As is well known, every Borel measure in a metric space S is regular, provided that S is the union of a sequence of open sets of finite measure. It seems, however, not yet to have been noticed that this theorem can be easily extended to all spaces with Urysohn' s f "F-property", i.e., spaces in which every closed set is a countable intersection of open sets (we call such spaces "F-spaces"). Indeed, various theorems are unnecessarily restricted to metric spaces, while weaker assertions are made about F-spaces. This seems to justify the publication of the following simple proof which extends the theorem stated above to F-spaces.