---- > [!definition] Definition. ([[absolutely continuous]]) > Suppose $\nu$ is a [[complex measure]] on a [[σ-algebra|measurable space]] $(X, \Sigma)$ and $\mu$ is a [[measure|(positive) measure]] on $(X, \Sigma)$. Then $\nu$ is called **absolutely continuous** with respect to $\mu$, denoted $\nu \ll \mu$, if $\mu(E)=0 \implies \nu(E)=0$ for every $E \in \Sigma$, i.e., $\text{NullSets}(\nu) \subset \text{NullSets}(\mu)$. ^definition > [!basicproperties] > [[absolutely continuous and singular implies the zero measure]] ^properties > [!basicexample] > > - If $\mu$ is a [[measure|(positive) measure]] [[Lp-norm|and]] $h \in \mathcal{L}^{1}(\mu)$, [[measure with a density|then]] $h \, d\mu \ll \mu$. Indeed, the [[integral]] over a null set is always zero. > > - If $\nu$ is a [[signed measure|real measure]], [[Jordan Decomposition Theorem|then]] $\nu^{+} \ll \nu$ and $\nu^{-} \ll \nu$. > > - If $\nu$ is a [[complex measure]], [[total variation measure|then]] $\nu \ll |\nu|$. > > - If $\nu$ is a [[complex measure]], then $\operatorname{Re}\nu \ll |\nu|$ and $\operatorname{Im}\nu \ll |\nu|$. > > - Every [[measure]] on a [[σ-algebra|measurable space]] is absolutely continuous wrt the [[measure|counting measure]] on $(X, \Sigma)$, since $\emptyset$ is null to every measure. ---- #### ---- #### 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 > ```