A probabilistic measure for the number of commuting subgroups in locally compact groups

We construct a probability measure which counts the pairs of closed commuting subgroups in locally compact groups, extending the ``subgroup commutativity degree'' of finite groups to pro-Lie groups. We use a topological approach via measure theory and the structure of the Chabauty spaces. An example of Andersen and Jessen shows some natural limitations in our approach, but for compact groups we present an alternative for the construction of the subgroup commutativity degree, using properties of the Chabauty spaces only.