The symmetric difference is a natural way to quantify the distance between measurable sets:
$$d(S,T)=measure([S\setminus T]\cup[T\setminus S])$$
This is a pseudo-metric because there may be different sets with symmetric distance of measure 0. The measurable sets can be partitioned to equivalence classes, such that symmetric difference between every pair in the same class is 0. For example, if we work in $R$, then there is a single class containing the open interval $(0,1)$, the closed interval $[0,1]$, as well as every set which contains one of those intervals plus or minus a countable set of other numbers.
Intuitively, not all members in a class are equal. Some members look much more "natural" and "representative" than the others. For example, the open interval $(0,1)$ looks much more representative of its class than, say, $(0,1)\cup \{397,501\}$ or $(0,1)\setminus \{0.5,0.23\}$. I am looking for a way to formalize this intuition. Specifically, I am looking for a definition (preferrably, a standard definition already used in mathematics) of "representative" elements of equivalence classes under the symmetric distance metric. My main interest lies in the Euclidean plane, but if there is a more general definition, this is even better.
EDIT: If there is no natural way to select a representative to all possible equivalence classes, is it possible to define a set of representatives at least for some of the equivalence classes? For example, consider (in $R^2$) all closed shapes defined by one or more continuous boundary curves. This includes, in particular, all closed discs, polygons, polygons with holes, closed half-planes, closed quarter-planes, etc. Does this set contain only a single representative from each class? (i.e. does the symmetric difference define a metric on this set?)
EDIT 2: What if we take only sets that are closures of open sets? This includes $[0,1]$ but excludes $(0,1)$, $[0,1]\cup\{x\}$, $(0,1)\setminus\{x\}$, etc. Does the symmetric difference define a metric on this set?
As I understand it, there is no "best" element to pick. We can pick any element as representative of the class. Depending on the problem we are working on there may be some which work out easier/nicer, but there is no way to formalise this.
Indeed in a more general setting, it is often required to check that a calculation using an equivalence class is independent of the representative element chosen.