Let $(\mathbb{N}, \mathcal{P}\mathbb{N}, \mu)$ be the measure space on the natural numbers with $\mu$ as the counting measure. Let $(Y, \mathcal{Y}, \nu)$ be an arbitrary measure space.
I want to show that $f: \mathbb{N} \times Y \to \mathbb{R}$ is measurable iff each section $f_n$ is $\mathcal{Y}$-measurable. In this context, a section $f_n: Y \to \mathbb{R}$ is defined via $f_n(y) = f(n,y), \quad y \in Y$ where $f: \mathbb{N} \times Y \to \mathbb{R}$.
Thoughts:
Given that $f$ is a measurable function, there's a theorem that allows me to conclude that every section of $f$ is measurable so the forward direction is done. For the backward direction, assuming each $f_n$ is $\mathcal{Y}$-measurable I have that $f_n^{-1}(\alpha, \infty] \in \mathcal{Y}$ for each $n \in \mathbb{N}$ and want to show this yields for $f^{-1}(\alpha, \infty] \in \mathcal{P}\mathbb{N} \times \mathcal{Y}.$
I know that I can write $f^{-1}(\alpha, \infty] = \{(n,y): f(n,y) > \alpha\}.$
Any ideas on how to prove this direction? General tips/strategies would be much appreciated.