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Let be a set and
a transformation of .
The notion of invariance by we are about to describe is stronger than the usual notion of invariance, and is especially useful in ergodic theory. Thus, in most applications,
is a measure space and is a measure-preserving transformation. Nevertheless, the definition of invariance and its properties are general and do not require any such assumptions.

Definition - A subset
is said to be invariant by , or -invariant, if
.

The fundamental property of this concept is the following: if is invariant by , then so is
.
Thus, when is invariant by we obtain by restriction two well-defined transformations
Hence, the existence of an invariant subset allows one to decompose the set into two disjoint subsets and study the transformation in each of these subsets.
Remark - When is a measure-preserving transformation in a measure space
one usually restricts the notion of invariance to measurable subsets
.
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