I am studying geometric measure theory, and I am having some trouble understanding how to deal with approximate tangent spaces. I would like an example/exhibition of an approximate tangent space in a regular setting (so that it does coincide with a classic vector subspace).
First of all, the definition. We say that a Radon measure $\mu$ defined on $\Omega\subset \mathbb{R}^n$ has a k-ats (k approximate tangent space) in $x\in \Omega$ if $\rho^{-k} \mu_{x,\rho} \rightharpoonup^* \theta \mathcal{H}^k |_{\pi}$ as $\rho \to 0$, where $\mu_{x,\rho}(A)= \mu(x + \rho A)$, $\mathcal{H}^k$ is the k-th dimensional Hausdorff measure, $\pi$ is a k-dimensional subspace of $\mathbb{R}^n$ and $\theta>0$ is the so called "multiplicity".
So, let's say I have these two examples: $S_1=S^{2} \subset \mathbb{R}^3$ and $S_2=\{y=x^2\} \cup \{y=-x^2\} \subset \mathbb{R}^2$. I want to calculate the approximate tangent space in $(0,1)$ of $S_1$ and in $(0,0)$ of $S_2$. First of all: who is $\mu$ in these cases? The standard Lebesgue measure? Or the curve measure given by the parametrization? I know intuitively that in the first case I should get the tangent plane in $(0,1)$ as a 2-atp and in the second case I should get the x-axis with multiplicity 2 (and if I put $n$ parabolas tangent to the origin, that would be the multiplicity), but how can I formalize this?
P.S. What could be an example of a set with non integer multiplicity at some point?
I think part of the confusion comes from the fact that the approximate tangent space is fundamentally about measures, whereas you are asking about the approximate tangent space of sets.
Now there is a canonical way of associating a measure to a set $S$ of dimension $k$, which is to take the measure $\mathcal{H}^k|_S$. Let $\mu = \mathcal{H}^1|_{S_1}$ be the one-dimensional Hausdorff measure restricted to the unit circle. Suppose we want to compute the approximate tangent space at the point $(0,1)$. The reasonable guess is that the space is $\pi = \textrm{span}\{(1,0)\}$. If this is the case, what we would need to do is show that $$ \rho^{-1} \mu_{x,\rho} \rightharpoonup^* \mathcal{H}^1|_\pi. $$ In other words, given a fixed continuous function $f\in C^0(\mathbb{R}^2)$ one needs to show that $$ \lim_{\rho\to 0} \rho^{-1} \int f(\rho^{-1}x,\rho^{-1} (y-1)) \,d\mu(x,y) = \int_{\mathbb{R}} f(x,0)\,dx. $$ I will sketch how one can prove this. First one reduces to the case that $f$ is a smooth function of compact support, say in the ball of radius $R$. Then the function $f_\rho(x,y) = f(\rho^{-1}x, \rho^{-1}(y-1))$ has support in the ball $B_{\rho R}((0,1))$ of radius $\rho R$ centered at $(0,1)$. Then one can express $\mu$ using the parametrization $$ \int f_\rho(x,y) \,d\mu(x,y) = \int_0^{2\pi} f_\rho(\cos(t),\sin(t))\,dt. $$ One then only needs to consider the case $t$ close to $\pi/2$, for which one has the bound $f_\rho(\cos(\pi/2 + \delta), \sin(\pi/2+\delta)) = f(0, \rho^{-1}\delta) + O(\delta^2)$ using a Taylor expansion and the smoothness of $f$. A change of variables and taking the limit gives the result.
In general to study the measure $\mathcal{H}^k|_S$ for some set $S$ one needs a way to represent integration of the measure in terms of integrals you already know how to compute. This is in general done with the area formula. The idea is that, given a map $\Phi:B_1^k\to \mathbb{R}^d$ from the unit ball $B_1^k\subset\mathbb{R}^k$, one can express integration against the Hausdorff measure on the image of the map $\Phi$ in terms of an integral over $B_1^k$ and the Jacobian of the map $\Phi$.
Now that we have looked at a familiar example, it is worth looking at unfamiliar examples. The simplest is the measure $\mu = f(x)dx$, where $dx$ is the Lebesgue measure on $\mathbb{R}^d$ and $f$ is some continuous function. The approximate tangent space to $\mu$ at $x$ is again going to be $\mathbb{R}^d$ with multiplicity $f(x)$. In fact, we can take $f\in L^1(\mathbb{R}^d)$ and obtain such a statement almost everywhere using the Lebesgue differentiation theorem.
Things can get much stranger because one can add measures of different dimensions. For example, if $S_1$ is a one-dimensional curve, $S_2$ is a two-dimensional surface, and $S_3$ is three-dimensional, it is perfectly reasonable to consider the measure $$ \nu = \mathcal{H}^1|_{S^1} + \mathcal{H}^2|_{S^2} + \mathcal{H}^3|_{S^3}. $$ The approximate tangent space of this measure at a point will depend on which surface it lies on, and if the point is at an intersection of surfaces the lowest-dimensional space will dominate.
One can even then multiply such a measure by a smooth function to get non-integer multiplicities.