Given a compact Lie group $G$ and a $G$-space $X$, useful invariants may obtained by studying the equivariant cohomology of $X$. There are various equivariant cohomology theories that may be defined, and in general return different information about $X$ and its $G$-action. My question is to how a certain pair of these theories may be related. Specifically:
Let $X$ be a $G$ space for $G$ a compact Lie group. Is there a spectral sequence relating the Borel cohomology $H^*_G(X)$ and the Bredon cohomology $\mathcal{H}^*_G(X)$ of $X$?