Theorem: For a probability space $(\Omega,A, \mathbb{P})$, a standard Borel space $(S, B)$ and a measurable map $X: \Omega \to S$ there exists a markov kernel $k$ s.t. $$\forall C \in B: P(X \in C | A_0)(\omega) = k(w, C)$$ Where $A_0$ is some sub-$\sigma$-algebra (I guess, the professor hasn't defined it)
I get that theorem and the proof. What I don't get is this apparantly directly following corollary:
Desintegration: Let $(S_1, B_1), (S_2, B_2)$ be standard Borel spaces, $S := S_1 \times S_2$ with the product-$\sigma$-field $Y$. Then for all prob. measures $\mu$ on $(S, Y)$ there exists a markov kernel $k$ from $S_1$ to $S_2$ s.t.
$$(*) \int_{S_1\times S_2} f(x,y) d\mu(x,y) = \int_{S_1}\int_{S_2}f(x,y)k(x, dy)d\mu_1$$
Where $\mu_1 = (\pi_1)_*\mu$, i.e $\mu_1(A) = \mu(A \times S_2)$
All it says is that this is a direct corollary from the prior theorem. I see that you can use $\pi_1$ as the measurable map, but equal to how I do it, I don't get why $(*)$ should hold. It would be a very great help if someone could try to proof the desintigration using the prior theorem as detailed as possible