Let be a signed measure space, and let $(A,B)$ be a Hahn decomposition for $\mu$ . We define $\mu^+$ and $\mu^-$ by$$\mu^+(E) = \mu(A\cap E)\quad \mbox{and} \quad \mu^-(E)=-\mu(B\cap E)$$ This definition is easily shown to be independent of the chosen Hahn decomposition.

It is clear that $\mu^+$ is a positive measure, and it is called the positive variation of $\mu$ . On the other hand, $\mu^-$ is a positive finite measure, called the negative variation of $\mu$ . The measure $|\mu|=\mu^+ + \mu^-$ is called the total variation of $\mu$ .

Notice that $\mu = \mu^+ - \mu^-$ . This decomposition of $\mu$ into its positive and negative parts is called the Jordan decomposition of $\mu$ .