Let $u$ be a $BV$ entropy-solution of the scalar conservation law $$u_t(x,t) + f(u(x,t))_x = 0$$ It is claimed that then (since the derivatives of a $BV$ function are measures) it holds in the sense of measures that
$$D_tu + \lambda D_xu = 0$$
where $$\lambda(t,x) = \begin{cases} f^{\prime}(u(t,x)) & if \quad (t,x)\quad \text{is continuity point of $u$} \\ \frac{f(u^+)-f(u^-)}{u^+-u^-}& if \quad (t,x) \quad \text{is a point of jump for $u$} \end{cases} $$
Now, i tried to prove this using the structure theorem for the derivatives of $BV$ functions, but I can't manage to recover the above formula. Does someone know how to do that? Thanks very much in advance.