Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measure on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of the map on $\Delta_n \times \mathbb{R}^n$, where $\Delta_n$ is the $n$-simples, taking $(k_1,\dots,k_n)\times (x_1,\dots,x_n)$ to $\sum_{i=1}^n k_j \delta_{x_i}$. Clearly this map is continuous, when $\mathcal{P}_n(\mathbb{R})$ is equipped with the Prokhorov metric.

However, is it clear that it admits a continuous selection? Ie.: a continuous right inverse (definetly not unique ofcourse)?

Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of the map on $\Delta_n \times \mathbb{R}^n$, where $\Delta_n$ is the $n$-simplex, taking $(k_1,\dots,k_n)\times (x_1,\dots,x_n)$ to $\sum_{i=1}^n k_j \delta_{x_i}$. Clearly this map is continuous, when $\mathcal{P}_n(\mathbb{R})$ is equipped with the Prokhorov metric.

However, is it clear that it admits a continuous selection? Ie.: a continuous right inverse (definitely not unique of course)?