Productivity of some classes of topological linear spaces

The Mackey theorem on products of bornological spaces is generalized to classes of topological linear spaces closed under quotients and inductive limits: either they are not countably productive or are nonmeasurably productive. The result can be shifted up to higher measurable cardinals. It is shown that the Mackey theorem is not valid for TLS-bornological spaces provided the real measurable cardinal does not coincide with the Ulam measurable cardinal (thus contradicting some results by Robertson and Adasch).