Signed measure

generalized measure, real valued measure

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL] $ \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\norm}[1]{\left\|#1\right\|} $

A signed measure is real-valued $\sigma$-additive function defined on a certain σ-algebra $\mathcal{B}$ of subsets of a set $X$. More generally one can consider vector-valued measures, i.e. $\sigma$-additive functions $\mu$ on $\mathcal{B}$ taking values on a Banach space $V$ (see Vector measure). The total variation measure of $\mu$ is defined on $B\in\mathcal{B}$ as: \[ \abs{\mu}(B) :=\sup\left\{ \sum \norm{\mu(B_i)}_V: \text{$\{B_i\}\subset\mathcal{B}$ is a countable partition of $B$}\right\} \] where $\norm{\cdot}_V$ denotes the norm of $V$. In the real-valued case the above definition simplifies as \[ \abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (B\setminus A)}\right). \] $\abs{\mu}$ is a measure and $\mu$ is said to have finite total variation if $\abs{\mu} (X) <\infty$. If $V$ is finite-dimensional the [[Radon-Nikodym theorem]] implies the existence of a measurable $f\in L^1 (\abs{\mu}, V)$ such that \[ \mu (B) = \int_B f \rd\abs{\mu} \] for all $B\in\mathcal{B}$. In the case of real-valued measures this implies that each such $\mu$ can be written as the difference of two nonnegative measures $\mu^+$ and $\mu^-$ which are mutually singular i.e. such that there are sets $B^+, B^-\in\mathcal{B}$ with \[ \mu^+ (X\setminus B^+) = \mu^- (X\setminus B^-) = \mu^+ (B^-) = \mu^- (B^+) = 0. \] This last statement is sometimes referred to as [[Hahn decomposition|Hahn decomposition theorem]]. The Hahn decomposition theorem can also be proved defining directly the measures $\mu^+$ and $\mu^-$ in the following way: \begin{align*} \mu^+ (B) &= \sup \{ \mu (A): A\in \mathcal{B}, A\subset B\}\\ \mu^- (B) &= \sup \{ -\mu (A): A\in \mathcal{B}, A\subset B\} \end{align*} $\mu^+$ and $\mu^-$ are sometimes called, respectively, positive and negative variations of $\mu$. Observe that $|\mu| = \mu^++\mu^-$. By the [[Riesz representation theorem]] the space of signed measures with finite total variation on the $\sigma$-algebra of Borel subsets of a locally compact Hausdorff space is the dual of the space of continuous functions (cp. also with Convergence of measures).

References

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