Principle of good quantities

The principle of good sets is a frequently used method of proof , especially in measure theory . It can be used to prove that a statement holds for all elements of a σ-algebra or other system of sets . Since, in general, the elements of a σ-algebra, such as Borel's σ-algebra , cannot be specified explicitly, and only one producer is known, such proofs often have to be used indirectly.

The principle

Let be a σ-algebra over a basic set . In order to show that all elements of have a given property, the set of all subsets of (or all elements of ) for which this property applies, i.e. all “good sets” is considered. Applies now ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}}$

1. ${\displaystyle {\mathcal {G}}}$contains a producer of and ${\displaystyle {\mathcal {A}}}$
2. ${\displaystyle {\mathcal {G}}}$ is a σ-algebra,

so it follows that the property applies to all . In other words: It only has to be shown that certain “good sets” are generated and that all “good sets” form a σ-algebra. ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$

Reason: If it is generated by a system of sets , because of the monotony and idempotency of the σ operator it follows from : ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}\subseteq {\mathcal {G}}}$

${\displaystyle {\mathcal {A}}=\sigma ({\mathcal {E}})\subseteq \sigma ({\mathcal {G}})={\mathcal {G}}.}$

If it is difficult to show for point 2 that it is closed against countable unions of arbitrary elements, the principle can be combined with a Dynkin system argument based on Dynkin's π-λ theorem . If the producer is stable on average , then it suffices to show that there is a Dynkin system, because in this case the following applies , where the Dynkin system produced by denotes. ${\displaystyle {\mathcal {G}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle \sigma ({\mathcal {E}})=\delta ({\mathcal {E}})\subseteq {\mathcal {G}}}$ ${\displaystyle \delta ({\mathcal {E}})}$ ${\displaystyle {\mathcal {E}}}$

example

If a mapping and a system of sets is made up of subsets of , then holds ${\displaystyle f\colon \Omega \to \Omega '}$ ${\displaystyle {\mathcal {E}}\subseteq {\mathcal {P}}(\Omega ')}$ ${\displaystyle \Omega '}$

${\displaystyle f^{-1}(\sigma ({\mathcal {E}}))=\sigma (f^{-1}({\mathcal {E}}))\,,}$

d. That is, the archetype of the σ-algebra generated by is the σ-algebra generated by the archetype of . ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$

In order to prove inclusion , the principle of good quantities can be applied, because it has to be shown that all have the property . So this is what ${\displaystyle f^{-1}(\sigma ({\mathcal {E}}))\subseteq \sigma (f^{-1}({\mathcal {E}}))}$ ${\displaystyle A\in {\mathcal {A}}:=\sigma ({\mathcal {E}})}$ ${\displaystyle f^{-1}(A)\in \sigma (f^{-1}({\mathcal {E}}))}$

${\displaystyle {\mathcal {G}}:=\{A\subseteq \Omega '\mid f^{-1}(A)\in \sigma (f^{-1}({\mathcal {E}}))\}}$

chosen as the set of good sets. The two conditions above are thus fulfilled:

1. For all true , that is . ${\displaystyle E\in {\mathcal {E}}}$ ${\displaystyle f^{-1}(E)\in \sigma (f^{-1}({\mathcal {E}}))}$ ${\displaystyle E\in {\mathcal {G}}}$
2. ${\displaystyle {\mathcal {G}}}$ is a σ-algebra: This is checked directly against the definition with the help of the calculation rules for archetypes.

This shows the inclusion.

In contrast, the reverse inclusion follows with a simple monotony argument. Since prototypes of σ-algebras are again σ-algebras, the following applies

${\displaystyle \sigma (f^{-1}({\mathcal {E}}))\subseteq \sigma (f^{-1}(\sigma ({\mathcal {E}})))=f^{-1}(\sigma ({\mathcal {E}}))\,.}$

Individual evidence

1. ↑ Jürgen Elstrodt : Measure and integration theory. 6th edition, Springer, Berlin 2009, ISBN 978-3-540-89727-9 , p. 19.
2. ↑ Norbert Kusolitsch: Measure and probability theory: An introduction. Springer, Vienna 2011, ISBN 978-3-7091-0684-6 , p. 24.
3. ↑ Dirk Werner : Introduction to Higher Analysis. 2nd edition, Springer, Berlin 2009, ISBN 978-3-540-79599-5 , p. 213.
4. ↑ Jochen Wengenroth: Probability Theory . Walter de Gruyter, 2008, ISBN 978-3-11-020358-5 , p. 11 .