Definition:Mutually Singular Measures

Definition
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a measure, signed measure or complex measure on $\struct {X, \Sigma}$.

Let $\nu$ be a measure, signed measure or complex measure on $\struct {X, \Sigma}$.

We say that $\mu$ and $\nu$ are mutually singular there exists $E \in \Sigma$ such that:


 * $\mu$ is concentrated on $E$ and $\nu$ is concentrated on $E^c$.

We write:


 * $\mu \perp \nu$

Also known as
We may also say that $\mu$ and $\nu$ are singular, $\nu$ is singular with respect to $\mu$ or $\mu$ is singular with respect to $\nu$.