equation with initial condition $Q^n$. Then the sequence $\{(P^n,\Theta^n)\}$ converges to $(P,\Theta)$.

-   Fix $Q\in {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ with ${{\mathcal H}}(Q) <+\infty$ and consider coefficients $b$, $\sigma$ together with sequences $b^n$, $\sigma^n$ all satisfying Assumption \[t:asb\]. Denote by $(P^n,\Theta^n)\in {{\mathcal S}}_\mathrm{be}$ the solution to the linear Boltzmann equation with initial condition $Q$ and coefficients $b^n$, $\sigma^n$. If $b_n\to b$ in $\pi$ probability, $\sigma^n\to \sigma$ in $\pi\times \pi$ probability, and $\lim_n \big\{ \log \pi \big[ e^{\gamma |b^n -b |}
        \big] +\log \pi\big[ e^{\gamma |\lambda^n -\lambda|}\big] \big\}=
        0$ for any $\gamma>0$ then the sequence $\{(P^n,\Theta^n)\}$ converges to $(P,\Theta)$.

While uniqueness will be proven by using the argument in [@Gi], the key ingredient for the continuity result is the following lemma. Its proof, whose details are omitted, is achieved by truncating with continuous and bounded functions and using the basic entropy inequality.

\[t:lem\] Let $\{P^n\}\subset {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ be a sequence converging to $P$ and satisfying the entropy bound $\sup_n
  {{\mathcal H}}(P^n) < +\infty$. Then $P^{n}(\phi) \to P(\phi)$ for any function $\phi$ having all exponential moments with respect to $\pi$. Moreover, if $\phi$ has all exponential moments and $\lim_n \log \pi \big[ e^{\gamma |\phi^n -\phi|} \big] =0$ for any 