Measurability of an uncountable union

Question

Let $\{X_i\}_{I \in I}$ be a family of real-valued random variables on $(\Omega,\mathcal{A},P)$ for an uncountable index set $I$, which is the sample space of $(I,\mathcal{F},Q)$.

On the product space $(I\times\Omega,\mathcal{F}\otimes\mathcal{A},P\times Q)$ define \begin{equation} Y((i,\omega)):=X_i(\omega), \ (i,\omega)\in I\times\Omega. \end{equation}

Is $Y$ a random variable, i.e. is $Y$ $(\mathcal{F}\otimes\mathcal{A})$-measurable?

Answer 1

Let $I$ be the interval $[0,1]$ with Lebesgue measure.

Let $\Omega$ be a single point.

Let $A$ be a non-measurable subset of $[0,1]$, and let $X_i={\bf 1}(i\in A)$. Then $Y^{-1}(\{1\})=A\times \Omega$, so $Y$ is not measurable.