= Solution
Work in the <Banach space> $\mathcal B_T$ of bounded strongly measurable maps $[0,T]\to L^2_{x,v}$, with norm $\|g\|_{\mathcal B_T}=\sup_{0\leq t\leq T}\|g(t)\|_2$. Keeping actual representatives at every time matches the pointwise-in-time mild formulation. The <Boltzmann Volterra operator> is bounded on this space, and the <factorial bound for a Volterra iterate> gives
$$
\|\tau^n\|_{\mathcal B_T\to\mathcal B_T}\leq\frac{(CT)^n}{n!}.
$$
Thus the <Volterra series for the linear Boltzmann equation> converges in <operator norm> for every finite $T$, even when $CT\geq1$. Set
$$
\boxed{f=\sum_{n=0}^\infty\tau^nF(f_0,a).}
$$
For its partial sums, $(I-\tau)\sum_{n=0}^N\tau^nF=F-\tau^{N+1}F$. The remainder tends to zero by the factorial estimate. Hence $(I-\tau)f=F$, precisely the required characteristic integral equation.
The norm bound in (b) gives \b[an explicit choice of the existence constant]:
$$
\boxed{\sup_{0\leq t\leq T}\|f(t)\|_2
\leq e^{CT}\|f_0\|_2,\qquad C_T=e^{CT}.}
$$
Also $f(0)=f_0$, since every term with $n\geq1$ vanishes at zero and the damping interval has length zero. This proves existence in the paper's weak, characteristic-integral sense. With merely measurable nonnegative $a$, that sense does not itself require a continuous initial trace; that trace follows under the additional local characteristic-integrability condition described in (b).
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