Tonelli-Fubini Theorem.
Let $(\mathbb{X},\mathscr{X},\mu)$ and $(\mathbb{Y},\mathscr{Y},\nu)$ be probability spaces and let $\mathscr{Z}$ be the $\sigma$-field product i.e. the $\sigma$-field generated by $\{A\times B : A\in\mathscr{X}, B\in\mathscr{Y}\}$. Let $f:\mathbb{X}\times\mathbb{Y}\rightarrow [0,+\infty]$ be $\mathscr{Z}$-mensurable, and let $\eta=\mu\times\nu$ be the product probability on $\mathscr{Z}$. Then it is true that
a) $f(x,\cdot)$ is $\mathscr{Y}$-mensurable $\forall x\in\mathbb{X}$, where $f(x,\cdot):\mathbb{Y}\rightarrow[0,+\infty]$ is defined by $y\mapsto f(x,y)$. Analogously, $f(\cdot,y)$ is $\mathscr{X}$-mensurable $\forall y\in\mathbb{Y}$, where $f(\cdot,y):\mathbb{X}\rightarrow[0,+\infty]$ is defined by $x\mapsto f(x,y)$.
b) $\phi$ is $\mathscr{Y}$-mensurable where $\phi:\mathbb{Y}\rightarrow[0,+\infty]$ is defined by $y\mapsto \int_{\mathbb{X}}f(\cdot,y) d\mu$. Analogously, $\psi$ is $\mathscr{X}$-mensurable, where $\psi:\mathbb{X}\rightarrow[0,+\infty]$ is defined by $x\mapsto \int_{\mathbb{Y}} f(x,\cdot) d\nu$.
c)$ \int_{\mathbb{X}\times\mathbb{Y}}f d(\nu\times\mu)=\int_{\mathbb{X}}\psi d\mu=\int_{\mathbb{Y}}\phi d\nu$
My question. If $\mathscr{B}\subset\mathscr{Z}$ is the sub-$\sigma$-field generated by $\{A\times\mathbb{Y} : A\in\mathscr{X}\}$ and $f:\mathbb{X}\times\mathbb{Y}\rightarrow [0,+\infty]$ is $\mathscr{B}$-measurable, then:
1) $f(\cdot,y_1)=f(\cdot,y_2)$ for all $y_1,y_2\in\mathbb{Y}$?
2) $\phi(y_1)=\phi(y_2)$ for all $y_1,y_2\in\mathbb{Y}$?