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In previous work with Harman, we introduced a new class of representationsfor an oligomorphic group $G$, depending on an auxiliary piece of data called ameasure. In this paper, we look at this theory when $G$ is the symmetry groupof the Cantor set. We show that $G$ admits exactly two measures $\mu$ and$\nu$. The representation theory of $(G, \mu)$ is the linearization of thecategory of $\mathbf{F}_2$-vector spaces, studied in recent work of the authorand closely connected to work of Kuhn and Kov\'acs. The representation theoryof $(G, \nu)$ is the linearization of the category of vector spaces over theBoolean semi-ring (or, equivalently, the correspondence category), studied byBouc--Th\'evenaz. The latter case yields an important counterexample in thegeneral theory.

On the representation theory of the symmetry group of the Cantor set