Let $(\Omega, \mathcal{A})$ be a measurable space and consider the space $(\Omega \times \mathbb{R}, \mathcal{A} ⊗ \mathcal{B}$).
Let $f : \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying:
- for every $t \in \mathbb{R}, \Omega \ni \omega \rightarrow f(\omega, t)$ is measurable (with respect to $\mathcal{A}$);
- for every $\omega \in \Omega, \mathbb{R} \ni t \rightarrow f(\omega, t)$ is continuous.
Prove that $f$ is measurable (with respect to $\mathcal{A} ⊗ \mathcal{B}$).
If we can define a function $f_n(\omega, n(t))$ which converges pointwise to $f(\omega, t)$ and is measurable with respect to $\mathcal{A} ⊗ \mathcal{B}$, then I can say that $f(\omega, t)$ also is measurable with respect to $\mathcal{A} ⊗ \mathcal{B}$.
I know that this $n(t)$ must consist of Borel subsets of $\mathbb{R}$, but I can't quite think of how to define such a function.