# Measurable functions for certain sub-sigma-fields are constant.

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.

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Yes. For fixed $a, b$, the set $\{(a,y) : f(a,y) = f(a,b)\}$ belongs to $\sigma$-algebra $\mathscr{A}$ and contains $(a,b)$ and therefore contains all $(a,y)$.