Let \(\Gamma \) be a torsion-free lattice of \(PU (p,1)\) with \(p \ge 2\) and let \((X,\mu _X)\) be an ergodic standard Borel probability \(\Gamma \)-space. We prove that any maximal Zariski dense measurable cocycle \(\sigma : \Gamma \times X \longrightarrow SU (m,n)\) is cohomologous to a cocycle associated to a representation of \(PU (p,1)\) into \(SU (m,n)\), with \(1 \le m \le n\). The proof follows the line of Zimmer’ Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, there cannot exist maximal measurable cocycles with the above properties when \(1< m < n\).

令\(\Gamma \)是\(PU (p,1)\)与\(p \ge 2\)的无扭晶格，并令\((X,\mu _X)\)是遍历标准Borel 概率\(\Gamma \) - 空间。我们证明了任何最大Zariski致密可测量闭链\（\西格玛：\伽玛\倍X \ longrightarrow SU（M，N）\）是cohomologous到关联于的表示的闭链\（PU（P，1）\）成\(SU (m,n)\) , 与\(1 \le m \le n\). 证明遵循 Zimmer 的超刚性定理，需要边界图的存在，我们在更一般的环境中证明了这一点。作为我们的结果的结果，当\(1< m < n\)时，不存在具有上述属性的最大可测量共环。