Consider the set $E$ of compactly supported Borel probability measures on the real line equipped with the Wasserstein infinity metric. A continuous function $f:E\to\mathbb R$ is called a U-statistic of degree $n$ if there's a measurable function $h:\mathbb R^n\to\mathbb R$ such that $$f(\mu)=\mathbb E[h(x_1, \dots, x_n)]$$ where $x_i\sim\mu$ are independent samples. For example, the second moment $f(\mu)=(\int xd\mu)^2$ is a U-statistic of degree $2$ with the function $h(x,y)=xy$ but it cannot be a U-statistic of degree $1$ because then we would have $h(x)=f(\delta_x)=x^2$ for every $x\in\mathbb R$ which yields a contradiction for $\mu=\frac 12\delta_a+\frac 12\delta_b$ where $a\neq b$.
The set of U-statistics of any degree seems to be quite general. Furthermore, U-statistics feel easier to work with than a general function of a probability measure. Is this set dense in the compact-open topology for the space of maps $E\to\mathbb R$?
