Question: Let (mathscr{A}) be a (sigma)-algebra, (mathscr{D}) be a Dynkin system and (mathscr{G} subset mathscr{H} subset mathscr{P}(X)) two collections of subsets of (X). Show that (i)

Let $\mathscr{A}$ be a $\sigma$-algebra, $\mathscr{D}$ be a Dynkin system and $\mathscr{G} \subset \mathscr{H} \subset \mathscr{P}(X)$ two collections of subsets of $X$. Show that

(i) $\delta(\mathscr{A})=\mathscr{A}$ and $\delta(\mathscr{D})=\mathscr{D}$;

(ii) $\delta(\mathscr{G}) \subset \delta(\mathscr{H})$;

(iii) $\delta(\mathscr{G}) \subset \sigma(\mathscr{G})$.

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