Abstract
We give two concrete examples of continuous valuations on dcpo’s to separate minimal valuations, point-continuous valuations, and continuous valuations:
(1)
Let
${\mathcal J}$
be the Johnstone’s non-sober dcpo, and μ be the continuous valuation on
${\mathcal J}$
with μ(U)=1 for nonempty Scott opens U and μ(U)=0 for
$U=\emptyset$
. Then, μ is a point-continuous valuation on
${\mathcal J}$
that is not minimal.
(2)
Lebesgue measure extends to a measure on the Sorgenfrey line
$\mathbb{R}_\ell$
. Its restriction to the open subsets of
$\mathbb{R}_\ell$
is a continuous valuation λ. Then, its image valuation
$\overline\lambda$
through the embedding of
$\mathbb{R}_\ell$
into its Smyth powerdomain
$\mathcal{Q}\mathbb{R}_\ell$
in the Scott topology is a continuous valuation that is not point-continuous.
We believe that our construction
$\overline\lambda$
might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpo’s.
Subspaces of the Sorgenfrey Line
