In the context of lattice approximation, the term "UV stability" seems to be used frequently. To me, it seems like

Uniform boundedness of the partition function in the limit where lattice spacing goes to zero.

For example, Theorem 2 in this article deals with such an estimate.

However, I am confused if UV stability leads to existence of a continuum limit by compactness argument. The definition of continuum limit is from p.379 of this book. This definition pertains to Schwinger functions, so I guess the continuum functional measure should be somehow reconstructed by the moment problem.

Here are my main confusions:

First, the space of (tempered) distributions with the strong dual topology is a Montel space, implying that a bounded sequence has a convergent subsequence. So, some kind of uniform boundedness seems necessary.

Nevertheless, it is not clear to me if UV stability of lattice approximations implies boundedness in the relevant space of distributions. For example, does Theorem 2 in this article imply such estimate in the space of 3D periodic distributions?

It is stated in p.394-395 of this book that Balaban constructed continuum limit of 4D lattice pure YM theories. However, the papers cited there are devoted to UV stability without explicit mention of continuum limit, so I am confused.

I fear that this post is not detailed enough, but still hope for any answer from experts in this field..