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

The notion of invariance by $T$ we are about to describe is stronger than the usual notion of invariance, and is especially useful in ergodic theory. Thus, in most applications, $(X, \mathfrak{B}, \mu)$ 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

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