---- Let $\mathbb{F}$ denote $\mathbb{R}$ or $\mathbb{C}$. > [!definition] Definition. ([[measure with a density]]) > Let $\mu$ be a [[measure|(positive) measure]] on a [[σ-algebra|measurable space]] $(X, \Sigma)$. Then [[Lp-norm|any]] $h \in \mathcal{L}^{1}(\mu)$ determines an $\mathbb{F}$[[signed measure|-]][[complex measure|valued]] [[complex measure|measure]] on $(X, \Sigma)$, denoted $h \, d\mu$, via $(h \, d\mu)(E \in \Sigma):= \int _{E} h \, d\mu .$ We call $h \, d\mu$ the **measure determined by $h$** or **measure with density $h$ (with respect to $\mu$)**. If $\nu$ is a [[measure]] which equals $h \, d\mu$, we conventionally write $d\nu=h\, d\mu$ even though the $d$ are spurious. > Note that if $h \in \mathcal{L}^{1}(\mu)$ is nonnegative, then $h \, d\mu$ is a [[finite measure|finite (positive) measure]]. ^definition > [!justification] > We have to show that $\nu :=h \, d\mu$ indeed defines a [[signed measure]] on $(X, \Sigma)$ when $\mathbb{F}=\mathbb{R}$ and a [[complex measure]] on $(X, \Sigma)$ when $\mathbb{F}=\mathbb{C}$. Suppose $E_{1},E_{2},\dots$ is a disjoint [[sequence]] of sets in $\Sigma$. Then $\nu (\bigsqcup_{k=1}^{\infty} E_{k})= \int \sum_{k=1}^{\infty} (\chi_{E_{k}}(x) h(x) )\, d\mu(x) = \sum_{k=1}^{\infty} \int \chi_{E_{k}}h(x) \, d\mu(x)= \sum_{k=1}^{\infty} \nu(E_{k}), $ where the middle equality uses the [[Dominated Convergence Theorem]] with majorant $|h|$ (valid since $h \in \mathcal{L}^{1}(\mu)$). ^justification ---- #### ---- #### 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 > ```