Generalization of approximate tangent spaces to subsets of arbitrary manifolds?

Question

I'm anything but an expert in Geometric Measure Theory, so please forgive me if I'm asking a trivial question.

Let $(M^n, g)$ be a smooth Riemannian manifold, $d \in \mathbb{N}$ and $A ⊂ M$ be a subset which is measurable w.r.t. the Hausdorff-$d$ measure $𝓗^d$ coming from the metric $g$. Embed $(M, g)$ isometrically into some $ℝ^N$. For $𝓗^d$-measurable subsets of $ℝ^N$ (like $A$) there exists the notion of approximate tangent spaces.

So suppose $A ⊂ M ⊂ ℝ^N$ has an approximate tangent space $T_x A$ at $x ∈ A$. Are the following statements true?

1. $T_x A$ is a subspace of $T_x M$.

2. $T_x A$ is independent of the isometric embedding.

3. (Assuming 2 is true), $T_x A$ is independent of the choice of $g$. (This requires that $𝓗^d$-measurability of $A$ is independent of the choice of $g$ which should be the case, unless I'm mistaken.)

Put differently, the question I'm asking is: In how far is the approximate tangent space of $A ⊂ M$ at $x$ an intrinsic concept, depending only on the structure of $A$ as a subset of $M$, and $M$ as a differentiable manifold? The main reason I'm asking is that I frequently need to switch between metrics (and thus isometric embeddings) and don't want to worry about approximate tangent spaces changing when I do that.

Answer 1

We say that $x\in A$ is a density point if $$ \lim_{r\to 0} \frac{\mathcal{H}^d(A\cap B(x,r))}{\omega_d r^d}=1. $$ It is well known that $\mathcal{H}^d$-almost all points of $A\subset M$ are density points. At the density points we simply have that $T_xA=T_xM$. The density point is independent of a choice of a metric or an isometric embedding.

For an overview of basic results in geometric measure theory I would recommend

F. Morgan, Geometric Measure Theory: A Beginner's Guide.