Lemma 22.5.6.label Let $(X, \cm, \mu)$ be a localisable measure space, $\cf = \bracs{A \in \cm|\mu(A) < \infty}$, $\bracs{(E_A, F_A)}_{A \in \cf}$ such that:
- (a)
For each $A \in \cf$, $E_{A}, F_{A} \in \cm$, $E_{A}, F_{A} \subset A$, and $E_{A} \cap F_{A} = \emptyset$.
- (b)
For each $A, B \in \cf$, $\mu((E_{A} \cap B) \Delta (E_{B} \cap A)) = 0$ and $\mu((F_{A} \cap B) \Delta (F_{B} \cap A)) = 0$.
Let $E$ and $F$ be essential suprema of $\bracsn{E_A}_{A \in \cf}$ and $\bracsn{F_A}_{A \in \cf}$, respectively, then
- (1)
For each $B \in \cf$, $\mu(E \cap F_{B}) = 0$.
- (2)
$\mu(E \cap F) = 0$.
Proof. (1): Let $A, B \in \cf$, then
so $F_{B}^{c}$ is an essential upper bound of $\bracs{E_A}_{A \in \cf}$. Since $E$ is an essential supremum of $\bracs{E_A}_{A \in \cf}$, $\mu(E \setminus F_{B}^{c}) = \mu(E \cap F_{B}) = 0$.
(2): For any $B \in \cf$, $\mu(F_{B} \setminus E^{c}) = \mu(F_{B} \cap E) = \mu(E \cap F_{B}) =0$. Thus $E^{c}$ is an essential upper bound of $\bracs{F_B}_{B \in \cf}$. Given that $F$ is an essential supremum of $\bracsn{F_B}_{B \in \cf}$, $\mu(F \cap E) = \mu(F \setminus E^{c}) = 0$.$\square$
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