I'd like to know if there exists a connected Lie group $G$ and a closed manifold $M$ such that there is a locally-free smooth action $G\times M\to M$ (i.e. the stabilizer of any point of $M$ is a discrete subgroup of $G$) with no invariant (Borel) probability measure.



locally-free Lie group action not preserving any measure 
Let $\Gamma$ be a co-compact lattice in $G=SL\_2(\mathbb{R})$, acting linearly on the real projective line $P^1(\mathbb{R})$. Take $M=G\times\_\Gamma P^1(\mathbb{R})$, a 4-dimensional closed manifold which is a circle bundle over $G/\Gamma$ (notation $\times\_\Gamma$ means we divide out by the diagonal action of $\Gamma$). The left $G$-action is locally free, as stabilizers are conjugate into $\Gamma$. Moreover, since the groupoid $G\ltimes M$ is equivalent to the groupoid $\Gamma\ltimes P^1(\mathbb{R})$ (the former being induced up from the latter), it is enough to check that there is no $\Gamma$-invariant Borel measure on $P^1(\mathbb{R})$, which is classical (see e.g. Cor. 3.2.2 in Zimmer's `Èrgodic theory and semisimple groups'', Birkhauser, 1984).


