Signed measure

generalized measure, real valued measure, charge

2020 Mathematics Subject Classification: Primary: 28A10 [MSN][ZBL] $ \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\norm}[1]{\left\|#1\right\|} $

Definition

The terminology signed measure denotes usually a real-valued $\sigma$-additive function defined on a certain σ-algebra $\mathcal{B}$ of subsets of a set $X$ (see Section 28 of [Ha]). 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 and Chapter 1 of [AFP]). Some authors consider also measures taking values in the extended real line: in this case it is assumed that the measure either does not take the value $\infty$ or does not take the value $-\infty$.

Total variation

The total variation measure of $\mu$ is defined on $B\in\mathcal{B}$ as: \[ \abs{\mu}(B) :=\sup\left\{ \sum \norm{\mu(B_i)}_V: \{B_i\}\subset\mathcal{B} \text{ is a countable partition of } B\right\} \] 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 (cp. with Theorem B of [Ha] for real-valued measures and [AFP] for the vector-valued case). $\mu$ is said to have finite total variation if $\abs{\mu} (X) <\infty$. This is in fact a restriction only if the measure is, apriori, taking values in the extended real-line and it is equivalent to say that the measure of any set $E\in\mathcal{B}$ is finite (cp. with Section 29 of [Ha]).

Upper and lower variations

In the case of real-valued measures one can introduce also the upper and lower variations: \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 also measures (cp. with Theorem B of Section 28 in [Ha]). $\mu^+$ and $\mu^-$ are sometimes called, respectively, positive and negative variations of $\mu$. Observe that $\mu = \mu^+ - \mu^-$ and $|\mu| = \mu^++\mu^-$.

Characterization of the total variation

For a real-valued measure the total variation can be characterized as \[ |\mu| (E) = \sup \left\{\int_E f\, d\mu\; :\; \mbox{ '"`UNIQ-MathJax27-QINU`"' is '"`UNIQ-MathJax28-QINU`"'-measurable and '"`UNIQ-MathJax29-QINU`"'}\right\}\, \] (see Section 29 of [Ha]). A similar characterization can be extended to measures taking values in a finite-dimensional Banach space.

Radon-Nikodym theorem and consequences

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 disjoint sets $B^+, B^-\in\mathcal{B}$ with $B^+\cup B^- = X$ and \[ \mu^+ (B^-) = \mu^- (B^+) = 0\, . \] This last statement is usually referred to as Jordan decomposition whereas the decomposition of $X$ into $B^+$ and $B^-$ is called Hahn decomposition theorem. In fact the measures $\mu^+$ and $\mu^-$ coincide with the upper and lower variations defined above (cp. with Theorem B of [Ha]).

Duality with continuous functions

By the Riesz representation theorem the space of signed measures with finite total variation on the $\sigma$-algebra of Borel subsets of a compact Hausdorff space is the dual of the space of continuous functions (cp. also with Convergence of measures). A similar duality statement can be generalized to locally compact Hausdorff spaces.

References

[AmFuPa]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
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[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[Ha]	P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Ma]	P. Mattila, "Geometry of sets and measures in euclidean spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005