σ-algebra of invariant events

The σ-algebra of invariant events is a special σ-algebra that is used in ergodic theory . There it is used, for example, to define the ergodicity or to formulate the individual ergodic set and the L ^p -ergodic set .

definition

Be a probability space and a measurable map . ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle T:\Omega \to \Omega }$

A is called an invariant event if is. ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle T^{-1}(A)=A}$

The set of all invariant events, so

${\displaystyle {\mathcal {I}}:=\{A\in {\mathcal {A}}\,|\,T^{-1}(A)=A\}}$,

is then called the σ-algebra of the invariant events .

properties

• The fact that there is actually a σ-algebra follows directly from the compatibility of the archetype operation with the set operations. ${\displaystyle {\mathcal {I}}}$
• A function from to is -measurable precisely when it -measurable and applies. ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {I}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle f\circ T=f}$

Quasi-invariant events

A weakening of the concept of an invariant event is a quasi-invariant event . Equality is only almost certainly required. Accordingly, a is called quasi-invariant if ${\displaystyle A\in {\mathcal {A}}}$

${\displaystyle \chi _{A}=\chi _{T^{-1}(A)}\quad P{\text{-fast sicher}}}$

applies. The quasi-invariant events also form a σ-algebra for dimension-preserving maps ; it is given by ${\displaystyle T}$

${\displaystyle {\mathcal {I}}_{P}:=\{A\in {\mathcal {A}}\,|\,\chi _{A}=\chi _{T^{-1}(A)}\ P{\text{-fast sicher}}\}}$.

In fact, distinguish the quasi-invariant events and the events invariant not, because it can be shown that for every one there, so is. An invariant set can therefore always be found for every quasi-invariant set, so that these only differ on a zero set. ${\displaystyle A\in {\mathcal {I}}_{P}}$ ${\displaystyle B\in {\mathcal {I}}}$ ${\displaystyle P(A\,\triangle \,B)=0}$

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 .
• Manfred Einsiedler, Klaus Schmidt: Dynamic Systems . Ergodic theory and topological dynamics. Springer, Basel 2014, ISBN 978-3-0348-0633-6 , doi : 10.1007 / 978-3-0348-0634-3 .