 Solution

Use the unit-period circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, with total measure one. The free characteristic flow map givesTranslations in $x$ preserve its periodic measure. The Tonelli theorem and the integral triangle inequality yieldThe Fubini's theorem therefore applies also to signed data, and the same translation givesThis holds for positive or negative time. The printed $L^1(\mathbb{R})$ in this subpart is a domain typo: the spatial variable is periodic, so the correct space is $L^1(\mathbb{T})$. For example, the smooth initial value $f_0(x,v)=e^{-v^2}$ gives the constant density $\sqrt{\pi }$, whose periodic extension is not integrable on the real line.