Signed measure

Signed measure is a term from the mathematical branch of measure theory . Like measure, it is a function defined on a system of sets, usually a σ-algebra , and differs from this only in that negative values ​​are also permitted. The signed dimension thus represents a generalization of the concept of dimension. Sometimes signed dimensions are also referred to as charge distributions , since they visually assign the charge contained in each part of a charged body.

In contrast to the usual dimensions, quantities of signed dimensions have more structure. For example, the set of all signed dimensions in a common measurement space forms a vector space with a standard .

definition

Let be a non-empty set and a set system on with . ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {C}}\subseteq 2^{\Omega }}$ ${\displaystyle \Omega }$ ${\displaystyle \emptyset \in {\mathcal {C}}}$

A quantity function of after or is called a signed measure if: ${\displaystyle \nu }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle [-\infty ,+\infty )}$ ${\displaystyle (-\infty ,+\infty ]}$

1. ${\displaystyle \nu (\emptyset )=0}$
2. For every disjoint family with and applies ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle A_{i}\in {\mathcal {C}}}$ ${\displaystyle \textstyle \bigcup _{i\in \mathbb {N} }A_{i}\in {\mathcal {C}}}$

${\displaystyle \nu \left(\bigcup _{i\in \mathbb {N} }A_{i}\right)=\sum _{i\in \mathbb {N} }\nu (A_{i})}$.

This property is called σ-additivity .

If the system of sets is a σ-algebra , it will be referred to in the following as . In particular, is then always included in. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \textstyle \bigcup _{i\in \mathbb {N} }A_{i}}$ ${\displaystyle {\mathcal {A}}}$

Comments on the definition

The convergence of the series is to be regarded as the unconditional convergence in , that is, its limit value is . ${\displaystyle \textstyle \sum _{i\in \mathbb {N} }\nu (A_{i})}$ ${\displaystyle {\bar {\mathbb {R} }}}$ ${\displaystyle \textstyle \nu \left(\bigcup _{i\in \mathbb {N} }A_{i}\right)}$

The restriction to either the image set or the image set is made in order to preserve the associativity of the addition. It also avoids the occurrence of undefined expressions such as . ${\displaystyle [-\infty ,+\infty )}$ ${\displaystyle (-\infty ,+\infty ]}$ ${\displaystyle -\infty +\infty }$

If the set is chosen as the pictorial space , the requirement can be dispensed with. This follows from the fact that and is a real number ${\displaystyle (-\infty ,+\infty )}$ ${\displaystyle \nu (\emptyset )=0}$ ${\displaystyle \nu (\emptyset )}$

${\displaystyle \nu (\emptyset )=\sum _{i\in \mathbb {N} }\nu (\emptyset )}$

applies.

Examples

The two examples given here are also the classic methods of constructing signed dimensions.

Difference of dimensions

If there are finite dimensions in the measuring space , then are ${\displaystyle \mu _{1},\mu _{2}}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$

${\displaystyle \nu _{1}=\mu _{1}-\mu _{2}{\text{ und }}\nu _{2}=\mu _{2}-\mu _{1}}$

signed dimensions on . The finiteness of one of the two dimensions can be dispensed with if one wants to allow the signed dimensions to assume the values or . ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle \mu _{1},\mu _{2}}$ ${\displaystyle +\infty }$ ${\displaystyle -\infty }$

Integrally induced signed dimensions

Signed measures also appear in integration theory, they are induced by an indefinite integral.

Be a measure space and a measurable function. Is positive (takes values in on) or quasi integrated , the integral exists with an indicator function and forever. The figure with ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle f\colon \Omega \rightarrow {\bar {\mathbb {R} }}}$ ${\displaystyle {\mathcal {A}}-{\mathcal {B}}({\bar {\mathbb {R} }})}$ ${\displaystyle f}$ ${\displaystyle [0,\infty ]}$ ${\displaystyle \textstyle \int _{\Omega }f\chi _{A}d\mu }$ ${\displaystyle \chi }$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle \textstyle \int fd\mu \colon {\mathcal {A}}\rightarrow {\bar {\mathbb {R} }}}$

