---- > [!definition] Definition. ([[singular measures]]) > Suppose $\nu$ and $\mu$ are [[complex measure|complex]] or [[measure|positive measures]] on a [[σ-algebra|measurable space]] $(X,\Sigma)$. Then $\nu$ and $\mu$ are called **singular** with respect to one another, denoted $\nu \perp \mu$, if there exists a [[partition]] $A \sqcup B=\Sigma$, $A,B \in \Sigma$, for which $\nu(E)=\nu(E \cap A)$ and $\mu(E)=\mu(E \cap B)$ for all $E \in \Sigma$. ^definition > [!basicproperties] > - $\mu(E \cap A)=0$ and $\nu(E\cap B)=0$ for all $E \in \Sigma$[^1] > - [[absolutely continuous and singular implies the zero measure]] ^properties [^1]: Indeed, $\mu(E \cap A)=\mu \big( \overbrace{ (E \cap A) \cap B }^{ \emptyset } \big)=0$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```