On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says

Under some assumptions, suppose a sequence of solutions $U_{\mu_k}$ to a conservation law with viscosity term

converges boundedly almost everywhereon $\mathbb R^m \times [0,T)$ to some function $U,$ and $\mu_k \to 0.$ Then $U$ is a weak solution to the conservation law without viscosity term. Moreover, $U$ is entropy admissible.

**Question:** What does "converges boundedly almost everywhere" mean?

Note: $U_\mu$ should be smooth according to theory of heat equations, but no other assumptions are made about $U_\mu.$