  Given a metric measure space $(X,d,\mathfrak{m})$ that satisfies the
Riemannian Curvature Dimension condition, $RCD^*(K,N),$ and a compact subgroup
of isometries $G \leq Iso(X)$ 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 $RCD^*(K,N)$ space. We also obtain some
applications to Lie group actions on $RCD^*(K,N)$ spaces. We look at
homogeneous spaces, symmetric spaces and obtain dimensional gaps for closed
subgroups of isometries.

Metric Geometry;Differential Geometry