Lemma 5.6.4.label Let $\cali{I}=\curl{(a,b]:-\infty<a\leq b<\infty}$ then
- (1)
$\cali{I}$ is an elementary family.
- (2)
The collection
\begin{align*}\cali{A}=\curl{\dcups{n}{j=1}I_j:\curl{I_j}_1^n\suf\cali{I},n\in\N^+}\end{align*}
is a ring.
Proof. (1): Let $(a,b],(c,d]\in\cali{I}$ and assume without loss of generality that $a<b$ and $c<d$ then
\begin{align*}(a,b]\cap(c,d]&=(\max\curl{a,c},\min\curl{b,d}]\in\cali{I}\\ (a,b]\del(c,d]&=(a,\min\curl{b,c}]\sqcup(\max\curl{a,d},b]\end{align*}
(2): By Proposition 4.4.2, $\cali{A}$ is a ring.$\square$
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