nLab invariant set

Idea

An invariant set is a subset which is invariant under a given action of a group or a monoid.

Definition

Let $X$ be a set, let $M$ be a monoid (or a group), and consider an action of $M$ on $X$. (For each $m\in M$, denote the resulting action on $X$ again by $m:X\to X$.)

A subset $S$ of $X$ is called invariant if and only if for each $m\in M$, $m^{-1}(S)=S$. Equivalently, if for every $x\in X$ we have that $x\in S$ if and only if $m(x)\in S$.

Note that the elements (“points”) of an invariant set need not themselves be fixed points: they might move under the action (but they remain in the set $S$). For example, for rotations in the plane, any circle centered at the origin is an invariant set, but each point in each nontrivial circle is not a fixed point.

Variations of the definition

Since the original definition was given for the case of group actions, which are invertible, there are a number of variants of the same definition which differ for the case of arbitrary monoids. In particular, sometimes one calls a subset $S\subseteq X$ invariant if and only if $m^{-1}(S)\subseteq S$, i.e. if for every $x\in S$, $m(x)\in S$ as well. (Without requiring that if $m(x)\in S$, then $x\in S$.) In other words, it is a set “which we cannot leave” under the specified action. Sometimes such a set is called an absorbing set instead. (But that term may itself be used for other concepts, to make matters even worse.)

Also, in different categories than Set, similar variations of the notion of invariant set are given:

• In topology, given a continuous action, one is often interested in invariant closed sets;
• In probability theory, one is often interested in invariant measurable sets;
• In differential geometry, one is often interested in invariant submanifolds,

…and so on.

Examples

• Each orbit of a group action is an invariant set. Moreover, a set is invariant if and only if it is a (possibly empty) union of orbits.

• In probability theory, an exchangeable event is a measurable subset which is invariant under finite permutations, and a tail event is invariant under shifts.

Invariant sigma-algebras

Let $X$ be a measurable set. Let $M$ be a monoid (or group), and consider an action of $M$ on $X$ via measurable functions $m:X\to X$.

A measurable subset $A\subseteq X$ is called invariant if and only if for every $m\in M$, $m^{-1}(A)=A$.

Invariant sets form naturally a sigma-algebra, often called the invariant sigma-algebra.

The exchangeable sigma-algebra and the tail sigma-algebra are examples of this. (Sometimes one uses the almost sure versions, see below.)

The invariant sigma-algebra gives a colimit of the action of $M$ in the category of Markov kernels (MP’23). More precisely, denote by $X_inv$ the set $X$ together with the invariant sigma-algebra, and denote by $q:X\to X_inv$ the kernel induced by the set-theoretic identity. Then the $(X_inv,q)$ forms a universal cocone (i.e. colimit), meaning that for every kernel $k:X\to Y$ satisfying $k\circ m=k$, (i.e. right-invariant under the action of $m$), there exists a unique kernel $u:X_inv\to Y$ making the following diagram commute, where $\delta_m$ is the kernel induced by $m$. The kernel $u$ has the same entries as $k$: by invariance, $k$ is indeed measurable for the sigma-algebra of invariant sets.

Note that $X_inv$ is not the colimit simply in the category of measurable functions: the colimit in that case also takes a set-theoretic quotient. It is however a colimit also in the category of zero-one kernel.

In the category of kernels, the set-theoretic quotient and $X_inv$ are isomorphic. This can be seen from the fact that Markov kernels are Kleisli morphisms of the Giry monad on Meas, and that left adjoints preserve colimits. The same can be said about zero-one kernels and the sobrification monad.

For stochastic actions

A monoid $M$ can also act stochastically on $X$, via Markov kernels $k_m:X\to X$ such that for all $x\in X$ and all measurable $A\subseteq X$,

$$k_m(A|x) \;=\; 1_A(x) \;=\; \begin{cases} 1 & x\in A ; \\ 0 & x\notin A , \end{cases}$$

and

$$k_{m\cdot n}(A|x) \;=\; \int_X k_m(A|x')\,k_n(d x'|x) .$$

(This is equivalently a functor $B M\to Stoch$, see the discussion at invariant measure.)

In this case, an invariant set is a measurable set $A$ such that for all $m\in M$ and $x\in X$,

$$k(A|x) \;=\; 1_A(x) \;=\; \begin{cases} 1 & x\in A ; \\ 0 & x\notin A . \end{cases}$$

These sets form again a sigma-algebra, generalizing the case of actions via functions.

This sigma-algebra is in general not a colimit in a category of kernels (EP’23, Example 3.9). See however the almost sure case below.

Almost surely invariant sets

For stationary actions, i.e. for actions of $M$ on a probability space $(X,p)$ via measure-preserving functions, (or equivalently, for measurable actions equipped with an invariant measure $p$, we can define a notion of almost surely invariant set: a measurable subset $A\subseteq X$ such that

$$p(A\setminus m^{-1}(A)) \;=\; p(m^{-1}(A)\setminus A) \;=\; 0 .$$

Equivalently, if for all measurable $S\subseteq X$,

$$p(S\cap m^{-1}(A)) \;=\; p(S\cap A) .$$

More generally, if we have measure-preserving kernels $k_m:(X,p)\to(X,p)$, we call almost surely invariant a measurable subset $A\subseteq X$ such that for all $m\in M$,

$$k_m(A|x) \;=\; 1_A(x) \;=\; \begin{cases} 1 & x\in A ; \\ 0 & x\notin A \end{cases}$$

for $p$-almost all $x\in X$. Equivalently, if for all measurable $S\subseteq X$,

$$\int_S k_m(A|x) \,p(d x) \;=\; p(S\cap A) .$$

These sets form again a sigma-algebra, called the almost surely invariant sigma-algebra (or sometimes, possibly confusingly, again invariant sigma-algebra).

Similarly to the deterministic case, this sigma-algebra is the colimit of the action of $M$ in the category of couplings (EP’23, Section 3.1).

Further properties

• By definition, every ergodic measure is zero-one on invariant sets.
• The ergodic theorem? says that, in some cases, empirical means of random variables on a stationary system tend to conditional expectations w.r.t. the invariant sigma-algebra.

See also

• group, monoid, action

• cone, limit, cocone, colimit

• exchangeability, invariant measure, tail event

• ergodic measure, ergodic decomposition theorem

• Wikipedia, Invariant sigma-algebra

References

• Tobias Fritz, Tomáš Gonda, Paolo Perrone, De Finetti’s theorem in categorical probability. Journal of Stochastic Analysis, 2021. (arXiv:2105.02639)

• Sean Moss, Paolo Perrone, A category-theoretic proof of the ergodic decomposition theorem, Ergodic Theory and Dynamical Systems, 2023. (arXiv:2207.07353)

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

category: probability