# Approximating Disjoint Measurable Sets by Elementary Sets

Say $$X$$ is a set with endowed with an algebra $$\mathcal{R} \subseteq 2^X$$ and a $$\sigma$$-additive measure $$\mu : \mathcal{R} \to \mathbb{R}$$. We can apply the Lebesgue extension process to define an outer measure $$\mu^* : 2^X \to \mathbb{R}$$ and then define Lebesgue-measurable sets. It's well known that a set $$A \subseteq X$$ is measurable if and only if for all $$\epsilon > 0$$ there exists some $$B \in \mathcal{R}$$ such that $$\mu^*(A \triangle B) < \epsilon$$.

Say we have a finite collection of pairwise disjoint measurable sets $$A_1,\ldots,A_n \subseteq X$$. For any $$\epsilon$$, I'd like to be able to find the approximations $$B_i \in \mathcal{R}$$ so that they're pairwise disjoint as well. Is this possible?

Yes.

Take any $$B_i \in \mathcal{R}$$ such that $$\mu^*(A_i \triangle B_i) \leqslant \varepsilon$$ and put $$B_i' = B_i \setminus \bigcup \limits_{j. Then $$B_i' \in \mathcal{R}$$ are the desired approximations: they are obviously disjoint and

$$\mu^*(B_i \triangle B_i') = \mu^* \left( B_i \cap \bigcup_{j

But

$$\mu^*(B_i \cap B_j) \leqslant \mu(B_i \triangle A_i) + \mu^*(A_i \cap A_j) + \mu^*(A_j \triangle B_j) \leqslant 2 \varepsilon$$

hence

$$\mu^*(B_i \triangle B_i') \leqslant (i-1) \cdot 2 \varepsilon \leqslant 2(n-1) \varepsilon.$$

So finally

$$\mu^*(A_i \triangle B_i') \leqslant \mu^*(A_i \triangle B_i) + \mu^*(B_i \triangle B_i') \leqslant \varepsilon + 2(n-1)\varepsilon = (2n-1)\varepsilon.$$

It suffices to replace $$\varepsilon$$ with $$\frac{\varepsilon}{2n-1}$$.

• Basically - just make them disjoint. Sep 26, 2021 at 17:19