---
tag: 21RJ8035KP
kind: statement
role: theorem
name: Ergodic decomposition
msc: [37A05, 28D05, 37A25]
uses: [0VKD1TP7F3, 0TVJZYB6SH, DYR1QFNGNJ, 059TJN0NZ1, B232EZ1FMB, AZRPHS478B, 3MSGFJM2NX, MBW9JCGZC1, J6NMBX3E55, NZZE9KCAD8, 1DHZ81S0YE]
about: [AZRPHS478B, B232EZ1FMB]
sources:
  - url: https://en.wikipedia.org/wiki/Ergodicity
---

## Statement

Let $X$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $T \colon X \to X$ be a continuous map.
Let $M(X, T)$ be the set of $T$-[[invariant|B232EZ1FMB]] [[Borel|DYR1QFNGNJ]] [[probability measures|059TJN0NZ1]] on $X$, with the topology of [[weak convergence of measures|MBW9JCGZC1]].
Let $E(X, T) \subseteq M(X, T)$ be the subset of measures for which $T$ is [[ergodic|AZRPHS478B]].

1. $M(X, T)$ is a convex compact set, and $E(X, T)$ is exactly the set of its [[extreme points|3MSGFJM2NX]].
2. For every $\mu \in M(X, T)$ there is a unique Borel probability measure $\tau$ on $M(X, T)$ with $\tau(E(X, T)) = 1$ and
$$\mu(A) = \int_{E(X, T)} \nu(A) \, d\tau(\nu) \quad \text{for every Borel set } A \subseteq X.$$

## Remarks

For part 1, suppose $\mu$ is not ergodic, and let $A$ be an invariant set with $0 < \mu(A) < 1$.
Then $\mu$ is the convex combination $\mu(A) \mu_A + (1 - \mu(A)) \mu_{X \setminus A}$ of the normalized restrictions, which are distinct invariant measures, so $\mu$ is not extreme.
Conversely, if $\mu$ is ergodic and $\mu = t \nu_1 + (1-t) \nu_2$ with $0 < t < 1$, then $\nu_1$ is absolutely continuous with respect to $\mu$ with an invariant density, which ergodicity forces to be constant, so $\nu_1 = \mu$.
Part 2 follows from Choquet theory, since $M(X, T)$ is a compact convex metrizable set whose barycentric decompositions are unique.
The decomposition can also be built from the conditional measures of $\mu$ on the σ-algebra of invariant sets.
Example: for the identity map, every Borel probability measure is invariant, and the ergodic ones are the point masses $\delta_x$.
The decomposition of $\mu$ is $\mu = \int \delta_x \, d\mu(x)$, that is, $\tau$ is the image of $\mu$ under $x \mapsto \delta_x$.
Example: for the rotation of the circle by a quarter turn, Lebesgue measure is the average over $x \in [0, 1/4)$ of the uniform measures on the four-point orbits of $x$.

## Explanations

### L1 human

Every invariant measure can be split, in a unique way, into ergodic pieces.
Each piece is a measure that cannot be split further, and the original measure is a weighted average of the pieces.
So to understand all invariant measures, it is enough to understand the ergodic ones.

### L2 human

Many theorems are first proved for ergodic systems, where time averages are constant, and then extended to all systems by the ergodic decomposition.
For example, [[Birkhoff's ergodic theorem|J6NMBX3E55]] for a non-ergodic measure gives a limit that is constant on each ergodic component.
Two distinct ergodic measures are mutually singular.
By part 1, the unique invariant measure of a [[uniquely ergodic|NZZE9KCAD8]] map is ergodic.
That $M(X, T)$ is not empty is the [[Krylov–Bogolyubov theorem|1DHZ81S0YE]].