Monotonous class

A monotonic class , also called a monotonic system , is a set system with special properties, which is used in measure theory to build further, more complex set systems on it.

definition

Be a non-empty set. A non-empty subset of is called a monotonic class, ${\displaystyle X}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle P(X)}$

if the limit value of every monotonically increasing or decreasing sequence of quantities is contained in again . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {M}}}$

This means in full:

• are sets from with ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {M}}}$

${\displaystyle A_{1}\subset A_{2}\subset A_{3}\subset \cdots }$,

then is too

${\displaystyle \lim _{n\to \infty }A_{n}=\bigcup _{n=1}^{\infty }A_{n}}$ in ${\displaystyle {\mathcal {M}}}$

• are sets from with ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {M}}}$

${\displaystyle A_{1}\supset A_{2}\supset A_{3}\supset \cdots }$,

then is too

${\displaystyle \lim _{n\to \infty }A_{n}=\bigcap _{n=1}^{\infty }A_{n}}$ in ${\displaystyle {\mathcal {M}}}$

Generated monotonic class

Hierarchy of the quantity systems used in measure theory

Sections of any number of monotonous classes are again monotonous classes. Thus, for any set system, the monotone class generated by can be defined as ${\displaystyle K}$ ${\displaystyle K}$

${\displaystyle {\mathcal {M}}_{K}:=\bigcap _{{\mathcal {M}}{\text{ ist Monotone Klasse}} \atop K\subset {\mathcal {M}}}{\mathcal {M}}}$.

This can be interpreted as an envelope operator .

Relationship to other systems of quantities

• Every monotonic class that contains the superset and for which the following applies: are contained in the monotonic class, so is also contained in the monotonic class, is a Dynkin system . ${\displaystyle X}$ ${\displaystyle B\subset A}$ ${\displaystyle A\backslash B}$
• The monotonic class produced by an algebra corresponds to the σ-algebra produced by the algebra .

Rings and σ-rings

Every ring that is a monotonic class is a σ-ring (and therefore also a δ-ring ). Because if the quantities are contained in the ring, so is it ${\displaystyle A_{1},A_{2},A_{3},\dots }$

${\displaystyle B_{n}:=\bigcup _{i=1}^{n}A_{i}}$

due to the properties of the ring contained in the system of quantities. However, the quantities form a monotonically increasing sequence of quantities , hence their limit value ${\displaystyle B_{n}}$

${\displaystyle \lim _{n\to \infty }B_{n}=\bigcup _{n=1}^{\infty }A_{n}}$

Due to the properties of the monotonic class, it is also contained in the system of sets, so this is closed with regard to countable unions. Thus the monotonic class generated by a ring is always a σ-ring.

Conversely, every σ-ring is always a monotone class due to its stability under countable unions and cuts.

literature

• 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 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .

Individual evidence

1. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 23.
2. ↑ Kusolitsch: Measure and probability theory. 2014, p. 21.

See also

• Theorem about monotonous classes