Motivation: Let $X$ be a continuous real random variable and let $Q$ be a continuous distribution. Does there exist $f$ such that $f(X)\overset{D}{=}Q$? that is, transformation of $X$ has distribution $Q$? Answer: Yes, if we define $f$ as an appropriate quantile function.
My question: Let $G$ be a space of all simple counting measures on $\mathbb{R}^d$ and let $\mathcal{G}$ be a sigma-algebra on $G$.
Consider a random variable $X$ with a values in $G$; that is, $X:\Omega\to G$.
Consider a (different) distribution $Q$ on $G$. $$\text{Does there exist a function }f:G\to G \text{ such that } f(X)\overset{D}{=}Q?$$ We can assume some 'nice' properties of $X$ and $Q$ if that helps (such as non-trivial distributions etc).