Finite reducibility of maximal infinite dimensional measurable cocycles of complex hyperbolic lattices

Given $Gamma < text{PU}(n,1)$ a torsion-free lattice and $(X,mu_X)$ a standard Borel $Gamma$-space, we introduce the notion of Toledo invariant of a measurable cocycle $sigma:Gamma times X rightarrow text{PU}(p,infty)$. Since that invariant has bounded absolute value, it makes sense to speak about maximality. We prove that any maximal measurable cocycle is finitely reducible, that is it admits a cohomologous cocycle with image contained in a copy of $text{PU}(p,np)$ inside $text{PU}(p,infty)$, which is a finite algebraic subgroup. Even if we do not provide a real rigidity result in this setting, our statement can be seen as the natural adaption of the results for representations due to Duchesne, Lecureux and Pozzetti. We conclude the paper by completing the analysis of maximal cocycles of complex hyperbolic lattices started in cite{sarti:savini} with a characterization of their algebraic hull.