Under what conditions can a collection of sets be turned into a pairwise disjoint collection of sets?

Question

I am looking at the construction of the Lebesgue measure, and the following doubt came up. When defining the outer Lebesgue measure, we let $$m\left(\bigcup_{k \in \mathcal{K}}I_k\right) = \sum_{k \in \mathcal{K}} m(I_k)$$ for any countable pairwise disjoint collection of intervals $\{I_k \mid k \in \mathcal{K}\}$, thus satisfying the countable additivity requirement.

Now I want to argue that this definition also works for any countable collection of intervals, not just pairwise disjoint ones. My understanding is that the standard construction is the following. Given a countable collection $\mathcal{A} = \{A_1, A_2, A_3, \dots\}$, we can construct $\mathcal{B} = \{B_1, B_2, B_3, \dots\}$ by $$B_n = A_n \cap \left(\bigcap^{n - 1}_{i = 1} (A_i)^c\right).$$ Then we have $\bigcup_i A_i = \bigcup_i B_i$. So back to the outer measure, if we have a countable collection $\mathcal{I}$ of intervals, then we can use the previous procedure to construct a countable pairwise disjoint collection $\mathcal{I}'$ such that $$m\left(\bigcup \mathcal{I}\right) = m\left(\bigcup \mathcal{I}'\right).$$

My question is whether my understanding of this is correct, and under which conditions the construction holds. I guess we have to have closure under intersection for the construction to hold, but is there anything else?

Answer 1

The property you use here is the following: If $A$ and $B$ are intervals, then the difference $A\setminus B$ can be written as the disjoint union of a finite number of intervals (countable would suffice), so $A \setminus B = \biguplus_{i=1}^n I_i$. Iterating this property, we see that each $B_n = (\cdots(A_n \setminus A_1) \setminus \cdots) \setminus A_{n-1}$ is a finite disjoint union of intervals, and hence $\bigcup_n B_n$ equals a disjoint union of a countable number of intervals.

This property, together with closure under intersection has a name:

Definition A set of sets $\mathcal S \subseteq \mathfrak P(X)$ is called semiring on $X$, if all of the following holds:

1. $\emptyset \in \mathcal S$
2. $\mathcal S$ is closed under finite intersection.
3. If $A, B \in \mathcal S$, then $A \setminus B$ is the finite union of disjoint elements from $\mathcal S$.

So, for your construction to work, you need a semiring of sets.