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.