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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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up vote 3 down vote accepted

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 $(a,y)$.

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Thank you, GEdgar. –  Elias Dec 22 '11 at 1:09
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