Lemma 18.7.3.label Let $\ci = \bracs{(a, b]| -\infty < a \le b < \infty}$ be the collection of all $h$-intervals, then:
- (1)
$\ci$ is an elementary family.
- (2)
The collection
\[\alg = \bracs{\bigsqcup_{j = 1}^n I_j \bigg | \seqf{I_j} \subset \ci, n \in \natp}\]is a ring.
Proof. (1): Let $(a, b], (c, d] \in \ci$ and assume without loss of generality that $a < b$ and $c < d$, then
- (P2)
$(a, b] \cap (c, d] = (\max(a, c), \min(b, d)] \in \ci$.
- (E)
$(a, b] \setminus (c, d] = (a, \min(b, c)] \sqcup (\max(a, d), b]$.
(2): By Proposition 17.4.2, $\alg$ is a ring.$\square$