${\displaystyle (\int fd\mu )(A):=\int _{\Omega }f\chi _{A}d\mu }$

defines the indefinite integral. ${\displaystyle \mu }$

• If it is positive, there is a measure. ${\displaystyle f}$ ${\displaystyle \textstyle \int fd\mu }$
• If integrable, then there is a finite signed measure, that is for . ${\displaystyle f}$ ${\displaystyle \textstyle \int fd\mu }$ ${\displaystyle \textstyle (\int fd\mu )(A)\in \mathbb {R} }$ ${\displaystyle A\in {\mathcal {A}}}$
• Is quasi-integrable, so is a signed dimension. ${\displaystyle f}$ ${\displaystyle \textstyle \int fd\mu }$

The shorthand is usually used for . ${\displaystyle \textstyle (\int fd\mu )(A)}$ ${\displaystyle \textstyle \int _{A}fd\mu }$

properties

Given are and . Is , so is always , because it applies . The finiteness of the right-hand side then follows from the σ-additivity. ${\displaystyle A,B\in {\mathcal {A}}}$ ${\displaystyle B\subset A}$ ${\displaystyle |\nu (A)|<\infty }$ ${\displaystyle |\nu (B)|<\infty }$ ${\displaystyle \nu (A)=\nu (A\setminus B)+\nu (B)}$

Is with disjoint and is ${\displaystyle A,(A_{i})_{i\in \mathbb {N} }\in {\mathcal {A}}}$ ${\displaystyle A_{i}}$

${\displaystyle A=\bigcup _{i\in \mathbb {N} }A_{i}{\text{ sowie }}|\nu (A)|<\infty }$,

so the series is absolutely convergent. Because it is always for every bijection ${\displaystyle \sum _{i=1}^{\infty }\nu (A_{i})}$ ${\displaystyle \pi \colon \mathbb {N} \to \mathbb {N} }$

${\displaystyle \bigcup _{i\in \mathbb {N} }A_{\pi (i)}=A=\bigcup _{i\in \mathbb {N} }A_{i}}$

and thus

${\displaystyle \sum _{i=1}^{\infty }\nu (A_{\pi (i)})=\sum _{i=1}^{\infty }\nu (A_{i})}$.

So the series converges unconditionally and therefore also absolutely .

Steadiness from above

If a ring is continuous from above , it is therefore true that for every monotonically decreasing sequence with , and ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \nu }$ ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle A_{i}\in {\mathcal {C}}}$ ${\displaystyle |\nu (A_{1})|<\infty }$ ${\displaystyle \textstyle \bigcap _{i\in \mathbb {N} }A_{i}\in {\mathcal {C}}}$

${\displaystyle \lim _{i\rightarrow \infty }\nu (A_{i})=\nu \left(\bigcap _{i\in \mathbb {N} }A_{i}\right)}$

applies. If a σ-algebra, then the property is always fulfilled. ${\displaystyle {\mathcal {C}}}$

Steadiness from below

A signed measure on a σ-algebra is steadily from below , that is a monotonically increasing sequence of sets of valid ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle \lim _{i\rightarrow \infty }\nu (A_{i})=\nu \left(\bigcup _{i\in \mathbb {N} }A_{i}\right)}$.

Derived terms

Positive and negative quantities

A set is called a positive set if that holds for every additional set with ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle B\subset A}$ ${\displaystyle B\in {\mathcal {A}}}$

${\displaystyle \nu (B)\geq 0}$.

Likewise, a set is called a negative set if that holds for every additional set with ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle B\subset A}$ ${\displaystyle B\in {\mathcal {A}}}$

${\displaystyle \nu (B)\leq 0}$.

The concept of zero quantity transfers directly from dimensions to signed dimensions.

