9. Approximation theorems

Theorem 1

$\mathcal{A}\subset \Im (\Omega )$ is an algebra. $\mathcal{F}=\mathcal{F}(\mathcal{A})$ is a $\sigma$-algebra generated by $\mathcal{A}$. There is a measure $\mu :\mathcal{F}⟶\overline{{\mathbb{R}}_{+}}$, if $A\in \mathcal{F}$ and $\mu (A)<+\infty$, then for any $\epsilon >0$, $\exists$ $E\in \mathcal{A}$ such that $$\mu (E\backslash A)+\mu (A\backslash E)<\epsilon$$