Throughout this question when I talk about relations, I am referring to relations on $\mathbb{R}$, i.e. subsets of $\mathbb{R}^2$
One of the first examples given when the topic of composition of relations is discussed, is that of the composition of a circle with itself. This is a circle:
And this is what you get when you compose the circle with itself (if the circle were represented by the relation $C=\{(x,y):x^2+y^2=1\}$, then this is $C\circ C$)
So the circle is not stable under composition. And neither is the unit disc. The unit square however, is. Below is its graph:
Another relation which is stable under composition is the diagonal, i.e. the relation
$$D = \{(x,x)|x\in \mathbb{R}\}$$
And of course the entire $\mathbb{R}^2$. Now we have already found one relation whose graph has non-zero finite measure, and which is stable under composition, namely the unit square. Are all such relations simply the unions of squares with their diagonals lying on the main diagonal? or is there some other non-trivial non-zero measure region which has this property?
The property in question, namely idempotence $R\circ R = R$, does not seem to be too restrictive. Indeed, consider a (nonempty) set $X$, let $A$ be a proper nonempty subset of $X$ and $S$ be any subset of $(X\setminus A)\times A$. Then
$$\mathcal{R}(A,S) = \Delta_A \uplus S$$
is an idempotent relation on $X$; here $\Delta_A = \{(x,x)\in X\times X\,|\, x\in A\}$ is the diagonal relation and $\uplus$ signifies the union being the union of disjoint subsets.
This way, at least any measurable subset $E$ of any rectangle on the plane can be transformed into an idempotent relation by translating to an "off-diagonal" position and adjoining the corresponding piece of the diagonal.