---- > [!definition] Definition. ([[signed measure]]) > A **signed measure** on a [[σ-algebra|measurable space]] $(X, \Sigma)$ is a [[measure|countably additive]] function $\nu: \Sigma \to \mathbb{R}$. > > The triple $(X, \Sigma, \nu)$ is called a **signed measure space**. > > The [[Banach space|sub-Banach space]] of signed measures on $(X, \Sigma)$ is denoted $\mathcal{M}_{\mathbb{R}}(\Sigma)$. ^definition > [!generalization] > - [[complex measure]] (so find more properties there) ^generalization > [!specialization] > - [[finite measure|Finite measures]] are the nonnegative-valued signed measures. Note that a general [[measure|(positive) measure]] $\mu$ is not a signed measure, because $\mu$ can take on the value $\infty \not \in \mathbb{R}$. ^specialization > [!basicexample] > > - Let $\lambda$ denote the [[Lebesgue measure]] on $[-1,1]$. Define $\nu$ on the [[Borel set|Borel subsets]] of $[-1,1]$ by $\nu(E)=\lambda(E \cap [0,1])-\lambda(E \cap [-1,0]).$Then $\nu$ is a signed measure. > - For $\mu_{1},\mu_{2}$ [[measure|(positive) measures]], their difference $\mu_{1}-\mu_{2}$ is a signed measure. > - If $\nu$ is a [[complex measure]], then $\text{Re }\nu$ and $\text{Im }\nu$ are signed measures ---- #### ---- #### 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 > ```