Definition:Singular Measure

Definition

Let $d \in \N$.

Let $\map \BB {\R^d}$ be the Borel $\sigma$-algebra on $\R^d$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R^d, \map \BB {\R^d} }$.

Let $\mu$ be a measure, signed measure or complex measure on $\struct {\R^d, \map \BB {\R^d} }$.

We say that $\mu$ is singular if and only if $\mu$ and $\lambda$ are mutually singular measures.

Sources

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