Let be a compact metric space and be a homeomorphism. Let be the set of -invariant measures and be the set of -invariant ergodic measures.

Let be an invariant measure. A distribution on is said to be the ergodic decomposition of if for each continuous function , the following holds:

.

The following approach is attributed to R. Mane.

Let be the set of generic points of and . Then is a Borel subset of and of full probability. Consider the map .

Proposition: The map is Borel.

So the pushforwad is a Borel distribution on . Moreover for each continuous map , we have

.

In particular letting , we have and

, where is the Birkhoff average and the last equality follows from Brikhoff ergodic theorem. So is the ergodic decomposition of .

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