If $X: \Omega_1 \times \Omega_2 \to \mathbb{R}$ is measurable then each of the slices $X_{\omega_1} : \Omega_2 \to \mathbb{R}$, $X_{\omega_1} : \Omega_2 \to \mathbb{R}$ defined by $X_{\omega_1}(\omega_2) = X(\omega_1,\omega_2)$ etc. are measurable.
I guess that the converse need not be true. Is it true that if $X_{\omega_1}$ is $\textit{continuous}$ for all $\omega_1$ while $X_{\omega_2}$ is $\textit{measurable}$ for all $\omega_2$ (e.g. for $\Omega_1 = \Omega_2 = \mathbb{R}^d$) then $X$ is measurable?
More generally, if $X : \Omega_1 \times \Omega_2 \cdots \Omega_n \to \mathbb{R}$ and $X_{\omega_1},X_{\omega_2},\cdots,X_{\omega_{n-1}}$ are continuous and $X_{\omega_n}$ is measurable can we say that $X$ is measurable?
Thanks, Phanindra