Definition 2

A subset $A$ of a locally compact space $X$ is said to be integrable with respect to a measure $\mu$ on $X$ (or, that it is $\mu$*-integrable)* if the characteristic function $\varphi_A$ of $A$ is integrable. The finite number $\mu(A) = \int \varphi_A\, d\mu$ is called the measure of $A$.

For every integrable set $A$, $|\mu|(A) = |\mu|^*(A)$ (No. 2, Prop. 1); for a set to be negligible, it is necessary and sufficient that it be of measure zero with respect to $|\mu|$.