Generalizing the Haar measure

[sethaxen]

I usually see the Haar measure defined as an measure on a group invariant to actions of the group on itself, but I have sometimes seen a Haar measure on a non-group manifold defined in terms of invariance of actions of some group on that manifold. By this definition, the Hausdorff measure on the Sphere is a bi-invariant Haar measure under the orthogonal actions.

Do we want to adopt this broader definition of a Haar measure? Basically it would mean storing a G-manifold, a group, and an action. We could have a convenient alias for the case where the G-manifold and the group are the same, and the action is the group operation.

[mateuszbaran]

Do you have a reference for the more general case? I only managed to find this: https://mathoverflow.net/questions/40329/how-to-define-the-quotient-of-a-measure-which-is-invariant-under-group-action and I have no idea what would the practical implications be. In particular, how do we define measures on quotient manifolds that don't have a nice embedding in R^n? Is this the right way?

[sethaxen]

Do you have a reference for the more general case?

I've just noticed it for the Stiefel, Sphere, and Grassmann manifolds. e.g. for the Sphere, there's the wikipedia page: https://en.wikipedia.org/wiki/Spherical_measure#Relationship_with_other_measures which defines a Haar measure on the sphere. And Chikuse, Statistics on Special Manifolds 2003 uses the term "invariant measure" to describe the measure on the Stiefel and Grassmann manifolds invariant to left-translation by O(n) and right-translation by O(k), but they also say "invariant measure" is synonymous with "Haar measure".

I have no idea what would the practical implications be. In particular, how do we define measures on quotient manifolds that don't have a nice embedding in R^n? Is this the right way?

To be honest, I don't know. All groups I have worked with do act on R^n, so I can see a potential use here. But e.g. one might want to define a measure on the sphere that is invariant to the action of a planar rotation around a specific axis. I don't think one could reasonably call this a Haar measure; it certainly isn't uniform in any sense, but it does define a symmetry. And I don't yet see how this would be useful.

So for the moment I'm inclined to keep Haar measure referring to specifically invariance to the action of the group on itself.