# Measure on group invariant under group action on metric space

This is a question very similar to I recently asked on mathexchange, but different enough to get its own entry in MO.

The setting is still the same. I consider the metric space $$\mathbb{R}$$ and the Lebesgue-measure $$\lambda$$ on $$\mathbb{R}$$ as well as the transitive group action $$+:(\mathbb{R},+)\times\mathbb{R}\to\mathbb{R}$$, for which the measure and the metric are both invariant. When integrating over a set $$M$$ with respect to $$\lambda$$ I can fix a point $$x\in\mathbb{R}$$ (the metric space rather than the group) and, since the action is transitive, integrate over a set $$M'$$ with respect to some measure $$\mu$$ on the additive group $$(\mathbb{R},+)$$, such that $$M'+x=M$$ and $$μ(M')=λ(M)$$. This is possible as well as completely independent of the choice of $$x$$ by choosing $$μ$$ to be the Lebesgue-measure on $$(\mathbb{R},+)$$.

Now I want to generalise this statement to arbitrary metric spaces and group actions. For this I need a radon-measure $$\mu$$ on $$X$$ and a locally compact group $$G$$ with group action $$G\times X\to X$$ such that the measure and metric are both invariant. Furthermore I require the action to be transitive and continuous.

In order to have the same properties like the example above on the real number line I need a measure $$\nu$$ on $$G$$ such that $$\nu(U)=\mu(Ux)$$ for every measurable $$U\subseteq G$$ and this should be independent of $$x\in X$$. When trying to simply construct such a measure I ran into problems because I couldn't guarantee that $$Ux$$ would be measurable for all $$U$$, only for compact $$U$$ (since the action is continuous and the image of $$G\to X,g\mapsto gx$$ would therefore be compact), but the compact subsets of $$G$$ do not form a $$\sigma$$-algebra, as far as I know. Also, I couldn't prove that $$Ux$$ and $$Uy$$ would have the same measure, which would make defining such a measure on $$G$$ impossible.

I now wonder whether the haar-measure on $$G$$ could actually fulfill this role or at least what kind of conditions I'd need so that the haar measure could work. I would much rather not require the action to be free since this would be too strong. Any ideas are much appreciated!

You are asking about the relationship between the Haar measures on the group $$G$$ and its homogeneous space $$X=G/H$$ (which, by the way, has nothing to do with metrics on either space). It is in great detail discussed in Nachbin's book The Haar integral.