Premeasure

A premeasure is a special set function in measure theory that is used to mathematically specify the intuitive concept of volume. In contrast to a measure , the domain of a premeasure does not have to be a σ-algebra .

definition

A set function of the set system is called premeasure if it fulfills the following properties: ${\displaystyle \mu :{\mathcal {C}}\to [0,\infty ]}$ ${\displaystyle {\mathcal {C}}}$

• It is ${\displaystyle \mu (\emptyset )=0}$
• It is σ-additive , that is, for every sequence of countably many pairwise disjoint sets from with : ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \textstyle \bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {C}}}$

${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\mu (A_{i}).}$

Alternatively, a premeasure can also be defined as an -additive content . A half-ring or a ring is usually chosen as the quantity system . A premeasure means finite if it applies to everyone . A premeasure is called -final if there is a decomposition of in such that applies to all . ${\displaystyle \sigma }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mu (A)<\infty }$ ${\displaystyle A\in {\mathcal {C}}}$ ${\displaystyle \sigma }$ ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mu (A_{i})<\infty }$ ${\displaystyle i\in \mathbb {N} }$

properties

The properties change depending on the quantity system on which a premeasure is defined. In addition to all of the properties mentioned here, all properties of content also apply to the corresponding set system.

In a half ring

If a half-ring , then it is possible to each premeasure on a unique premeasure to that of the generated ring construct. See also the section on sequels. ${\displaystyle {\mathcal {C}}={\mathcal {H}}}$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \mu '}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {R}}}$

In the ring

If there is a ring , then the following properties apply: ${\displaystyle {\mathcal {C}}={\mathcal {R}}}$ ${\displaystyle 1\iff 2\iff 3\Longrightarrow 4\iff 5}$

1. ${\displaystyle \mu }$ is a premeasure.
2. σ-subadditive (Sigma-subadditive), the following applies:for every sequence of setsinwith
   ${\displaystyle \mu \left(\bigcup _{j\in \mathbb {N} }A_{j}\right)\leq \sum _{j\in \mathbb {N} }\mu (A_{j})}$ ${\displaystyle (A_{j})_{j\in \mathbb {N} }}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \bigcup _{j\in \mathbb {N} }A_{j}\in {\mathcal {C}}}$
3. Continuity from below : If there isan opposingincreasing sequence of sets, then is. ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle A\in {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \mu (A_{i})\uparrow \mu (A)}$
4. Continuity from above : If there isadecreasing sequence of sets from with, then is. ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle A\in {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \mu (A_{1})<\infty }$ ${\displaystyle \mu (A_{i})\downarrow \mu (A)}$
5. Continuity against : ${\displaystyle \emptyset }$ If there is a decreasing sequence of sets against , then is . ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle \emptyset }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \mu (A_{i})\downarrow 0}$

These properties are also often used as characterization. If the premeasure is finite, then equivalence applies to all properties.

Continuability

From half rings to rings

For every pre-measure on the half - ring, one can construct a pre-measure on the ring produced by . Due to the properties of a half ring, there are pairwise disjoint sets with for all . By going through ${\displaystyle \mu }$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \mu '}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle A\in {\mathcal {R}}}$ ${\displaystyle A_{1},A_{2},\dotsc ,A_{m}\in {\mathcal {H}}}$ ${\displaystyle \textstyle A=\bigcup _{j=1}^{m}A_{j}}$ ${\displaystyle \mu '}$

${\displaystyle \mu '(A):=\sum _{j=1}^{m}\mu (A_{j})}$

defined, you get a clearly defined continuation . The continuation is -finite exactly if -finite. ${\displaystyle \mu '}$ ${\displaystyle \mu '}$ ${\displaystyle \sigma }$ ${\displaystyle \mu }$ ${\displaystyle \sigma }$

To a measure

According to Carathéodory's theorem of expansion of measures , a premeasure on a ring can be continued to a measure on the σ-algebra generated by the ring . To do this, an external dimension is first constructed from the pre-dimension . Those sets that are measurable with regard to this external measure form a σ-algebra . The restriction of the external measure to this σ-algebra is then a measure that corresponds to the premeasure. It also contains the ring and thus also the σ-algebra generated by the ring . ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \nu }$ ${\displaystyle {\mathcal {A}}_{\nu }}$ ${\displaystyle \nu |_{{\mathcal {A}}_{\nu }}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {A}}_{\nu }}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \sigma ({\mathcal {R}})}$

In addition, is a complete measure space and is the completion of . ${\displaystyle (\Omega ,{\mathcal {A}}_{\nu },\nu |_{{\mathcal {A}}_{\nu }})}$ ${\displaystyle \nu |_{{\mathcal {A}}_{\nu }}}$ ${\displaystyle \nu |_{\sigma ({\mathcal {R}})}}$

If the pre-measure is σ-finite, the uniqueness of the continuation follows with the uniqueness theorem for measures .

Examples

Lebesgue's premeasure

The most important premise is the so-called Lebesgue premise

${\displaystyle \lambda ([a,b))=b-a}$.

on the half-ring of the half-open intervals on the real numbers. It can also be generalized to higher dimensions. The Lebesgue measure and then the Lebesgue integral are constructed from it. ${\displaystyle [a,b)}$

Lebesgue-Stieltjessches premise

Another important pre-measure is the Lebesgue-Stieltjes pre-measure , from which the Lebesgue-Stieltjes measure and the Lebesgue-Stieltjes integral are derived:

${\displaystyle \mu _{F}([a,b))=F(b)-F(a)}$,

where is a growing right-hand continuous real-valued function. If the right-hand side is not continuous, then it is the Stieltjesian content . For it agrees with the Lebesgue premise. Every finite premeasure on the real numbers can be represented as Lebesgue-Stieltjesche premise with a suitable function . ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle F(x)=x}$ ${\displaystyle F}$

literature

• Jürgen Elstrodt: Measure and integration theory . 6th edition. Springer, Berlin / Heidelberg / New York 2009, ISBN 978-3-540-89727-9 .
• Achim Klenke: Probability Theory. 2nd Edition. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8