20.4 Mutually Singular

Definition 20.4.1 (Mutually Singular).label Let $(X, \cm)$ be a measurable space and $\mu, \nu$ be signed/vector measures, then $\mu$ and $\nu$ are mutually singular, denoted $\mu \perp \nu$, if there exists $U, V \in \cm$ such that:

1. (1)

   $U$ is $\nu$-null.

2. (2)

   $V$ is $\mu$-null.

3. (3)

   $X = U \sqcup V$.