----- [[absolutely continuous|Absolute continuity]] and [[singular measures|singularity]] are two extreme possibilities for the relationship between two [[complex measure|complex measures]]. > [!proposition] Proposition. ([[absolutely continuous and singular implies the zero measure]]) > Suppose $\mu$ is a [[measure|(positive) measure]] on a [[σ-algebra|measurable space]] $(X, \Sigma)$. Then the *only* measure that is both [[absolutely continuous]] and [[singular measures|singular]] wrt $\mu$ is the $0$ measure: $\nu \ll \mu \text{ and }\nu \perp \mu \iff \nu=0.$ ^proposition > [!proof]- Proof. ([[absolutely continuous and singular implies the zero measure]]) > Suppose $\nu$ satisfies both $\nu \ll \mu$ and $\nu \perp \mu$; let $A,B \in \Sigma$, $A \sqcup B=X$, witness the latter. Let $E \in \Sigma$. Then $\mu(E \cap A)= \mu \big( (E \cap A) \cap B \big)=\mu(\emptyset)=0.$ Thus $\nu(E \cap A)=0$ since $\nu \ll \mu$. Hence $\nu(E)=\nu(E \cap A)=0$. ----- #### ---- #### 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 > ```