Continuity of measurable cocycles

Suppose $G\curvearrowright X$ is a Polish group action, $H$ is a Polish group and $G\times X\oversetψ\longrightarrow H$ is a cocycle that is continuous in the second variable. If $ψ$ is either Baire measurable or is $λ\times μ$-measurable with respect to a Haar measure $λ$ on $G$ and a fully supported $σ$-finite Borel measure $μ$ on $X$, then $ψ$ is jointly continuous.

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