invariant by a measure-preserving transformation

invariantpropertiesproperty

Let $X$ be a set and $T:X⟶X$ a transformation of $X$.

The notion of invariance by $T$ we are about to describe is stronger than the usual notion of invariance (http://planetmath.org/invariant), and is especially useful in ergodic theory. Thus, in most applications, $(X,𝔅,\mu )$ is a measure space and $T$ is a measure-preserving transformation. Nevertheless, the definition of invariance and its properties are general and do not require any such assumptions.

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Definition - A subset $A\subseteq X$ is said to be invariant by $T$, or $T$-invariant, if $T^{-1}(A)=A$.

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The fundamental property of this concept is the following: if $A$ is invariant by $T$, then so is $X\setminus A$.

Thus, when $A$ is invariant by $T$ we obtain by restriction two well-defined transformations

${T|}_A:A⟶A$

${T|}_{X\setminus A}:X\setminus A⟶X\setminus A$

Hence, the existence of an allows one to decompose the set $X$ into two disjoint subsets and study the transformation $T$ in each of these subsets.

Remark - When $T$ is a measure-preserving transformation in a measure space $(X,𝔅,\mu )$ one usually restricts the notion of invariance to measurable subsets $A\in 𝔅$.