Let $X$ be a compact metric space, $T:X \to X$ continuous, $M_T(X)$ the set of borel measure that are $T$-invariant and $E_T(X)\subseteq M_T(X)$ the set of ergodic measures. The ergodic decomposition theorem tell us that for every $\mu \in M_T(X)$ exists a probability measure $\tau : \mathcal{B}(M_T(X))\to [0,1]$ such that for every continuos function $f \in C(X)$ $$\int_X f d\mu = \int_{E_T(X)} \left( \int_X f d\nu \right) d\tau(\nu)$$ The latter can be extend to every measurable function $f$.

Let now $F: M_T(X)\to \mathbb{R}$ be a $\mathcal{B}(M_T(X))$-measurable function.

It is true that $F(\mu)= \int_{E_T(X)} F(\nu)d\tau(\nu)$ ?

An interesting case is when $F(\nu)=h_\nu(T)$ but I would like to know the general case as well.

Thanks!