Invariant sigma-algebra

In mathematics, especially in probability theory and ergodic theory, the invariant sigma-algebra is a sigma-algebra formed by sets which are invariant under a group action or dynamical system. It can be interpreted as of being "indifferent" to the dynamics.

The invariant sigma-algebra appears in the study of ergodic systems, as well as in theorems of probability theory such as de Finetti's theorem and the Hewitt-Savage law.

Definition

Strictly invariant sets

Let $(X,{\mathcal {F}})$ be a measurable space, and let $T:(X,{\mathcal {F}})\to (X,{\mathcal {F}})$ be a measurable function. A measurable subset $S\in {\mathcal {F}}$ is called invariant if and only if $f^{-1}(S)=S$.

Almost surely invariant sets

Let $(X,{\mathcal {F}})$ be a measurable space, and let $T:(X,{\mathcal {F}})\to (X,{\mathcal {F}})$ be a measurable function. A measurable subset (event) $S\in {\mathcal {F}}$ is called almost surely invariant if and only if its indicator function $1_{S}$ is almost surely equal to the indicator function $1_{f^{-1}(S)}$.

Properties

• An invariant measure $p$ is ergodic if and only if for every invariant subset $A\in {\mathcal {I}}$, $p(A)=0$ or $p(A)=1$.

See also

• Invariant set
• De Finetti theorem
• Hewitt-Savage zero-one law
• Kolmogorov zero-one law
• Exchangeable random variables
• Ergodic system

References

• Viana, Marcelo; Oliveira, Krerley (2016). Foundations of Ergodic Theory. Cambridge University Press. ISBN 978-1-107-12696-1.

• Billingsley, Patrick (1995). Probability and Measure. John Wiley & Sons. ISBN 0-471-00710-2.

• Durrett, Rick (2010). Probability: theory and examples. Cambridge University Press. ISBN 978-0-521-76539-8.

• Douc, Randal; Moulines, Eric; Priouret, Pierre; Soulier, Philippe (2018). Markov Chains. Springer. ISBN 978-3-319-97703-4.

• Klenke, Achim (2020). Probability Theory: A comprehensive course. Springer. ISBN 978-3-030-56401-8.