We will be working with subsets of the plane $\Bbb R^2$. Some definitions:
An arc or a Jordan arc is an embedded copy of $[0,1]$ in the plane. The $1$-dimensional Hausdorff measure essentially gives you the "length" of a subset of a plane, and is defined in full generality here. And an ambient isotopy between two subsets $A,B\subseteq\Bbb R^2$ is a map $F:[0,1]\times\Bbb R^2\to\Bbb R^2$ such that $F_0$ is the identity map, each $F_t$ is a homeomorphism, and $F_1(A)=B$.
Given a finite union of arcs in the plane, are they ambiently isotopic to something with finite 1-dimensional Hausdorff measure?
Note that this isn't true if I upgrade it to countably many arcs, even if I force it to be compact, because I'm pretty sure the topologist's sine curve is the countable union of arcs and you can't make it have finite length through ambient isotopy.
If $A_i$, $1\le i\le n$ are my arcs and $K=\bigcup_{i=1}^n A_i$ is their union, I'm sort of imagining $K$ to look like it is filled with a whole lot of "bubbles" - like the Apollonian gasket, but hopefully more tame (the Apollonian gasket has Hausdorff dimension 1.3057). Crucially, the complement of $K$ might have infinitely many (bounded) components, which complicates things.
This feels like it should be true but is way too complicated for me to reason through. So... is it true?