Let $(X,d)$ be an $\mathcal{H}^n$-rectifiable metric space, i.e. there exits a collection of Lipschitz maps from measurable subsets of $\mathbb{R}^n$ to $X$ such that $ \mathcal{H}^n(X \backslash \cup_i f_i(A_i)) = 0 $.

Is it true that for any subset $A \subset X$, $$ \mathcal{H^n}(A) = \mathcal{H}^n_\infty (A) \ .$$

The claim is true on $X = \mathbb{R}^n$. Note that the same equality fails horribly if we consider $\mathcal{H}^k$ for $k<n$ -- think of an infinitely long curve inside a bounded set.

If this helps: I am interested in small scales, so, you might consider asymptotic behavior as $\text{diam} (A) \to 0$.

Thanks for your consideration.