Difference between revisions of "Signed measure"
m |
m (→Characterization of the total variation) |
||
(10 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
− | ''generalized measure'', ''real valued measure'' | + | ''generalized measure'', ''real valued measure'', ''charge'' |
− | {{MSC| | + | {{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\|} | ||
+ | $ | ||
− | + | ===Definition=== | |
− | + | 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 $ | + | 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: | ||
\[ | \[ | ||
− | \abs{\mu}(B) :=\sup\left\{ \sum \ | + | \abs{\mu}(B) :=\sup\left\{ \sum \norm{\mu(B_i)}_V: \{B_i\}\subset\mathcal{B} \text{ is a countable partition of } B\right\} |
\] | \] | ||
− | where $\ | + | 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 | ||
\[ | \[ | ||
− | \abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (X\ | + | \abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (B\setminus A)}\right). |
+ | \] | ||
+ | $\abs{\mu}$ 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 | ||
+ | \[ | ||
+ | |\mu| (E) = \sup \left\{\int_E f\, d\mu\; :\; f \mbox{ is } \mu\text{-measurable and } |f|\leq 1\, \right\}\, | ||
\] | \] | ||
− | + | (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 | ||
\[ | \[ | ||
− | \mu (B) = \int_B f | + | \mu (B) = \int_B f \rd\abs{\mu} |
\] | \] | ||
− | 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 | + | \[ |
− | sets $B^+, B^-\in\mathcal{B}$ with $ | + | \mu^+ (B^-) = |
− | + | \mu^- (B^+) = 0\, . | |
− | + | \] | |
− | + | This last statement is usually referred to as [[Jordan decomposition]] whereas the decomposition of $X$ into $B^+$ and $B^-$ is called [[Hahn decomposition|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}}). | |
− | |||
− | |||
− | |||
− | |||
− | $\mu^+$ and $\mu^-$ | ||
− | |||
− | 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 | + | 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=== | |
{| | {| | ||
|- | |- | ||
|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| | + | |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 10:52, 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\|} $
Contents
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 |
Signed measure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Signed_measure&oldid=27267