An extension result for continuous valuations

It is shown, by a simple and direct proof, that if a bounded valuation on a monotone convergence space is the supremum of a directed family of simple valuations, then it has a unique extension to a Borel measure. In particula r, this holds for any directed complete partial order with the Scott topolo gy. It follows that every bounded and continuous valuation on a continuous directed complete partial order can be extended uniquely to a Borel measure . The last result also holds for sigma-finite valuations, but fails for dir ected complete partial orders in general.