Approximation of (sigma)-algebras. Let (mathscr{G}) be a Boolean algebra in (X), i.e. a family of sets such

Question:

Approximation of $\sigma$-algebras. Let $\mathscr{G}$ be a Boolean algebra in $X$, i.e. a family of sets such that $X \in \mathscr{G}$ and $\mathscr{G}$ is stable under the formation of finite unions, intersections and complements. Let $\mathscr{A}=\sigma(\mathscr{G})$ and $\mu$ be a finite measure on $(X, \mathscr{A}, \mu)$. Let $A \triangle B:=(A \backslash B) \cup(B \backslash A)$ denote the symmetric difference of $A, B \subset X$.

(i) For every $\epsilon>0$ and $A \in \mathscr{A}$ there is some $G \in \mathscr{G}$ such that $\mu(A \triangle G) \leqslant \epsilon$.

[ $\{A \in \mathscr{A}: \forall \epsilon>0 \exists G \in \mathscr{G}: \mu(A \triangle G) \leqslant \epsilon\}$ is a Dynkin system.]

(ii) Let $\mu, u$ be finite measures on $(X, \mathscr{A})$. For every $\epsilon>0$ and $A \in \mathscr{A}$ there is some $G \in \mathscr{G}$ such that $\mu(A \triangle G) \leqslant \epsilon$ and $u(A \triangle G) \leqslant \epsilon$.

(iii) Assume that $X=\mathbb{R}^{n}, \mathscr{A}=\mathscr{B}\left(\mathbb{R}^{n}ight)$ and $\mu=\lambda^{n}$. A set $A \in \mathscr{A}$ satisfies $\mu(A)=0$ if, and only if, for every $\epsilon>0$ there is a sequence $\left(I_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{J}$, such that $A \subset \bigcup_{n} I_{n}$ and $\mu\left(\bigcup_{n} I_{n}ight) \leqslant \epsilon$.

Remark. Compare the result of this exercise with Theorem 27.25 .

Data from theorem 27.25