CONTINUITY OF MEASURABLE COCYCLES

Abstract Suppose $G\curvearrowright X$ is a Polish group action, H is a Polish group, and $G\times X\mathop {\overset {\psi }\longrightarrow } H$ is a cocycle that is continuous in the second variable. If $\psi $ is either Baire measurable or is $\lambda \times \mu $ -measurable with respect to a Haar measure $\lambda $ on G and a fully supported $\sigma $ -finite Borel measure $\mu $ on X , then $\psi $ is jointly continuous.