# Two ergodic decomposition theorems

Let $$X$$ be a compact metric space and $$T\colon X \to X$$ a continuous map. Additionally, let $$\mathcal{M}^T(X)$$ be the set of $$T$$-invariant probability measures on $$X$$ and $$\mathcal{E}^T(X)$$ the set of $$T$$-ergodic probability measures. Ergodic decomposition theorem claims that for every $$\mu \in \mathcal{M}^T(X)$$ there is a probability measure $$\lambda$$ on $$\mathcal{M}^T(X)$$ such that $$\lambda$$ is supported on $$\mathcal{E}^T(X)$$ and $$\int_X f \ d\mu = \int_{\mathcal{E}^T(X)} \Bigl( \int_X f \ d\nu \Bigr) \ d\lambda(\nu) ~\text{for every}~f \in C(X).$$ This is usually proved via Choquet's theorem.

However, one can also find in literature a related statement (also known as ergodic decomposition theorem) claiming that there is a map $$\beta \colon X \to \mathcal{E}^T(X)$$ such that $$\mu(A) = \int_X \beta_x(A) \ d\mu(x) ~\text{for every measurable}~A \subset X.$$ I am wondering if there is a proof of this statement directly from the first one (i.e. which avoids the measure disintegration machinery one usually employs to prove it).

• The second statement follows from the first by writing $\beta_{X}(A)$ explicitly as conditional expectation over smallest the $T$-invariant sigma algebra and noticing the $T$-invariant sigma algebra will be the supports of the various $T$-ergodic measures... Notice that your inner integral in the first equation is a bit misleading, as one actually only integrates over supp($\nu$), which gives you the conditional expectation...
– Asaf
Apr 13 at 19:19
• Hm, could you provide few more details (or maybe a reference)? For example, it is not clear to where continous function $f$ from the first statement come into play. Apr 14 at 8:08
• The continuous function assumption is only to ensure measurability wrt any kind of Borel measure. That's the most natural choice of functions to consider. You can read about conditional measures in manfred's book in the GTM series (probably the standard ergodic theory book nowadays) or also a very precise treatment in the notes of manfred and elon in the Pisa proceedings (Clay institute). There's also a treatment in this tiny book of Dan Rudolph.
– Asaf
Apr 14 at 13:46
• I will add on obvious remark - if $A$ is Borel (and one should take a tiny bit of care of the sigma algebras here and the measure theory which is involved), then one may approximate it with continuous functions, in the standard manner...
– Asaf
Apr 14 at 18:01