= Solution
Use the unit-period <circle> $\mathbb T=\mathbb R/\mathbb Z$, with total measure one. The free <characteristic flow map> gives
$$
f_t(x,v)=f_0(x-tv,v).
$$
Translations in $x$ preserve its periodic measure. The <Tonelli theorem> and the <integral> <triangle inequality> yield
$$
\boxed{\|\rho_t\|_{L^1(\mathbb T)}
\leq\int_{\mathbb T}\int_{\mathbb R}|f_0(x-tv,v)|\,dv\,dx
=\|f_0\|_{L^1(\mathbb T\times\mathbb R)}.}
$$
The <Fubini's theorem> therefore applies also to signed data, and the same translation gives
$$
\boxed{\int_{\mathbb T}\rho_t(x)\,dx=\int_{\mathbb T}\int_{\mathbb R}f_0(x,v)\,dv\,dx=\rho_\infty.}
$$
This 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.
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