# What does it mean by "converges boundedly"?

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 everywhere on $$\mathbb R^m \times [0,T)$$ 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 $$U_\mu.$$

• The natural guess is that there is some norm and a costant $M$ such that $U_{\mu_k} \to U$ pointwise a.e. and $||U_{\mu_k}|| <M$ for all $\mu_k$. Jul 14 at 14:19
• I would guess similar things. But then it is not clear which norm to use. This term is used several times, but without definition. Jul 14 at 14:21

This should mean that $$U_{\mu_k} \to U$$ almost everywhere on $$\mathbb{R}^m \times [0,T)$$, and moreover the sequence of functions $$U_{\mu_k}$$ is uniformly bounded: $$\sup_k \sup_{(x,t) \in \mathbb{R}^m \times [0,T)} |U_{\mu_k}(x,t)| < \infty.$$ I suppose that the functions $$U_{\mu_k}$$ take their values in $$\mathbb{R}$$ or $$\mathbb{C}$$ or some other obvious normed space.
In context, it is probably fine to replace the sup over $$(x,t)$$ with the essential supremum.