we show item [(i)]{}. We write $\Theta=\Theta^P+\tilde{\Theta}$ and we observe that, in view of the balance equation , $\tilde\Theta(\zeta)=0$ if $\zeta(t,x,v,v')=z(t,x,v')-z(t,x,v)$ for some function $z$. By choosing in the variational formula $\zeta=z'-z$, where $z=z(t,x,v)$ and $z'=z(t,x,v')$, we get $$\begin{split}
    {{\mathcal R}} (P,\Theta^P +\tilde\Theta)
    \geq &  \sup_{z,\alpha} \bigg\{
    \Theta^P (z'-z) - 
    \int_0^T\!{\mathop{}\!\mathrm{d}}t \iiint \! P_t({\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v') \, 
    \sigma(v,v') 
    \\ & \qquad \qquad
    \times \big[ {\mathop{\rm ch}\nolimits}(z'-z) -1 \big] \big[ \alpha(t,x,v,v') +
    \alpha(t,x,v',v)^{-1} \big]
    \bigg\}\\
    = & {{\mathcal R}} (P,\Theta^P),
\end{split}$$ where the last equality follows by direct computation. We conclude by uniqueness.

We next prove item [(ii)]{}. Remark \[t:rem\] implies that $$\label{apb}
    \varlimsup_{n\to \infty} \, \sup_{t\in [0,T]} \, 
    {{\mathcal H}}(P_t^n) \le {{\mathcal H}}(Q).$$ In view of the lower semicontinuity of ${{\mathcal H}}$ and the observation after Lemma \[t:lem\] regarding ${{\mathcal E}}$ and ${{\mathcal R}}$, using the uniqueness it is enough to show precompactness of the sequence $\{(P^n,\Theta^n)\}\subset {{\mathcal S}}_\mathrm{be}$. Observe indeed that, by Assumption \[t:asb\] and Lemma \[t:lem\], we can take the limit $n\to \infty$ in the balance equation .

To prove precompactness of $\{\Theta^n\}$, observe that from and the variational representation it follows $$\sup_n\, \sup_{\zeta\colon \|\zeta\|_\infty\le 1} 
    \Theta^n(\zeta) < +\infty$$ and we conclude by the Banach-Alaoglu theorem.

The bound implies, by 