Let $(X,\mathfrak{M})$ be a measurable space.
Let $(\Omega,\mathscr{F},P),(\Omega',\mathscr{F}',P')$ be probability spaces and $G:(\Omega,\mathscr{F})\rightarrow (\mathbb{R}^\mathfrak{M}, \otimes_{A\in \mathfrak{M}} \mathscr{B}_{\mathbb{R}})$ be a measurable function and $H:(\Omega',\mathscr{F}')\rightarrow (\mathbb{R}^\mathfrak{M}, \otimes_{A\in \mathfrak{M}} \mathscr{B}_{\mathbb{R}})$ be a measurable function.
Suppose that $G_*P=H_*P'$ and $G(x),H(y)$ are probability measures on $(X,\mathfrak{M})$ for every $x,y$.
If we know that $G(x)$ is a discrete measure for almost every $x$, can we conclude that $H(x)$ is a discrete measure?