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Following the work of Burger, Iozzi and Wienhard for representations, in thispaper we introduce the notion of maximal measurable cocycles of a surfacegroup. More precisely, let $\mathbf{G}$ be a semisimple algebraic$\mathbb{R}$-group such that $G=\mathbf{G}(\mathbb{R})^\circ$ is of Hermitiantype. If $\Gamma \leq L$ is a torsion-free lattice of a finite connectedcovering of $\text{PU}(1,1)$, given a standard Borel probability $\Gamma$-space$(\Omega,\mu_\Omega)$, we introduce the notion of Toledo invariant for ameasurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$.  The Toledo remains unchanged along $G$-cohomology classes and its absolutevalue is bounded by the rank of $G$. This allows to define maximal measurablecocycles. We show that the algebraic hull $\mathbf{H}$ of a maximal cocycle$\sigma$ is reductive and the centralizer of $H=\mathbf{H}(\mathbb{R})^\circ$is compact. If additionally $\sigma$ admits a boundary map, then $H$ is of tubetype and $\sigma$ is cohomologous to a cocycle stabilizing a unique maximaltube-type subdomain. This result is analogous to the one obtained forrepresentations.  In the particular case $G=\text{PU}(n,1)$ maximality is sufficient to provethat $\sigma$ is cohomologous to a cocycle preserving a complex geodesic.  We conclude with some remarks about boundary maps of maximal Zariski densecocycles.

Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups