Let $(\Omega,\mathscr{F},P)$ be a probability space and $(X,\mathfrak{M})$ be a measurable space.

DefinitionLet $G:\Omega\times \mathfrak{M}\rightarrow [0,\infty]$ be a function such that $G_x:\mathfrak{M}\rightarrow [0,\infty]$ is a probability measure for all $x\in\Omega$ and $G^A:(\Omega,\mathscr{F})\rightarrow [0,\infty]$ is measuable for all $A\in\mathfrak{M}$.

Then, we say $G$ is a

Markov kernel.

Let $G,H$ be markov kernels such that $(G^{A_1},...,G^{A_n}) = (H^{A_1},...,H^{A_n})$ in distribution for all finite sequence $A_i$'s.

If $G_x$ is discrete for $P$-almost every $x$, then how do I prove that $H_x$ is discrete for $P$-almost every $x$?

(That $G_x$ is discrete means that there exists a function $f_x$ such that $G_x(A)=\int_A f_x d\mu_c$ for all $A\in\mathfrak{M}$, where $\mu_c$ is the counting measure on $X$).