= Solution
Let $E=L^1(\mathbb R_x^d\times\mathbb R_v^d)$, and denote the <free-transport semigroup> by $(U_tg)(x,v)=g(x-tv,v)$. For fixed $v$, the spatial translation has unit <Jacobian determinant>, so the <Tonelli theorem> gives
$$
\boxed{\|U_tf_0\|_1=\int\int|f_0(x-tv,v)|\,dx\,dv
=\int\int|f_0(y,v)|\,dy\,dv=\|f_0\|_1.}
$$
Thus the given free term is an <isometry> on $E$. The operators form a <strongly continuous semigroup>: continuity first holds for smooth compactly supported functions by dominated convergence, and density plus the isometry extends it to every $L^1$ function.
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