Definition. If there exists a sequence of Borel measurable sets $X_1, ..., X_n\subseteq \mathbb{R}^2$ with positive Lebesgue measure and maps $f_i:X_i\to X_{i+1}$ such that each $f_i$ is either a homeomorphism or a bi-measurable measure preserving bijection then say that $X_1$ and $X_n$ are equivalent, denoting it by $X_1\sim X_n$.
Question: Can we classify compact connected subspaces with positive Lebesgue measure of $\mathbb{R}^2$ under this notion of equivalence?
Source: Me. Since classification of such spaces topologically seems impossible in general, I introduced measure.