooth 1-dimensional submanifold, and there is no classical tangent space at the origin. On the other hand, by zooming in at the origin the set M_2 is approximately equal to two straight lines that overlap in the limit. It would be reasonable to say it has an approximate tangent space \mathbb{R} \times \{0\} with multiplicity two.
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 \mu be a Radon measure on \mathbb{R}^n. We say that an
m-dimensional subspace P \subset \mathbb{R}^n is the approximate tangent space to \mu at a point x with multiplicity \theta(x) \in (0,\infty), denoted T_x \mu = P with multiplicity \theta(x), if :\mu_{x,\lambda} \rightharpoonup \theta(x) \; \mathcal{H}^m \llcorner P as \lambda \downarrow 0 in the sense of Radon measures. The right-hand side is a constant multiple of
m-dimensional
Hausdorff measure restricted to P. This definition generalizes the one for sets as one can see by taking \mu := \mathcal{H}^n \llcorner M for any M as in that section. It also accounts for the reflected paraboloid example above because for \mu := \mathcal{H}^1 \llcorner M_2 we have T_{(0,0)} \mu = \mathbb{R} \times \{0\} with multiplicity two. == Relation to rectifiable sets ==