Equivariant maps for measurable cocycles with values into higher rank Lie groups

Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose has dimension $n$ it no factor isometric either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given closed $n$-dimensional manifold $N$, $\Gamma=\pi_1(N)$ its fundamental $Y$ universal cover. Consider representation $\rho:\Gamma \rightarrow G$ with measurable $\rho$-equivariant map $\psi:Y \mathcal{X}_G$. Connell-Farb described way construct $F:Y\rightarrow \mathcal{X}_G$ which is smooth, uniformly bounded Jacobian. In this paper we extend construction context cocycles. More precisely, if $(\Omega,\mu_\Omega)$ standard Borel probability $\Gamma$-space, $\sigma:\Gamma \times \Omega cocycle. We $F: Y $\sigma$-equivariant, whose slices are smooth they have For such equivariant maps define also notion volume prove sort mapping degree theorem in particular context.