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.
be a measurable space, and let
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
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).
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.
be the countable cartesian power of
, equipped with the product sigma-algebra.
as the space of infinite sequences of elements of
by permuting the components, and so we have an action of the countable group
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.
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
, the product measure
assigns to each exchangeable event probability either zero or one.
, every exchangeable random variable on
It also plays a role in the de Finetti theorem.
, consider the countably infinite cartesian product
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
Every shift-invariant event is a tail event, but the converse is not true.