Transport only translates

Let u = u(x, t) be the solution of the following initial value problem: $u_t+2024u_x=0$ for $x\in \mathbb{R},t>0$, and $u(x,0)=u_0(x)$ for $x\in \mathbb{R}$, where $u_0:\mathbb{R}\to \mathbb{R}$ is an arbitrary $C^1$ function. Consider the following statements: $S_1$: If $A_t:=$ {$x\in \mathbb{R}:u(x,t)<1$} and |$A_t$| denotes the Lebesgue measure of $A_t$ for every t ≥ 0, then |$A_t$| = |$A_0$|, $\forall t>0.S_2$: If $u_0$ is Lebesgue integrable, then for every t > 0, the function x ↦ u(x, t) is Lebesgue integrable. Then