Signed dimensional space

If a σ-algebra is over the basic set and a signed measure, then the triple is called a signed measure space . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Omega }$ ${\displaystyle \nu }$ ${\displaystyle (\Omega ,{\mathcal {A}},\nu )}$

Finally signed measure

A signed measure means finally, if for everyone . This is equivalent to, or finite, the variation of . ${\displaystyle \nu }$ ${\displaystyle |\nu (A)|<\infty }$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle |\nu (\Omega )|<\infty }$ ${\displaystyle \nu }$

σ-finite signed measure

A signed measure is called σ-finite if there is a sequence of sets such that ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle \Omega =\bigcup _{n\in \mathbb {N} }A_{n}}$

and for everyone . This is equivalent to the variation of being a σ-finite measure . ${\displaystyle |\nu (A_{n})|<\infty }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \nu }$

Regular signed size

A finite signed measure on a Hausdorff space , provided with Borel's σ-algebra, is called regular if the variation of the signed measure is a regular measure .

Important statements

Hahn-Jordan decomposition

→ Main article : Hahn-Jordan decomposition

The Hahn-Jordan decomposition provides a division of a signed measure. Either the basic set is clearly broken down into a positive set and a negative set (Hahn's decomposition theorem), or the signed measure is divided into two (ordinary) measures, at least one of which is finite and which together result in the signed measure (Jordan's decomposition theorem ).

For every signed measure there is a positive set and a negative set , so that and is. ${\displaystyle \mu }$ ${\displaystyle P}$ ${\displaystyle N}$ ${\displaystyle N\cup P=\Omega }$ ${\displaystyle N\cap P=\emptyset }$

There are also measures (the so-called positive variation and the negative variation ), of which at least one is finite, which are singular to each other and for which applies. ${\displaystyle \mu ^{+},\mu ^{-}}$ ${\displaystyle \mu =\mu ^{+}-\mu ^{-}}$

It then applies

${\displaystyle \mu ^{+}(A)=\mu (P\cap A),\quad ,\mu ^{-}(A)=\mu (N\cap A)}$.

The measure is then called the variation of , the number the total variation norm of the signed measure. ${\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}$ ${\displaystyle \mu }$ ${\displaystyle |\mu |(\Omega )}$

Radon-Nikodym theorem

→ Main article : Radon-Nikodym theorem

If a σ-finite measure is in the measurement space and is a signed measure that is absolutely continuous with respect to ( ), then has a density function with respect to , that is, there is a measurable function , so that ${\displaystyle \mu }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu \ll \mu }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle f\colon X\to \mathbb {R} }$

${\displaystyle \nu (E)=\int _{E}f\,\mathrm {d} \mu }$for everyone . ${\displaystyle E\in {\mathcal {A}}}$

Lebesgue decomposition theorem

→ Main article : Lebesgue decomposition theorem

If a σ-finite measure is in the measurement space and is a σ-finite signed measure, then there is exactly one decomposition , where signed measures are so that is absolutely continuous with respect to and is singular with respect to . ${\displaystyle \mu }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \nu }$ ${\displaystyle \nu =\tau +\pi }$ ${\displaystyle \tau ,\pi }$ ${\displaystyle \tau }$ ${\displaystyle \mu }$ ${\displaystyle \pi }$ ${\displaystyle \mu }$

Theorem by Vitali-Hahn-Saks

→ Main article : Theorem of Vitali-Hahn-Saks

Vitali-Hahn-Saks' theorem says that the quantitative limit value of a sequence of signed measures defines a signed measure.

Spaces signed measure

In contrast to the dimensions, the signed dimensions form a real vector space on a common measurement space if they are finite. In particular, every real linear combination of signed dimensions is also a signed dimension. The dimensions then form a convex cone in this vector space. Important convex subsets are the probability measures and the sub-probability measures .

If one provides the vector space of the finite signed measures with the total variation norm as the norm , one obtains a normalized space . This room is even complete , so it is a Banach room .

This room can also be provided with an order structure, this is defined as

${\displaystyle \mu \leq \nu \;\iff \,\mu (A)\leq \nu (A)\quad {\text{für alle }}A\in {\mathcal {A}}}$.

This turns the finite, signed dimensions into a Riesz space and even a Banach association . It is also neatly complete .

Regular signed measures also appear in functional analysis , for example, as a dual space of the infinitely vanishing continuous functions, the so-called C ₀ functions .

Applications

With signed dimensions, for example, distributions of positive and negative charges in a substance can be modeled.

literature

• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .
• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .