Lemma 22.4.5.label Let $(X, \cm, \cf, \mu)$ be a scaffolded measure space and $f \in L^{+}(X)$, then for any $E \in \cm$,
\[\int_{E} f d\mu = \sup_{A \in \cf}\int_{E \cap A}f d\mu\]
Proof. For any $F \in \cm$,
\[\int_{E}\one_{F} d\mu = \mu(E \cap F) = \sup_{A \in \cf}\mu(A \cap E \cap F) = \sup_{A \in \cf}\int_{E \cap A}\one_{F} d\mu\]
so by linearity, the above holds for all simple functions in $L^{+}(X)$.
Now, for each simple function $\phi \in \Sigma^{+}(X)$ with $\phi \le f$,
\[\int_{E} \phi d\mu = \sup_{A \in \cf}\int_{E \cap A}\phi d\mu \le \sup_{A \in \cf}\int_{E \cap A}f d\mu\]
As the above holds for all $\phi \in \Sigma^{+}(X)$ with $\phi \le f$,
\[\int_{E} f d\mu = \sup_{A \in \cf}\int_{E \cap A}f d\mu\]
$\square$
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