$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(\Omega, \cA)$ and $(E, \cE)$ be measurable space. We endow $\Omega \times E$ with the product $\sigma$-algebra $\cA \otimes \cE$. We endow $\bR$ with its Borel $\sigma$-algebra $\cB (\bR)$. Let $f: \Omega \times E \to \bR$ be measurable. Let $\mu$ be a $\sigma$-finite measure on $(E, \cE)$. We define the set
$$
D := \bigg \{ \omega \in \Omega : \int_E |f(\omega, x)| \diff \mu (x) < \infty \bigg \}.
$$

>Is $D \in \cA$?

Thank you so much for your elaboration!