Theorem 0.7. Walters' book of Ergodic Theory

Question

The following is from Walters' book of Ergodic Theory without a proof :

Theorem 0.7. Let $(X, \mathcal{B}, m)$ be a probability space and let $\mathcal{A}$ be an algebra of subsets of $X$ with $\mathcal{B}(\mathcal{A})=\mathcal{B}$. Then for each $ \epsilon > 0$ and each $B \in \mathcal{B}$ there is some $A \in \mathcal{A}$ with $m(A \Delta B) < \epsilon$.

What does it mean $\mathcal{B}(\mathcal{A})=\mathcal{B}$?

$\mathcal{A}$ is an algebra not a $\sigma$-algebra with some condition on it, can I mimic the proof of existence of a Borel set to approximate a Lebesgue measurable set for this situation too?

Can someone please guide me through an understandable proof of Theorem 0.7.?

Answer 1

Hint: The proof is rather lengthy, but the idea is simple. Consider the collection of subsets: $$ {\cal C} = \{ S \in {\cal B} \ | \ \forall \epsilon>0, \exists A\in {\cal A} : \mu(S\Delta A)<\epsilon\}.$$ This collection clearly contains ${\cal A}$. Now (the lengthy part) show that ${\cal C}$ is a $\sigma$-algebra, whence contains ${\cal B}$ and you are done.

Answer 2

[Note: this answer was written before a proof of the theorem was requested.]

This is not notation that I've seen before, but $\mathcal{B}(\mathcal{A})$ refers to the $\sigma$-algebra generated by $\mathcal{A}$; i.e., $$\mathcal{B}(\mathcal{A}) = \bigcap_{\mathcal{F} \supset \mathcal{A}, \mathcal{F}\text{ is a }\sigma\text{-algebra}}\mathcal{F}\text{.}$$ In most probability texts, I've usually seen $\sigma(\mathcal{A})$ as opposed to $\mathcal{B}(\mathcal{A})$.