Any topological group structure on the Cantor set \(\mathcal{C}\simeq2^{\mathbb{N}}\) is the strict projective limit of finite groups, and conversely, the limit of a countable projective system of finite groups is a topological group on the Cantor set \(\mathcal{C}\). The main results of the paper are these two results: (1) any two topological group structures on \(\mathcal{C}\) have the same Haar measure; and (2) the natural map of \(\mathcal{C}\) onto the unit interval sends the Haar measure into the Lebesgue measure. To prove (1) the authors show that any strict projective system of finite groups can be replaced by a system of finite abelian groups so that each finite group is replaced by an abelian group of the same cardinality. Since any two finite groups of the same cardinality have the same Haar measure this implies that the Haar measures on the limits are the same. Each abelian finite group is a product of cyclic groups which allows to define a total order on these groups relative to which the projection maps from larger to smaller groups are monotone. These orders induce a total order on the limit \(\mathcal{C}\). It follows that the natural map from \(\mathcal{C}\) onto the unit interval is monotone and Lawson continuous, if we view \(\mathcal{C}\) as a continuous lattice. Then the proof of (2) is an application of the domain theory.
For the entire collection see [Zbl 1287.68011].