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](https://arxiv.org/pdf/2009.01156) 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](https://webspace.science.uu.nl/~ferna107/papers/root.pdf). 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](https://arxiv.org/pdf/2009.01156) imply such estimate in the space of 3D periodic distributions?


It is stated in [p.394-395 of this book](https://webspace.science.uu.nl/~ferna107/papers/root.pdf) 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..