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 particular, this holds for any directed complete partial order with the Scott topology. 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 σ-finite valuations, but fails for directed complete partial orders in general.
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