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''generalized measure'', ''real valued measure''
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''generalized measure'', ''real valued measure'', ''charge''
  
{{MSC|28A33}}
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{{MSC|28A10}}
 
[[Category:Classical measure theory]]
 
[[Category:Classical measure theory]]
 
{{TEX|done}}
 
{{TEX|done}}
 +
$
 +
\newcommand{\abs}[1]{\left|#1\right|}
 +
\newcommand{\norm}[1]{\left\|#1\right\|}
 +
$
  
$\newcommand{\abs}[1]{\left|#1\right|}$
+
===Definition===
An real-valued $\sigma$-additive function defined on a certain [[Algebra of sets|σ-algebra]] $\mathcal{B}$ of subsets of
+
The terminology signed measure denotes usually a real-valued $\sigma$-additive function defined on a certain [[Algebra of sets|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 $\mathcal{B}$
+
a set $X$ (see Section 28 of {{Cite|Ha}}). More generally one can consider vector-valued measures, i.e. $\sigma$-additive functions $\mu$ on $\mathcal{B}$
taking values on a Banach space $B$ (see [[Vector measure]]). The total variation measure of $\mu$ is defined on $B\in\mathcal{B}$ as:
+
taking values on a Banach space $V$ (see [[Vector measure]] and Chapter 1 of {{Cite|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 \abs{\mu(B_i)}_B: \text{$\{B_i\}\subset\mathcal{B}$ is a countable partition of $B$}\right\}
+
\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 $\abs{\cdot}_B$ denotes the norm of $B$.
+
where $\norm{\cdot}_V$ denotes the norm of $V$.
 
In the real-valued case the above definition simplifies as
 
In the real-valued case the above definition simplifies as
 
\[
 
\[
\abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (X\setminus B)}\right).
+
\abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (B\setminus A)}\right).
 +
\]
 +
$\abs{\mu}$ is a measure (cp. with Theorem B of {{Cite|Ha}} for real-valued measures and {{Cite|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 {{Cite|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 {{Cite|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\; :\; f \mbox{ is } \mu\text{-measurable and } |f|\leq 1\, \right\}\,
 
\]
 
\]
$\abs{\mu}$ is a measure and $\mu$ is said to have finite total variation if $\abs{\mu} (X) <\infty$.  
+
(see Section 29 of {{Cite|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
 
If $V$ is finite-dimensional the [[Radon-Nikodym theorem]] implies the existence
 
of a measurable $f\in L^1 (\abs{\mu}, V)$ such that
 
of a measurable $f\in L^1 (\abs{\mu}, V)$ such that
 
\[
 
\[
\mu (B) = \int_B f d\abs{\mu}\qquad \mbox{for all $B\in\mathcal{B}$.}
+
\mu (B) = \int_B f \rd\abs{\mu}
 
\]
 
\]
In the case of real-valued measures this implies that each such $\mu$ can be written as the difference
+
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
of two nonnegative measures $\mu^+$ and $\mu^-$ which are mutually singular (i.e. such that there are
+
\[
sets $B^+, B^-\in\mathcal{B}$ with $\mu^+ (X\setminus B^+)= \mu^- (X\setminus B^-)
+
\mu^+ (B^-) =  
=\mu^+ (B^-)=\mu^- (B^+)=0$). This last statement is sometimes referred to as Hahn decomposition theorem.
+
\mu^- (B^+) = 0\, .
The Hahn decomposition theorem can also be proved defining directly the measures $\mu^+$ and $\mu^-$
+
\]
in the following way:
+
This last statement is usually referred to as [[Jordan decomposition]] whereas the decomposition of $X$ into $B^+$ and $B^-$ is called [[Hahn decomposition|Hahn decomposition theorem]]. In fact the measures $\mu^+$ and $\mu^-$ coincide with the upper and lower variations defined above (cp. with Theorem B of {{Cite|Ha}}).
\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  
+
===Duality with continuous functions===
variation on the $\sigma$-algebra of [[Borel set|Borel subsets]] of a locally compact
+
By the [[Riesz representation theorem]] the space of signed measures with finite total variation on the $\sigma$-algebra of [[Borel set|Borel subsets]] of a compact
Hausdorff space is the dual of the space of continuous functions (cp. also with [[Convergence of measures]]).
+
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====
+
===References===
 
{|
 
{|
 
|-
 
|-
 
|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}}  
 
|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}}  
 +
|-
 +
|valign="top"|{{Ref|Bi}}||  P. Billingsley, "Convergence of  probability measures" , Wiley (1968)  {{MR|0233396}} {{ZBL|0172.21201}}
 
|-
 
|-
 
|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}}
 
|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}}
Line 50: Line 72:
 
|valign="top"|{{Ref|DS}}||  N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1'''  , Interscience (1958) {{MR|0117523}}
 
|valign="top"|{{Ref|DS}}||  N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1'''  , Interscience (1958) {{MR|0117523}}
 
|-
 
|-
|valign="top"|{{Ref|Bi}}|| P. Billingsley, "Convergence of probability measures" , Wiley (1968) {{MR|0233396}} {{ZBL|0172.21201}}
+
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 
|-
 
|-
 
|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}}
 
|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}}
 
|-
 
|-
 
|}
 
|}

Latest revision as of 11:52, 16 August 2013

generalized measure, real valued measure, charge

2010 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 \[ |\mu| (E) = \sup \left\{\int_E f\, d\mu\; :\; f \mbox{ is } \mu\text{-measurable and } |f|\leq 1\, \right\}\, \] (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
[Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[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
[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
How to Cite This Entry:
Signed measure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Signed_measure&oldid=27267
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article