Difference between revisions of "Signed measure"
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[[Category:Classical measure theory]] | [[Category:Classical measure theory]] | ||
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− | $\newcommand{\abs}[1]{\left|#1\right|}$ | + | $ |
+ | \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 [[Algebra of sets|σ-algebra]] $\mathcal{B}$ of subsets of | A signed measure is real-valued $\sigma$-additive function defined on a certain [[Algebra of sets|σ-algebra]] $\mathcal{B}$ of subsets of | ||
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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 $B$ (see [[Vector measure]]). The total variation measure of $\mu$ is defined on $B\in\mathcal{B}$ as: | ||
\[ | \[ | ||
− | \abs{\mu}(B) :=\sup\left\{ \sum \ | + | \abs{\mu}(B) :=\sup\left\{ \sum \norm{\mu(B_i)}_B: \text{$\{B_i\}\subset\mathcal{B}$ is a countable partition of $B$}\right\} |
\] | \] | ||
− | where $\ | + | where $\norm{\cdot}_B$ denotes the norm of $B$. |
In the real-valued case the above definition simplifies as | In the real-valued case the above definition simplifies as | ||
\[ | \[ | ||
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of a measurable $f\in L^1 (\abs{\mu}, V)$ such that | of a measurable $f\in L^1 (\abs{\mu}, V)$ such that | ||
\[ | \[ | ||
− | \mu (B) = \int_B f | + | \mu (B) = \int_B f \rd\abs{\mu} |
\] | \] | ||
− | 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 sets $B^+, B^-\in\mathcal{B}$ with |
− | of two nonnegative measures $\mu^+$ and $\mu^-$ which are mutually singular | + | \[ |
− | sets $B^+, B^-\in\mathcal{B}$ with | + | \mu^+ (X\setminus B^+) = |
− | + | \mu^- (X\setminus B^-) = | |
− | The Hahn decomposition theorem can also be proved defining directly the measures $\mu^+$ and $\mu^-$ | + | \mu^+ (B^-) = |
+ | \mu^- (B^+) = 0. | ||
+ | \] | ||
+ | This last statement is sometimes referred to as Hahn decomposition theorem. The Hahn decomposition theorem can also be proved defining directly the measures $\mu^+$ and $\mu^-$ | ||
in the following way: | in the following way: | ||
\begin{align*} | \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\}\\ |
− | \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*} | \end{align*} | ||
$\mu^+$ and $\mu^-$ are sometimes called, respectively, positive and negative variations of $\mu$. | $\mu^+$ and $\mu^-$ are sometimes called, respectively, positive and negative variations of $\mu$. |
Revision as of 13:08, 31 July 2012
generalized measure, real valued measure
2010 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 $B$ (see Vector measure). The total variation measure of $\mu$ is defined on $B\in\mathcal{B}$ as: \[ \abs{\mu}(B) :=\sup\left\{ \sum \norm{\mu(B_i)}_B: \text{'"`UNIQ-MathJax11-QINU`"' is a countable partition of '"`UNIQ-MathJax12-QINU`"'}\right\} \] where $\norm{\cdot}_B$ 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 \[ \mu (B) = \int_B f \rd\abs{\mu} \] 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 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
[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 |
Signed measure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Signed_measure&oldid=27276