P-trivial σ-algebra

A P-trivial σ-algebra is a special set system in stochastics , which is characterized by the fact that each subset of the set system (or each event ) is assigned the probability 0 or 1. So the events are almost certain or almost impossible. P-trivial σ-algebras occur in stochastics, for example, within the framework of the 0-1 laws . They are also used in ergodic theory , for example when it comes to the question of whether a dimensionally preserving dynamic system is also ergodic .

definition

A probability space is given . A σ-algebra is called a P-trivial σ-algebra if it holds for all that either is or . ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle {\mathcal {O}}\subset {\mathcal {A}}}$ ${\displaystyle O\in {\mathcal {O}}}$ ${\displaystyle P(O)=0}$ ${\displaystyle P(O)=1}$

Elementary examples

• The trivial σ-algebra is always also P-trivial. This follows from the definition of the probability measure , since there is always and required. ${\displaystyle \{\Omega ,\emptyset \}}$ ${\displaystyle P(\Omega )=1}$ ${\displaystyle P(\emptyset )=0}$
• If there are two mutually singular probability measures , then there is a disjoint decomposition of the basic set. So it is and , so that and . Then σ-algebra is both -trivial and -trivial. Because of the elementary calculation rules for probabilities, and , the probabilities of the basic set and the empty set are again given by the definition of a probability measure. ${\displaystyle P_{1},P_{2}}$ ${\displaystyle \Omega =N_{1}\cup N_{2}}$ ${\displaystyle N_{1}\cap N_{2}=\emptyset }$ ${\displaystyle P_{1}(N_{1})=0}$ ${\displaystyle P_{2}(N_{2})=0}$ ${\displaystyle \{\Omega ,\emptyset ,N_{1},N_{2}\}}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{1}(N_{2})=1}$ ${\displaystyle P_{2}(N_{1})=1}$

Application examples

Usually the proof that a set system is P-trivial is not easy to use, so some of these statements have proper names. They are counted among the 0-1 laws because they make statements about which events will occur with probability 0 or 1. Classic examples are:

• The Kolmogorowsche zero-one law . It says that the terminal σ-algebra of a sequence of independent σ-algebras is P-trivial.
• The Hewitt-Savage zero-one law . It says that the interchangeable σ-algebra of a sequence of independently identically distributed random variables is P-trivial.

properties

On a probability space , a P-trivial σ-algebra is independent of any other system of sets . This can be derived from elementary calculation rules for probabilities. ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle {\mathcal {O}}\subset {\mathcal {A}}}$ ${\displaystyle {\mathcal {M}}\subset {\mathcal {A}}}$

An important conclusion from this is: If P is trivial, then the following applies for the conditional expectation , because and are independent of each other. This conclusion is used, for example, for the individual ergodic set and the L ^p -ergodic set . ${\displaystyle {\mathcal {O}}}$ ${\displaystyle \operatorname {E} (X|{\mathcal {O}})=\operatorname {E} (X)}$ ${\displaystyle \sigma (X)}$ ${\displaystyle {\mathcal {O}}}$

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 .