I try to understand the disintegration of measures by some trivial examples. From Chapter 6 in https://link.springer.com/book/10.1007/978-0-387-87859-1, the disintegration theorem is that the product measure $\pi$ on $(X\times X, \mathcal{F}\times \mathcal{F})$ has a $\mu$-a.s. unique disintegration w.r.t the first coordinate, which can be broken in the form of $$\pi(dx,dy)=\mu(dx)K(x,dy)$$ where $\mu$ is the probability measure on $X$ and $K(\cdot,\cdot)$ is the transition probability kernel from $X$ to $X$. We sometimes write $K(x,dy)=\pi_x(dy)$ represents the Borel measurable function $x\mapsto \pi_x$.
In my example, if we take $\pi=Ber(p_1)\times Ber(p_2)$ (product measure of two Bernoulli distribution) and then $\mu\sim Bernouli(p_1)$ and $\nu\sim Bernoulli(p_2)$. I want to get what is $\pi_x(dy)=K(x, dy)$ from the disintegration theorem w.r.t. the first coordinate for $x=0,1$.