---- > [!definition] Definition. ([[complex measure]]) > A **complex measure** on a [[σ-algebra|measurable space]] $(X, \Sigma)$ is a [[measure|countably additive]] function $\nu: \Sigma \to \mathbb{\mathbb{C}}$. > > The triple $(X, \Sigma, \nu)$ is called a **complex measure space**. > > The set of all complex measures on $(X, \Sigma)$ forms a [[Banach space]] with respect to the **total variation norm** $\|\nu\|:=|\nu|(X),$ > where $\nu$ denotes the [[total variation measure]] associated to $\nu$. [[normed vector space|This]] [[vector space|space]] is denoted $\mathcal{M}_{\mathbb{C}}(\Sigma)$. ^definition > [!specialization] > [[signed measure|Signed measures]] are complex measures, as are *[[finite measure|finite]]* [[measure|measures]] $\mu$. ^specialization > [!basicproperties] > - $\mu(\emptyset)=0$; > - *(absolute convergence)* $\sum_{k=1}^{\infty}|\nu(E_{k})|<\infty$ for every disjoint [[sequence]] $E_{1},E_{2},\dots$ of sets in $\Sigma$. > > > [!proof]- Proof. > > Suppose $E_{1},E_{2},\dots$ is a disjoint sequence of sets in $\Sigma$. First suppose $\nu$ is a [[signed measure|real measure]]. Thus $\nu(\bigsqcup_{\{ k:\nu(E_{k})>0 \}} E_{k})=\sum_{\{ k : \nu(E_{k}>0) \}}\nu(E_{k})=\sum_{\{ k: \nu(E_{k})>0 \}}|\nu(E_{k})|$ and $-\nu(\bigsqcup_{\{ k: \nu(E_{k})<0 \}})= - \sum_{\{ k : \nu(E_{k})<0 \}}\nu(E_{k})=\sum_{\{ k : \nu(E_{k})<0 \}} |\nu(E_{k})|.$ > > Because $\nu(E) \in \mathbb{R}$ for every $E \in \Sigma$, the RHS of the two displayed equations are each finite. Their sum $\sum_{k=1}^{\infty} |\nu(E_{k})|$ is thus finite. For suppose $\nu$ is a [[complex measure]]. Then $\sum_{k=1}^{\infty} |\nu(E_{k})| \leq \sum_{k=1}^{\infty} (|(\text{Re }\nu)(E_{k})| + (\text{Im }\nu)(E_{k}))<\infty,$ > > where the last inequality follows by reducing to the proven case of real measures. > >- Commuting with set differences, inclusion-exclusion, continuity from below, and continuity from above all hold with the same proofs as in [[measure|(positive) measure > [!basicexample] > - For $\mu_{1},\mu_{2}$ [[measure|(positive) measures]], their difference $\alpha _{1}\mu_{1}+\alpha_{2}\mu_{2}$ is a complex measure for all $\alpha_{1},\alpha_{2} \in \mathbb{C}$. > - The total variation norm of a [[finite measure|finite (positive) measure]] $\mu$ is $\|\mu\|=\mu(X)$, as follows from monotonicity of (positive) measure. > - If $\mu$ is a (positive) measure, $h \in \mathcal{L}^{1}(\mu)$, [[measure with a density|and]] $d\nu=h \, d\mu$, then $\|d \nu\|=\|h\|_{1}$, as follows from the example in [[total variation measure]]. > [!justification] > - [ ] we have to check that the TV norm is indeed a norm (especially that it is finite-valued) > - [ ] then we have to check completeness > These are both good exercises for later; have to get moving for now toward Radon-Nikodym ^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 > ```