How are defined the variation and total variation of a positive measure?

Question

How are defined the variation and total variation of a positive measure?

If $\mu : (X, \mathcal{B}(X)) \to \mathbb{R}$ is a signed measure, then $$\text{the variation of $\mu$ is} \; |\mu| = \mu^+ + \mu^-$$ and $$\text{the total variation of $\mu$ is} \; \| \mu\| = |\mu| (X).$$

How are defined $|\mu|$ and $\| \mu \|$ when $\mu \geq 0$?

Thank you!

Answer 1

When $\mu \geqslant 0$, then $\mu^+ = \mu$ and $\mu^- = 0$. So then $\lvert\mu\rvert = \mu$ and $\lVert\mu\rVert = \mu(X)$.