---- > [!theorem] Theorem. ([[Radon-Nikodym Theorem]]) > It is trivial [[Lp-norm|that if]] $h \in \mathcal{L}^{1}(\mu)$ and $d\nu=h\, d\mu$ [[absolutely continuous|then]] $\nu \ll \mu$. Remarkably, a converse holds upon assuming $\mu$ is [[finite measure|σ-finite]]. > > - Suppose $\mu$ is a [[measure|(positive)]] *[[finite measure|σ-finite]]* [[measure]] on a [[σ-algebra|measurable space]] $(X, \Sigma)$. > - Suppose $\nu$ is a [[complex measure]] on $(X, \Sigma)$ such that $\nu \ll \mu$. > > Then there exists $h \in \mathcal{L}^{1}(\mu)$ such that $d\nu=h \, d\mu$. > > $h$ is called the **Radon-Nikodym derivative** of $\nu$ wrt $\mu$ and denoted $\frac{d\nu}{d\mu}$. It is unique up to null modifications (i.e., is unique as an element of $L^{1}$.) > [!proposition] Corollary. (If $\nu$ is a [[complex measure]] then $d\nu=h\, d |\nu|$ for some $h$ on the disc. ) > > - Suppose $\nu$ is a [[signed measure|real measure]] on a [[σ-algebra|measurable space]] $(X, \Sigma)$. Then there exists a [[measurable function]] $h: X \to \pm 1$ [[measure with a density|such that]] $d \nu=h\, d |\nu|$. > - Suppose $\nu$ is a [[complex measure]] on a [[σ-algebra|measurable space]] $(X, \Sigma)$. Then there exists a [[measurable function]] $h:X \to \mathbb{D}$ such that $d\nu=h\, d |\nu|.$ > [!proof]- Proof of Corollary. > Since $\nu \ll |\nu|$ always, Radon-Nikodym applies to give $h \in \mathcal{L}^{1}(|\nu|)$ such that $d \nu=h\, d |\nu|$. By the example in [[total variation measure]], $d |\nu|=|h|\, d |\nu|$, which implies $|h|=1$ [[almost-everywhere]]. Refining so that $|h|=1$ everywhere (which doesn't change the [[integral]]) finishes the job. > > $|\nu|(E)=\int _{E} |h| \, d |\nu|=\|h\|_{L^{1}(E, \nu)}$. > [!proof]- Proof. ([[Radon-Nikodym Theorem]]) > ~ ---- #### ----- #### 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 > ```