P}(U).$$ From , we have that, for $\theta=1/\delta_R$, $\lambda+\theta {\mathcal T}^*(x,\Psi)\lambda$ is a positive measure. Since by $\langle {\mathcal T}^*(x,\Psi)\lambda, 1\rangle=\langle \lambda, {\mathcal T}(x,\Psi)1\rangle=0$, we get and we conclude the proof.

The discrete problem and statement of the main result {#section:discrete}
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We consider a particle system of $N$ agents evolving according to $$\label{T130}
\begin{cases}
\dot x_t^i = v_{\Lambda_t^N}(x_t^i,\lambda_t^i), \\
\dot \lambda_t^i = {\mathcal T}^*(x_t^i,\Lambda_t^N)\lambda_t^i,
\end{cases}
\qquad\text{for $i=1,\ldots,N$, $t\in[0,T]$,}$$ where $x^i\in{\mathbb{R}^{d}}$, $\lambda^i\in{\mathcal P}(U)$ for each $i\in\{1,\ldots,N\}$, and $$\label{T131}
\Lambda_t^N\coloneqq \frac1N\sum_{i=1}^N \delta_{(x_t^i,\lambda_t^i)}$$ is the empirical measure associated with the system. Recalling the definition of $b$ in , the evolution can be written in compact form as $$\label{T132}
y_t^i=b_{\Lambda_t^N}(y_t^i), \qquad\text{for $i=1,\ldots,N$, $t\in[0,T]$.}$$ We first discuss the well posedness of system for every choice of an initial datum $\bar y^i=(\bar x^i,\bar \lambda^i)$, for $i=1,\ldots,N$.

\[T133\] Assume that for every $y\in Y$ and $\Psi\in{\mathcal P}_1(Y)$ the velocity $v_\Psi\colon Y\to{\mathbb{R}^{d}}$ satisfies (v1)-(v3) and the operator ${\mathcal T}(x,\Psi)\colon {\operatorname{Lip}}(U)\to{\operatorname{Lip}}(U)$ satisfies (T0)-(T3). Then, for every choice of $\bar y^i\in Y$, $i=1,\ldots,N$, the system has a unique solution.

We introduce the vector-valued variable ${\boldsymbol{y}}\coloneqq(y^1,\ldots,y^N)\in Y^N\subset\overline Y^N$, which we endow with the norm $$\label{T136}
\lVert {\boldsymbol{y}}\rVert_{\overline Y^N}\coloneqq \frac1N\sum_{i=1}^N \lVert y^i\rVert_{\overline Y},$$ and the associated empirical measure $\Lambda^N\coloneqq\frac1N \sum_{i=1}^N \delta_{y^i}$, 