Extension of valuations on locally compact sober spaces.
by
Mauricio Alvarez-Manilla
Imperial College, London.

We show that every continuous valuation defined on the lattice of open sets of a locally compact sober space extends uniquely to a Borel measure. A valuation \nu is a real function defined on a lattice (0, 1, \/ , /\ , L) and satisfying: \nu(0)=0 (strictness), A <= B implies \nu(A) <= \nu(B) (monotonicity) and \nu(A \/ B)+\nu(A /\ B)=\nu(A)+\nu(B) (modularity). A valuation \nu is continuous if \nu( \/ _{i in I}A_i)=sup_{i in I}\nu(A_i) whenever \/ _{i in I}A_i is directed and exists.

In the sequel we derive a maximal point space construction for any locally compact sober space (X, G). That is, we show that there exists a continuous poset (\Lambda X, \sqsubseteq ) such that X embeds as the subset of maximal elements of \Lambda X where the relative Lawson topology of \Lambda X induces the patch topology of X.

Note: Some authors prefer to use the term quasicompact for sets that satisfy the Heine-Borel property and reserve the term compact for Hausdorff spaces. We do not make such a distinction.

Date received: May 24, 1999