# Theorem 0.7. Walters' book of Ergodic Theory

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.?

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.
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 $$\mathcal{B}(\mathcal{A})$$.