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Let $(X, \mathcal X , \mu)$ and $(Y, \mathcal Y, \nu)$ are two meaure spaces and let $\mathcal Z$ is the sigma algebra that generated by $P=AxB$ sets where $A \in \mathcal X$ and $B \in \mathcal Y$. $\mathcal {Z_0}$ is family of finite unions of $P$ sets.

I’ve shown that $\mathcal {Z_0}$ is an algebra. I’d like to learn that is $\mathcal Z$ sigma algebra generated by $\mathcal {Z_0}$ algebra? How can I show it? I couldn’t understanf these concepts well. Thanks for any help.

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Let $\mathcal A$ be the sigma algebra generated by $\mathcal Z_0$. Let us show that $\mathcal A$ is contained in $\mathcal Z$ and that $\mathcal A$ is contains in $\mathcal Z$. Since $\mathcal Z$ is a sigma algebra and it contains sets of the form $A \times B$ it also contains finite unions of these. Hence it contains $\mathcal A$ (because the latter is the smallest sigma algebra containing the sets $A \times B$).

On the other hand $\mathcal A$ is a sigma algebra which contains all finite unions of the sets $A\times B$ and hence it also contains sets of the type $A\times B$. Since $\mathcal Z$ is the smallest sigma algebra containing these sets we get $\mathcal Z \subseteq\mathcal A$

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