# How do I prove that a measure given in this situation is discrete?

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

Definition

Let $$G:\Omega\times \mathfrak{M}\rightarrow [0,\infty]$$ 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 $$X$$).