Topological equivalence of Haar measures on compact groups (Q5930788)

The existence of homeomorphisms establishing an isometry of normalized Haar measures on metrizable compact groups is studied. In the case of \(0\)-dimensional groups, a complete answer is given in terms of the indices of open normal subgroups. The main lemma is: the Haar measures in \(0\)-dimensional groups \(X\) and \(Y\) are homeomorphic if and only if the sets \(N(X)\), \(N(Y)\) are alternate (for a compact group \(X\), \(N(X)\) is the set of all indices of open subgroups of \(X\)).