$ \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!