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m (moved Charge to Signed measure: It is a more common terminology.)
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''generalized measure''
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{{MSC|28A33}}
  
An extended real-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c0217701.png" />-additive set function that is defined on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c0217702.png" />-algebra of Borel subsets of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c0217703.png" /> and that is finite on compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c0217704.png" />. The difference between two measures one of which is finite on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c0217705.png" /> is a charge; conversely, all charges may be obtained in this way: for any charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c0217706.png" /> there exists a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c0217707.png" /> into two disjoint Borel sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c0217708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c0217709.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177013.png" />. The measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177015.png" /> are independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177017.png" /> and are known respectively as the positive and negative variations of the charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177018.png" />; the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177019.png" /> is called the total variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177020.png" />. With this notation, the so-called Hahn–Jordan decomposition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177021.png" /> holds, so that the properties of charges may be phrased in terms of measure theory.
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[[Category:Classical measure theory]]
  
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.S. Landkof,  "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR></table>
 
  
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''generalized measure'', ''real valued measure''
  
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$\newcommand{\abs}[1]{\left|#1\right|}$
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An real-valued $\sigma$-additive function defined on a certain $\sigma$-algebra $\mathcal{B}$ of subsets of
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a set $X$. More generally one can consider vector-valued measures, i.e. $\sigma$-additive functions $\mu$ on $\mathcal{B}$
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taking values on a Banach space $B$. The total variation measure of $\mu$ is defined on $B\in\mathcal{B}$ as:
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\[
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\abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}_B: \text{$\{B_i\}\subset\mathcal{B}$ is a countable partition of $B$}\right\}
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\]
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where $\abs{\cdot}_B$ denotes the norm of $B$.
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In the real-valued case the above definition simplifies as
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\[
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\abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (X\setminus B)}\right).
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\]
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$\abs{\mu}$ is a measure and $\mu$ is said to have finite total variation if $\abs{\mu} (X) <\infty$.
  
====Comments====
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If $V$ is finite-dimensional the [[Radon-Nikodym theorem]] implies the existence
A charge is also called a signed measure [[#References|[a1]]], a real measure or a signed content. It can, more generally, be defined on a ring of subsets of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177022.png" />, or, alternatively, on a [[Riesz space|Riesz space]] of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177023.png" />, see [[#References|[a2]]].
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of a measurable $f\in L^1 (\abs{\mu}, V)$ such that
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\[
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\mu (B) = \int_B f d\abs{\mu}\qquad \mbox{for all $B\in\mathcal{B}$.}
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\]
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In the case of real-valued measures this implies that each such $\mu$ can be written as the difference
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of two nonnegative measures $\mu^+$ and $\mu^-$ which are mutually singular (i.e. such that there are
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sets $B^+, B^-\in\mathcal{B}$ with $\mu^+ (X\setminus B^+)= \mu^- (X\setminus B^-)
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=\mu^+ (B^-)=\mu^- (B^+)=0$). This last statement is sometimes referred to as Hahn decomposition theorem.
  
Any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177024.png" /> as above is called a Hahn decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177025.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177026.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177027.png" />, defined above, is also called the Jordan decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021770/c02177028.png" />.
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By the [[Riesz representation theorem]] the space of signed measures with finite total
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variation on the Borel $\sigma$-algebra of a locally compact
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Hausdorff space is the dual of the space of continuous functions (cp. also with [[Convergence of measures]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Jacobs,   "Measure and integral" , Acad. Press  (1978)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|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.  {{MR|1857292}}{{ZBL|0957.49001}}
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|-
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|valign="top"|{{Ref|Bo}}|| N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley  (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}}  {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}}  {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}}  {{ZBL|1095.28001}} {{ZBL|0156.06001}}
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|-
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|valign="top"|{{Ref|DS}}||  N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}}
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|-
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|valign="top"|{{Ref|Bi}}||  P. Billingsley, "Convergence of probability measures" , Wiley (1968)  {{MR|0233396}} {{ZBL|0172.21201}}
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|-
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|valign="top"|{{Ref|Ma}}|| P. Mattila, "Geometry of sets and measures in euclidean spaces.  Cambridge Studies in Advanced Mathematics, 44. Cambridge University  Press, Cambridge, 1995. {{MR|1333890}} {{ZBL|0911.28005}}
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|-
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|}

Revision as of 22:09, 27 July 2012

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL]

generalized measure, real valued measure

$\newcommand{\abs}[1]{\left|#1\right|}$ An real-valued $\sigma$-additive function defined on a certain $\sigma$-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 $B$. The total variation measure of $\mu$ is defined on $B\in\mathcal{B}$ as: \[ \abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}_B: \text{$\{B_i\}\subset\mathcal{B}$ is a countable partition of $B$}\right\} \] where $\abs{\cdot}_B$ denotes the norm of $B$. In the real-valued case the above definition simplifies as \[ \abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (X\setminus B)}\right). \] $\abs{\mu}$ 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 \[ \mu (B) = \int_B f d\abs{\mu}\qquad \mbox{for all $B\in\mathcal{B}$.} \] 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 theorem. By the [[Riesz representation theorem]] the space of signed measures with finite total variation on the Borel $\sigma$-algebra of a locally compact Hausdorff space is the dual of the space of continuous functions (cp. also with Convergence of measures).

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
[Bo] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[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
How to Cite This Entry:
Signed measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Signed_measure&oldid=27218
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article