# Measures invariant by a subgroup action

Given a locally compact group $G$ it's a well known fact the "uniqueness" of the left haar measure, i.e. a measure that satisfies $\mu(A)=\mu(xA),\forall x\in G$. }

I want to know what it's known about the space of measures that satisfies $\mu(A)=\mu(xA)$, where $x\in H$ and $H$ a closed subgroup of $G$. Any reference will be very helpful.

Thanks!

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What properties do you want the measures to satisfy? I imagine you want them to be Borel, but do you also want them to be locally finite or something? A large class of such measures can be obtained by simply translating the Haar measure of $H$ to each coset in $H\backslash G$, possibly multiplying by different constants, but that can quickly turn ugly if e.g. $H$ is finite and $G$ is perfect, and certainly isn't comprehensive. You might want to look for a generalization, where you have a group acting on a set and ask about measures preserved by the action... I don't know where to look, though. – tomasz Aug 9 '12 at 0:51
Yes, I'm looking for locally finite measures, the problem I was firstly concerned about is related the measures in $\R^2$ that are invariant by translations in one direction. The idea I had is the same you wrote, use the lebesgue measure on each leave of the foliation by lines in the direction you chose or a multiple of the lebesgue measure, finally you write your measure similar to $\int f(t) L(A\cap Rt) dh(t) = \mu(A)$ where Rt represent the lines in the direction of the translation and f(t) the constant by which this differs to lebesgue in R (L is the lebesgue measure and h any measure) – apvelozo Aug 9 '12 at 2:43