= Solution
Let $f_1,f_2$ have the same initial data and satisfy the characteristic integral equation. Their difference $g=f_1-f_2$ satisfies $g=\tau g$, hence $g=\tau^ng$ for every $n$.
Fix any $T_0<T$ and let $M=\sup_{0\leq s\leq T_0}\|g(s)\|_2<\infty$. The <factorial bound for a Volterra iterate> gives
$$
\|g(t)\|_2\leq\frac{(CT_0)^n}{n!}M
\qquad(0\leq t\leq T_0).
$$
The scalar factor tends to zero, so $g(t)=0$ throughout this interval. Since $T_0<T$ is arbitrary, \b[the weak solution of the <linear Boltzmann equation> is unique on $[0,T)$]. The same argument gives uniqueness at $T$ whenever solutions are defined there by the integral formula.
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