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 $T^{-1}(S)=S$. Equivalently, if for every $x\in X$, we have that $x\in S$ if and only if $T(x)\in S$.

More generally, let $M$ be a group or a monoid, let $\alpha :M\times X\to X$ be a monoid action, and denote the action of $m\in M$ on $X$ by $\alpha _{m}:X\to X$. A subset $S\subseteq X$ is $\alpha$-invariant if for every $m\in M$, $\alpha _{m}^{-1}(S)=S$.

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 $1_{T^{-1}(S)}$.

Similarly, given a measure-preserving Markov kernel $k:(X,{\mathcal {F}},p)\to (X,{\mathcal {F}},p)$, we call a set $S\in {\mathcal {F}}$ almost surely invariant if and only if $k(S\mid x)=1_{S}(x)$ for almost all $x\in X$.

As for the case of strictly invariant sets, the definition can be extended to an arbitrary group or monoid action.

In many cases, almost surely invariant sets differ by invariant sets only by a null set (see below).

Sigma-algebra structure

Both strictly invariant sets and almost surely invariant sets are closed under taking countable unions and complements, and hence they form sigma-algebras. These sigma-algebras are usually called either the invariant sigma-algebra or the sigma-algebra of invariant sets, both in the strict case and in the almost sure case, depending on the author. For the purpose of the article, let's denote by ${\mathcal {I}}$ the sigma-algebra of strictly invariant sets, and by ${\tilde {\mathcal {I}}}$ the sigma-algebra of almost surely invariant sets.

Properties

• Given a measure-preserving function $T:(X,{\mathcal {A}},p)\to (X,{\mathcal {A}},p)$, a set $A\in {\mathcal {A}}$ is almost surely invariant if and only if there exists a strictly invariant set $A'\in {\mathcal {I}}$ such that $p(A\triangle A')=0$.

• Given measurable functions $T:(X,{\mathcal {A}})\to (X,{\mathcal {A}})$ and $f:(X,{\mathcal {A}})\to (\mathbb {R} ,{\mathcal {B}})$, we have that $f$ is invariant, meaning that $f\circ T=f$, if and only if it is ${\mathcal {I}}$-measurable. The same is true replacing $(\mathbb {R} ,{\mathcal {B}})$ with any measurable space where the sigma-algebra separates points.

• An invariant measure $p$ is (by definition) 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
• Invariant measure
• 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.