Twisted Haar measures on locally compact G-spaces? Suppose $G$ is a locally compact group, $X$ is a locally compact space equipped with a nice right action $r_g$ of $G$ (I suspect every orbit being compact qualifies as "nice" here, but this probably isn't the greatest generality possible), and $(V,\rho)$ is a finite-dimensional left representation of $G$. Is there any characterization, in terms of some kind of data on the orbit space, of locally-defined $V$-valued measures on $X$ (in the sense of a system of measures on every compact subset of $X$ that behave well under restriction) which respect $G$'s action in the sense that $\left(\left(r_g\right)_*\mu\right)(E)=\rho_g\mu(E)$ for every precompact Borel set $E$? What if we let $V$ be a Banach space or similar? In the case where $X=G$ and $V$ is trivial, these are just Haar measures, but I've been unable to find any discussion of this sort of problem in greater generality.
(Here, all spaces are Hausdorff and all measures are Radon.)
 A: The answer seems a bit silly, which explains why I couldn't find any discussion. Some of what I say may not be strictly correct, but I think it is qualitatively correct. Suppose every orbit is closed in $X$, and suppose $X/G$ is also LCH.
One can construct invariant measures as follows. Take a measure $\mu$ on $X/G$, and take measures $\mu_O$ on every orbit $O$ of $X$ such that for every $f\in C_c(X)$, the function $O\mapsto\int_Of\,d\mu_O$ is in $L^1(\mu)$. Then the Radon measure $\nu$ defined by $\int f\,d\nu=\int_{X/G}\int_Of\,d\mu_O\,d\mu$ is invariant if and only if $\mu$-a.e. $\mu_O$ is invariant.
I think the converse, that every positive invariant measure arises this way, can be argued (probably assuming second-countability) by disintegration and extension theorems, though I'm not up to working through the details at this particular moment. This reduces the untwisted case to studying the homogeneous case, which can be classified in terms of modular functions; see Loomis' book, pages 132-133, or Bourbaki's section on homogeneous spaces.
The twisted case, I think, is a straightforward adaption of these ideas with a little more finagling involving stabilizers; I'll think about this in the morning. You may need to assume the representation is unitary.
If $X/G$ is not Hausdorff, we might have no good invariant measure. For example, if $K$ is a local field, the multiplicative action of $K^*$ on $K$ has a two-point non-Hausdorff quotient space. A standard fact of number theory in this case is if the representation $\rho$ is sufficiently bad, the twisted-invariant measure is not finite near zero, so you cannot integrate normally. One can still define a kind of twisted-invariant integral (this is the local theory in Tate's thesis), but in most modern treatments it lives in a much sketchier space of distributions.
