Definition 16.5.2 (Total Variation of Signed Measures). Let $(X, \cm)$ be a measure space, $\mu$ be a signed measure on $(X, \cm)$, and
\[|\mu|(A) = \sup\bracs{\sum_{j = 1}^n |\mu(A_j)| \bigg | \seqf{A_j} \subset \cm, A = \bigsqcup_{j = 1}^n A_j}\]
then:
For any $A \in \cm$,
\[|\mu|(A) = \nu(A) = \sup\bracs{\sum_{i \in I}|\mu(A_i)| \bigg | \seqi{A} \subset \cm, A = \bigsqcup_{i \in I}A_i}\]$|\mu|$ is a measure on $(X, \cm)$.
Let $\mu = \mu^{+} - \mu^{-}$ be the Jordan decomposition of $\mu$, then $|\mu| = \mu^{+} + \mu^{-}$.
and the measure $|\mu|$ is the total variation of $\mu$.
Proof. (1) and (2): Same as Definition 11.2.1.
(3): For any $A \in \cm$, $|\mu(A)| \le \mu^{+}(A) + \mu^{-}(A)$, so $(\mu^{+} + \mu^{-}) \ge |\mu|$. On the other hand, let $X = P \sqcup N$ be a Hahn decomposition of $\mu$, then
\[|\mu|(A) \ge \mu(A \cap P) + \mu(A \cap N) = \mu^{+}(A) + \mu^{-}(A)\]
so $(\mu^{+} + \mu^{-}) \le |\mu|$.$\square$