# Approximating a σ\sigma-algebra by a generating algebra

Theorem. Let $$(X,B,μ)(X,\mathcal B,\mu)$$ a finite measure space, where $$μ\mu$$ is a positive measure. Let $$A⊂B\mathcal A\subset \mathcal B$$ an algebra generating $$B\cal B$$.

Then for all $$B∈BB\in\cal B$$ and $$ε>0\varepsilon>0$$, we can find $$A∈AA\in\cal A$$ such that $$μ(AΔB)=μ(A∪B)−μ(A∩B)<ε.\mu(A\Delta B)=\mu(A\cup B)-\mu(A\cap B)<\varepsilon.$$

I don't think there is a proof in this site.

It's a useful result for several reasons:

• We know what the algebra generated by a collection of sets is, but not what the generated $$σ\sigma$$-algebra is.
• The map $$ρ:B×B→R+\rho\colon \cal B\times\cal B\to \Bbb R_+$$, $$ρ(A,A′)=μ(AΔA′)\rho(A,A')=\mu(A\Delta A')$$ gives a pseudo-metric on $$B\cal B$$. This makes a link between generating for an algebra and dense for the pseudo-metric.
• We say that a $$σ\sigma$$-algebra is separable if it's generated by a countable class of sets. In this case, the algebra generated by this class is countable. An with the mentioned result, we can show that $$Lp(μ)L^p(\mu)$$ is separable for $$1≤p<∞1\leq p<\infty$$, which makes a link between the two notions.
• In ergodic theory, we have to test mixing conditions only an a generating algebra, not on all the $$σ\sigma$$-algebra.

Proof: Let
We have to prove that $\cal S$ is a $\sigma$-algebra, as it contains by definition $\cal A$.

• $X\in\cal S$ since $X\in\cal A$.
• If $A\in\cal S$ and $\varepsilon>0$, let $A'\in\cal A$ such that $\mu(A\Delta A')\leq \varepsilon$. Then $\mu(A^c\Delta A'^c)=\mu(A\Delta A')\leq \varepsilon$ and $A'^c\in\cal A$.
• First, we show that $\cal A$ is stable by finite unions. By induction, it is enough to do it for two elements. Let $A_1,A_2\in\cal S$ and $\varepsilon>0$. We can find $A'_1,A'_2\in\cal A$ such that $\mu(A_j\Delta A'_j)\leq \varepsilon/2$. As

and $A'_1\cup A'_2\in\cal A$, $A_1\cup A_2\in \cal A$.

Now, let $\{A_k\}\subset\cal S$ pairwise disjoint and $\varepsilon>0$. For each $k$, let $A'_k\in\cal A$ such that $\mu(A_k\Delta A'_k)\leq \varepsilon 2^{-k}$.
Let $N$ be such that $\mu\left(\bigcup_{j\geq N+1}A_j\right)\leq \varepsilon/2$ (such a choice is possible since $\bigcup_{j\geqslant 1}A_j$ has a finite measure and $\mu\left(\bigcup_{j\geq n+1}A_j\right)\leq \sum_{j\geq n+1}\mu\left(A_j\right)$ and this can be made arbitrarily small). Let $A':=\bigcup_{j=1}^NA'_j\in\cal A$. As