Let be a compact metric space, be a homeomorphism preserving some probability measure . The stable set is defined by the formula

For convenience denote the collection of null sets as .

[**Lemma 6.3.2.** Brin M., Stuck G. *Introduction to dynamical systems*]

Let be an -invariant measurable function. Then there exists a subset such that for each , is constant on .

*Proof.* Pick a sequence of continuous function on such that

both and .

By Birkhoff Ergodic Theorem, the limit exists for So there exists such that the limit exists for each . Pick such a point . Then for each , the limit also exists at and has the same value as at . (both are stable-saturated)

By the invariance of and , we have

By Dominated Convergent Theorem, Passing to a subsequence if necessary, we assume that . So there exists such that for each the limit exists and equals to

Now let . Then and for each point , each point , for each and

1. the limits exist (hence are equal): .

2. they equal to respectively: .

So we have for each and .

A more interesting application:

**Corollary 1. **Let , and be an invariant subset. Then there exists a null set such that for each , . (equivalently, )

*Proof. *The characteristic function is invariant and measurable. Let be given by Lemma. Then if , we have for each . So .

Similarly result holds for unstable sets. The results can be imporved if the stable/unstable foliations are absolutely continuous. This is the case for nonuniformly hyperbolic invariant sets.

**Corollary 2.** Let for some as above. Let be an invariant nonuniform hyperbolic subset. Then there exists a null set such that for each , and .

Equivalently, we have for -a.e. , and .