Corollary 22.4.4.label Let $(X, \cm, \mu)$ be a measure space, $\cf \subset \bracs{A \in \cm|\mu(A) < \infty}$ be an ideal, and
\[\mu_{\cf}: \cm \to [0, \infty] \quad E \mapsto \sup\bracs{\mu(A \cap E)|A \in \cf}\]
then
- (1)
$\mu_{\cf}$ is a measure on $(X, \cm)$.
- (2)
$\cf$ is a scaffold for $\mu_{\cf}$/
and $\mu_{\cf}$ is the $\cf$-scaffolded part of $\mu$.
Proof. For each $A \in \cf$ and $E \in \cm$, let $\mu_{A}(E) = \mu(E \cap A)$, then $\bracsn{\mu_A}_{A \in \cf}$ is a family of measures satisfying Lemma 22.4.3. Therefore $\mu_{\cf}$ as defined is a measure, and $\cf$ is a scaffold for $\mu_{\cf}$.$\square$
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