Lemma 23.3.4.label Let $(X, \cm)$ be a measurable space, $K \in \RC$, $\mu: \cm \to K$ be a signed/complex measure, then:
- (1)
If $K = \real$, then $|\mu|(X) \le 2\sup_{A \in \cm}\norm{\mu(A)}_{E}$.
- (2)
If $K = \complex$, then $|\mu|(X) \le 4\sup_{A \in \cm}\norm{\mu(A)}_{E}$.
Proof. (1, scalar): Let $X = P \sqcup N$ with $P, N \in \cm$ be a Hahn decomposition of $\mu$, then
\[|\mu|(X) = |\mu(X \cap P)| + |\mu(X \cap N)| \le 2\sup_{A \in \cm}\norm{\mu(A)}_{E}\]
(2, scalar): By (1),
\[|\mu|(X) \le |\text{Re}(\mu)|(X) + |\text{Im}(\mu)|(X) \le 4\sup_{A \in \cm}\norm{\mu(A)}_{E}\]
$\square$
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