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).

continousfunction $f$ from the first statement come into play. $\endgroup$