Let $\mathcal{M}_f(\mathbb{N})$ the space of finite measures on $\mathbb{N}$, ie $\mu = \sum_i a_i \delta_i$ with $a_i\in \mathbb{N}, \sum_i a_i = M < \infty$. Let $\mathcal{M}_1(\mathbb{N})$ be the space of probability measures on $\mathbb{N}$. I'm considering the following functionals : \begin{align*} F \colon \mathcal{M}_f(\mathbb{N}) &\to \mathcal{M}_1(\mathbb{N})\\ \mu &\mapsto F(\mu):= \sum_i F_i(\mu)\delta_i \end{align*} Again $F_i>0$, $\sum F_i = 1$. And we add the condition that for any subset $A$ of $N$ $F(\mu)(A) = 0 \iff \mu(A) = 0$.
My question is the following : Is there a metric subspace $(S,d_S)$ of $ \mathcal{M}_f(\mathbb{N}) $ and again a metric subspace $(S',d_{S'})$ of $\mathcal{M}_1(\mathbb{N})$ such that the restriction of $F$ over $S$ is lipschitz over $S'$ that is:
\begin{equation} d_{S'}(F(\mu),F(\nu)) \leq C d_{S}(\mu,\nu) \end{equation} What I am asking is: Do I need to add supplementary restrictions of $\mu$ and $F(\cdot)$ for $F$ to be lipschitz?
Any suggestions/ideas would be welcome.
Context: I have been working on a measure valued EDO where a term popped up with $F$ defined like. I am trying to figure out uniqueness conditions for the solutions so I need $F$ to be somehow Lipschitz. It to be working without any further restrictions when $\mu$ is square integrable ($\sum i^2 a_i < \infty$), but I don't know how to approach the proof. I've tried the hands on, calculatory approach but it hasnt worked so far...