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 $\mu$ and $\lambda$ are mutually singular measures.