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 if and only if 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 $\nu$.

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.3$: Singularity