Define a coupling $\pi\in \Pi(\mu,\nu)$ on the product space $(X\times X,\mathcal{F}\times\mathcal{F})$. let $\pi_x$ be the disintegration of $\pi$ with respect to the $\mu$, i.e. there exists a Borel measurable function $x\mapsto\pi_x$ such that $$\pi(dx,dy)=\mu(dx)K(x,dy)$$ where $K(\cdot,\cdot)$ is a probability transition kernel.

If $\pi_x=\delta_{f(x)}$ ($\delta$ is the Dirac measure), can we say that there exists a measurable function $f$ such that $$ \nu=f_{\#}\mu $$ where $f_{\#}\mu$ is the pushforward of measure $\mu$, i.e. $\mu(f^{-1}(A))=\nu(A)$ for any Borel measurable sets $A\subset X$.