We consider the problem of extending a continuous valuation to a Borel measure on a \(T_0\) topological space. Let \((X,\Omega X)\) denote any \(T_0\) topological space. We use the term valuation to designate a nonnegative, extended real valued function \(\nu\), defined on the lattice of open sets and satisfying: \(\nu(\emptyset)=0\) (strictnenss), \(G\subseteq H\) implies \(\nu(G) \subseteq \nu(H)\) (monotonicity), \(\nu(G)+\nu (H)=\nu(G\cup H)+\nu (G\cap H)\) (modularity). We say that a valuation \(\nu\) is (Scott) continuous if for every increasing net of open sets \(\{G_i\}_{i\in I}\) we have \(\nu(\bigcup_{i\in I} G_i)= \sup_{i\in I}(G_i)\). A valuation is \(\sigma\)-finite if there exists a sequence of open sets \((G_i)_{i\in\mathbb{N}}\) such that \(X=\bigcup_{i\in\mathbb{N}} G_i\) and \(\nu (G_i)<\infty\) for all \(i\in\mathbb{N}\). All valuations are assumed to be \(\sigma\)-finite. By a simple valuation we designate (the restriction to the open sets of) a finite linear combination of Dirac measures. The main result of this paper states that if \(X\) is a monotone convergence space and \(\nu\) is the supremum of an increasing net of simple valuations, then \(\nu\) can be extended uniquely to a \(\tau\)-smooth Borel measure. Particular instances of monotone convergence spaces are sober spaces endowed with their specialisation ordering and directed complete partially ordered (dcpo) sets endowed with the Scott topology. As an important corollary we have that all continuous valuations on a continuous dcpo extend uniquely to \(\tau\)-smooth Borel measures. We notice that all locally compact Hausdorff spaces fall in the previous category; yet for these spaces the extension result is well known. We show, via a counterexample, that not all continuous valuations on a dcpo with the Scott topology can be extended to a measure.