Let $\gamma:[0,1]\to \mathbb{R}^2$ be a rectifiable curve and $\Gamma=\gamma[0,1]$ be its image.
Is it possible to cover $\Gamma$ by a countable collection of sets $N,R\_1,R\_2,\dots$ such that $N$ has vanishing 1-dimensional Hausdorff measure and each $R\_i$ admits a bi-Lipschitz embedding into $\mathbb R$?


If yes, is there an analogue of this statement for $k$-rectifiable sets in $\mathbb{R}^n$?



Decomposition of rectifiable curves in $\mathbb R^2$
Regardless of its dimension, a $k$-rectifiable set $M \subset \mathbf{R}^n$ can be covered by a collection $(N\_j \mid j \geq 0 )$ of sets, with $\mathcal{H}^k(N\_0) = 0$ and where the $N\_j$, $j \geq 1$ are $k$-dimensional embedded $C^1$ submanifolds of $\mathbf{R}^n$. Here by 'covered' I mean that $M \subset \cup\_{j \geq 0} N\_j$.


This is Lemma 11.1 in Leon Simon's lecture notes on geometric measure theory.


