I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct a counterexample refuting a result that had already been published in a peer-reviewed article.

Problem.Find measure spaces $ (X,\mathcal{S},\mu) $ and $ (Y,\mathcal{T},\nu) $, at least one of which isnot$ \sigma $-finite, and an $ (\mathcal{S} \otimes \mathcal{T}) $-measurable function $ f: X \times Y \to \mathbb{R}_{\geq 0} $ with the following properties:

- The function $ f(x,\bullet): Y \to \mathbb{R}_{\geq 0} $ belongs to $ {L^{1}}(Y,\mathcal{T},\nu) $ for every $ x \in X $.
- The function $ \left\{ \begin{matrix} X & \to & \mathbb{R}_{\geq 0} \\ x & \mapsto & \displaystyle \int_{Y} f(x,\bullet) ~ \mathrm{d}{\nu} \end{matrix} \right\} $ is
not$ \mathcal{S} $-measurable.

Does anyone know if this problem can even be solved? Thank you very much for your time!