Invariant measures and lower Ricci curvature bounds

Given a metric measure space $(X,d,\mathfrak{m})$ that satisfies the Riemannian Curvature Dimension condition, $RC{D}^{\ast }(K,N),$ and a compact subgroup of isometries $G\le Iso(X)$ we prove that there exists a $G-$invariant measure, ${\mathfrak{m}}_G,$ equivalent to $\mathfrak{m}$ such that $(X,d,{\mathfrak{m}}_G)$ is still a $RC{D}^{\ast }(K,N)$ space. We also obtain some applications to Lie group actions on $RC{D}^{\ast }(K,N)$ spaces. We look at homogeneous spaces, symmetric spaces and obtain dimensional gaps for closed subgroups of isometries.