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 .
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 .
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.
In the ring
If there is a ring , then the following properties apply:
is a premeasure.
σ-subadditive (Sigma-subadditive), the following applies:for every sequence of setsinwith
Continuity against : If there is a decreasing sequence of sets against , then is .
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
defined, you get a clearly defined continuation . The continuation is -finite exactly if -finite.
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 .
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.
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 .
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