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The main concern of this paper is to present some improvements
to results on the existence or non-existence of countably additive Borel measures
that are not Radon measures on Banach spaces taken with their weak topologies, on
the standard axioms (ZFC) of set-theory. However, to put the results in perspective we
shall need to say something about consistency results concerning measurable cardinals.
It is proved that if a Frechet space has property, then also has property, for .
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