Let $(\Omega,\mathcal{F})$ be a measurable space and $X, Y$ be separable Banach space. Consider a Caratheodory mapping $\varphi:\Omega\times X\to Y$, i.e $\forall x\in X$, $\varphi(\cdot,x)$ is measurable and $\forall \omega\in \Omega$, $\varphi(\omega,\cdot)$ is continuous. Then it is well-known that $\varphi$ is $\mathcal{F}\otimes \mathcal{B}(X)$-measurable.
Now consider the map $g:\Omega\times X\to \Omega\times Y$ defined by $g(\omega,x)=(\omega,\varphi(\omega,x))$. I want to know if $Ran(g)=\{(\omega,\varphi(\omega,x))| \omega\in \Omega,x\in X\}$ is in $\mathcal{F}\otimes\mathcal{B}(Y)$. But I can only prove that $g$ is measurable.