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