Definition 4

One says that a complex measure $\theta$ on $T$ is concentrated on a subset $M$ of $T$, or that $M$ carries $\theta$, if $\mathbf{C}M$ is locally negligible for $\theta$.

The set $M$ carries $\theta$ if and only if it carries $|\theta|$. It is equivalent to say that $M$ carries $\theta$, or that $M$ is $\theta$-measurable and $\theta = \varphi_M \cdot \theta$. If $\theta$ is concentrated on $M$, then every measure with base $|\theta|$ is concentrated on $M$.