Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 7 / 3 / a

Let $E=L^1({\mathbb{R}}_x^d\times {\mathbb{R}}_v^d)$, and denote the free-transport semigroup by $(U_{t}g)(x,v)=g(x-tv,v)$. For fixed $v$, the spatial translation has unit Jacobian determinant, so the Tonelli theorem givesThus 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.