Let $(\mathbb {X}\times \mathbb{Y}, \mathscr{X} \otimes \mathscr{Y}) $ space measurable product obtained measurable spaces $(\mathbb{X}, \mathscr{X}) $ and $(\mathbb{Y}, \mathscr{Y}) $. Let $ \mathscr{A} $ to $ \sigma $-algebra formed by the cylinders of $ \mathscr{X} \otimes \mathscr{Y} $ based on $ \mathscr{X} $, ie $ \mathscr{A} \triangleq \{X \times \mathbb{Y}: X\in \mathscr{X}\} $ agreed that $ \emptyset\times\mathbb{Y}=\emptyset $.
My question: if the $f:\mathbb{X} \times \mathbb{Y}\rightarrow [0, +\infty]$ is $ \mathscr{A}$-measurable then $f (x, y_1 ) = f (x, y_2) $ for all $ x \in \mathbb{X} $ and for all $ y_1, y_2 \in \mathbb{Y} $?.
Note: This is a reformulation of my question, 'consequences of Fubini-Tonelli theorem' because I think unnecessary steps to answer the question.