σ-continuity

In mathematics, σ-continuity is a property of set functions , i.e. functions that take sets as arguments ("input") rather than points. A distinction is made between σ-continuity from below (or short continuity from below), σ-continuity from above (or short continuity from above) and continuity . These types of continuity play a role in stochastics and measure theory , where they belong to the elementary properties of probability measures and measures . ${\displaystyle \emptyset }$

definition

A quantity ring is given on which a content is explained. ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \mu :{\mathcal {M}}\to [0,\infty ]}$

The set function is then called ${\displaystyle \mu }$

• σ-continuous from below in , if for every monotonically increasing set sequence off is always . ${\displaystyle A}$ ${\displaystyle A_{n}\uparrow A}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \lim _{n\to \infty }\mu (A_{n})=\mu (A)}$
• σ-continuous from above in , if for every monotonically decreasing set sequence out with is always for all . ${\displaystyle A}$ ${\displaystyle A_{n}\downarrow A}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \mu (A_{n})<\infty }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \lim _{n\to \infty }\mu (A_{n})=\mu (A)}$

Her name is now

• σ-continuous from below if it is σ-continuous from below for all . ${\displaystyle A\in {\mathcal {M}}}$
• σ-continuous from above if it is σ-continuous from above for all . ${\displaystyle A\in {\mathcal {M}}}$
• ${\displaystyle \emptyset }$-continuous if it is continuous from above in the empty set . ${\displaystyle \emptyset }$

The definitions apply identically to the more specific standard case of a measure on a σ-algebra .

comment

In the case of finite set functions such as probability measures and finite measures , the finiteness criterion can be dispensed with when defining the σ-continuity from above, since is always . In the general case, however, this is not possible. For example, consider the set function ${\displaystyle \mu (X)<\infty }$

${\displaystyle \mu :{\mathcal {P}}(\mathbb {N} )\to [0,\infty ]}$

defined by

${\displaystyle \mu (A):=\#A}$,

the so-called counting measure ( here denotes the set of elements in the set ) is the set sequence ${\displaystyle \#A}$ ${\displaystyle A}$

${\displaystyle A_{n}:=\{n,n+1,n+2,\dots \}}$

falling against the empty crowd, but it is

${\displaystyle 0=\mu (\emptyset )\neq \lim _{n\to \infty }\mu (A_{n})=\infty }$.

use

The continuity of a set function is an important tool in many proofs, since it allows one to infer the approximation of the function values ​​from the approximation of the sets. In addition, it can be used to specify equivalent characterizations of the σ-additivity of content and thus criteria under which these premeasures are and can thus be continued to measure.

literature

• 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 .
• Hans-Otto Georgii : Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .