Suppose $\mathscr S$ is a semiring on $X$, and $\mathscr F=\sigma(\mathscr S)$ is the $\sigma$-field generated by $\mathscr S$. If $A\in\mathscr S$, denote by $\mathscr S_A=\{A\cap B:B\in\mathscr S\}$ and $\mathscr F_A=\{A\cap B:B\in\mathscr F\}$. My problem is : Is $\mathscr F_A$ the $\sigma$-field generated by $\mathscr S_A$, considering both of them as families of sets in $A$ and how to prove it?
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To show $\sigma(\mathscr{S}_A) \subseteq \mathscr{F}_A$, it suffices to show that $\mathscr{F}_A$ is a $\sigma$-field. This should not be hard.
To show the reverse inclusion, consider the set $\mathscr{C} = \{B \in \mathscr{F} : A \cap B \in \sigma(\mathscr{S}_A)\}$. Trivially, $\mathscr{S} \subset \mathscr{C}$. Show that $\mathscr{C}$ is a $\sigma$-field and conclude that it contains $\mathscr{F}$.
Nate Eldredge
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Thanks a lot for your answer. But I am still a little confused that why $\mathscr C$ is a $\sigma$-field containing $\mathscr F$ implies $\mathscr F_A\subset\sigma(\mathscr S_A)$? – Attendre Apr 21 '20 at 15:51
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@Attendre: Every set in $\mathscr{F}_A$ is of the form $A \cap B$ for $B \in \mathscr{F}$. Since $\mathscr{F} \subseteq \mathscr{C}$, this means that $B \in \mathscr{C}$ and so $A \cap B \in \sigma(\mathscr{S}_A)$. – Nate Eldredge Apr 21 '20 at 16:24
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