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

edit

Strictly invariant sets

edit

Let   be a measurable space, and let   be a measurable function. A measurable subset   is called invariant if and only if  .[1][2][3] Equivalently, if for every  , we have that   if and only if  .

More generally, let   be a group or a monoid, let   be a monoid action, and denote the action of   on   by  . A subset   is  -invariant if for every  ,  .

Almost surely invariant sets

edit

Let   be a measurable space, and let   be a measurable function. A measurable subset (event)   is called almost surely invariant if and only if its indicator function   is almost surely equal to the indicator function  .[4][5][3]

Similarly, given a measure-preserving Markov kernel  , we call an event   almost surely invariant if and only if   for almost all  .

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

edit

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 events, both in the strict case and in the almost sure case, depending on the author.[1][2][3][4][5] For the purpose of the article, let's denote by   the sigma-algebra of strictly invariant sets, and by   the sigma-algebra of almost surely invariant sets.

Properties

edit
  • Given a measure-preserving function  , a set   is almost surely invariant if and only if there exists a strictly invariant set   such that  .[6][5]
  • Given measurable functions   and  , we have that   is invariant, meaning that  , if and only if it is  -measurable.[2][3][5] The same is true replacing   with any measurable space where the sigma-algebra separates points.
  • An invariant measure   is (by definition) ergodic if and only if for every invariant subset  ,   or  .[1][3][5][7][8]

Examples

edit

Exchangeable sigma-algebra

edit

Given a measurable space  , denote by   be the countable cartesian power of  , equipped with the product sigma-algebra. We can view   as the space of infinite sequences of elements of  ,

 

Consider now the group   of finite permutations of  , i.e. bijections   such that   only for finitely many  . Each finite permutation   acts measurably on   by permuting the components, and so we have an action of the countable group   on  .

An invariant event for this sigma-algebra is often called an exchangeable event or symmetric event, and the sigma-algebra of invariant events is often called the exchangeable sigma-algebra. A random variable on   is exchangeable (i.e. permutation-invariant) if and only if it is measurable for the exchangeable sigma-algebra.

The exchangeable sigma-algebra plays a role in the Hewitt-Savage zero-one law, which can be equivalently stated by saying that for every probability measure   on  , the product measure   on   assigns to each exchangeable event probability either zero or one.[9] Equivalently, for the measure  , every exchangeable random variable on   is almost surely constant.

It also plays a role in the de Finetti theorem.[9]

Shift-invariant sigma-algebra

edit

As in the example above, given a measurable space  , consider the countably infinite cartesian product  . Consider now the shift map   given by mapping   to  . An invariant event for this sigma-algebra is called a shift-invariant event, and the resulting sigma-algebra is sometimes called the shift-invariant sigma-algebra.

This sigma-algebra is related to the one of tail events, which is given by the following intersection,

 

where   is the sigma-algebra induced on   by the projection on the  -th component  .

Every shift-invariant event is a tail event, but the converse is not true.

See also

edit

Citations

edit

References

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