We know that the Borel group cohomology (group cohomology of measurable functions) of a group $G$, ${\cal H}_B^d(G,Z)$, is given by the cohomology of the classifying space: ${\cal H}_B^d(G,Z)=H^d(BG,Z)$. However, ${\cal H}_B^d(G,R/Z)\neq H^d(BG,R/Z)$, since for example: $H^d(BU(1),R/Z) = R/Z$ for even $d$ and $H^d(BU(1),R/Z) = 0$ for odd $d$; while ${\cal H}_B^d(BU(1),R/Z)=0$ for even $d$ and ${\cal H}_B^d(BU(1),R/Z)=Z$ for odd $d$. Instead, we have ${\cal H}^d(G,R/Z) = H^{d+1}(BG,Z)$.
My question is that do we have any relations between ${\cal H}_B^*(G,M)$ and $H^*(BG,M')$ where $G$ can be continuous, $M$ can be $Z_n$, and $M'$ can be different from $M$? In particular, I would like to know how ${\cal H}_B^*(G,Z_n)$ is related to $H^*(BG,M')$ when $G$ is continuous.
A related question group cohomology and cohomology of classifying space is closed. I hope it can be reopen.
Best Answer
Yes, there is a relation between these cohomology groups. The results you are referring to are contained for instance in Austin-Moore "Continuity properties of measurable group cohomology" http://arxiv.org/abs/1004.4937 and in parts already in Wigner "Algebraic cohomology of topological groups" [http://projecteuclid.org/euclid.bams/1183532101].
To understand the relation in general one can consider topological group cohomology and different models of it. This has the draw-back that it restricts the groups to which the results apply (for instance to finite-dimensional Lie groups), but has the advantage that the different models for it provide many useful relations to other concepts from algebraic topology, where classifying space cohomology is one instance of.
To make it short, the measurable cohomology (which I think you refer to with "Borel group cohomology") of a compact group $G$ with coefficients in a finite-dimensional connected abelian Lie group $A$ is the cohomology of the classifying space with a degree shift: $H^n_{B}(G,A)\cong H^{n+1}(BG,\pi_1(A))$. This explains the phenomenon you describe. More on this can be found in [http://arxiv.org/abs/1110.3304] (see in particular Remark 4.13 for the relation to measurable cohomology).