  We consider weak solutions with finite entropy production to the scalar
conservation law \begin{equation} \partial_t u+\mathrm{div}_x F(u)=0 \quad
\mbox{in }(0,T)\times \mathbb{R}^d. \end{equation} Building on the kinetic
formulation we prove under suitable nonlinearity assumption on $f$ that the set
of non Lebesgue points of $u$ has Hausdorff dimension at most $d$. A notion of
Lagrangian representation for this class of solutions is introduced and this
allows for a new interpretation of the entropy dissipation measure.
