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



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> 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.
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**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 $U\_\mu.$



What does it mean by "converges boundedly"?
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.


This is standard terminology; see for instance [Section 10.5](https://books.google.com/books?id=1qd6DwAAQBAJ&lpg=PA458&ots=39gz2Zw3JG&dq=converge%20%22boundedly%22&pg=PA458#v=onepage&q=converge%20%22boundedly%22&f=false) of:



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> *Ghorpade, Sudhir R.; Limaye, Balmohan V.*, [**A course in calculus and real analysis**](http://dx.doi.org/10.1007/978-3-030-01400-1), Undergraduate Texts in Mathematics. Cham: Springer (ISBN 978-3-030-01399-8/hbk; 978-3-030-01400-1/ebook). ix, 538 p. (2018). [ZBL1403.26001](https://zbmath.org/?q=an:1403.26001).
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In context, it is probably fine to replace the sup over $(x,t)$ with the essential supremum.


