Problems about the "sub"$\sigma-$field generated by a set in a semiring.

Question

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?

Answer 1

• 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 $\mathscr{F}$.