Some negative results on existence of Sazonov topology in \(l\)-adic Frechet spaces (Q1176779)

Let \(F\) be a local field and \(V\) a topological Hausdorff \(F\)-linear space. Let \(H_ M\) be the normalized Haar measure of an \(R_ F\)- submodule \(M\) of \(V\) and \[ {\mathcal H}(V)=\{H_ M:\;M\text{ is a compact \(R_ F\)-submodule of }V\}. \] Then the family \({\mathcal H}(V)\) may be regarded as a substitute of the Haar measure, if \(\dim_ F V=+\infty\). In this paper the author investigates the family \({\mathcal H}(V)\) in an \(\ell\)-adic Fréchet space and shows how one can determine some \(\ell\)- adic Fréchet spaces which are not Sazonov ones. The author gives an example of an \(\ell\)-adic locally convex space without Sazonov's property.