Trace of a system of quantities

Trace of a set system is a term from mathematics and is particularly often used in measure theory and stochastics . It describes the reduction of set systems to a smaller basic set and is closely related to the term track topology .

definition

Let an arbitrary system of sets on the basic set and a set be given . Then is called ${\displaystyle {\mathcal {M}}\subset {\mathcal {P}}(\Omega )}$ ${\displaystyle \Omega }$ ${\displaystyle E\subset \Omega }$

${\displaystyle {\mathcal {M}}|_{E}:=\{M\cap E\,|\,M\in {\mathcal {M}}\}}$

the trace or restriction of on . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle E}$

comment

In general, the trace of a set system is no longer of the same type as the original set system. Dynkin systems are an example of this . Classes of sets, the track is again from the same class, are half-rings , lot of rings , Mengenalgebren and σ-rings and σ-algebras .

example

Let , be a corresponding σ-algebra and , then the trace σ-algebra of over . ${\displaystyle \Omega =\{1,2,3\}}$ ${\displaystyle {\mathcal {A}}=\{\emptyset ,\{1\},\{2,3\},\Omega \}}$ ${\displaystyle E=\{1,2\}}$ ${\displaystyle {\mathcal {A}}|_{E}=\{\emptyset ,\{1\},\{2\},E\}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle E}$

literature

• Jürgen Elstrodt : Measure and integration theory. 4th, corrected edition. Springer, Berlin et al. 2005, ISBN 3-540-21390-2
• Achim Klenke: Probability Theory. 2nd Edition. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8

Individual evidence

1. ↑ Achim Klenke: Probability Theory. (PDF) p. 10 , accessed on January 4, 2015 .