= Solution
With $g=0$, pull the bounded measurable initial value back along the <characteristic flow map>:
$$
u(t,x)=u_0(X(0;t,x)).
$$
The flow is measurable and invertible, so $u$ is measurable and $\|u\|_\infty\leq\|u_0\|_\infty$. Choose smooth $u_0^{(k)}$ converging to $u_0$ in $L^1_{\mathrm{loc}}$ with uniformly bounded essential suprema, and define
$$
u^{(k)}(t,x)=u_0^{(k)}(X(0;t,x)).
$$
Part a makes each $u^{(k)}$ a classical, hence weak, solution. On every compact subset of spacetime, the $C^1$ change-of-variables formula for the flow and its locally bounded <Jacobian determinant> give $u^{(k)}\to u$ in $L^1$. Passing to the limit in the weak identity by <dominated convergence theorem>[dominated convergence] proves that $u$ is a bounded weak solution with initial datum $u_0$.
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