  We consider mixed powerdomains combining ordinary nondeterminism and
probabilistic nondeterminism. We characterise them as free algebras for
suitable (in)equation-al theories; we establish functional representation
theorems; and we show equivalencies between state transformers and
appropriately healthy predicate transformers. The extended nonnegative reals
serve as `truth-values'. As usual with powerdomains, everything comes in three
flavours: lower, upper, and order-convex. The powerdomains are suitable convex
sets of subprobability valuations, corresponding to resolving nondeterministic
choice before probabilistic choice. Algebraically this corresponds to the
probabilistic choice operator distributing over the nondeterministic choice
operator. (An alternative approach to combining the two forms of nondeterminism
would be to resolve probabilistic choice first, arriving at a domain-theoretic
version of random sets. However, as we also show, the algebraic approach then
runs into difficulties.)
  Rather than working directly with valuations, we take a domain-theoretic
functional-analytic approach, employing domain-theoretic abstract convex sets
called Kegelspitzen; these are equivalent to the abstract probabilistic
algebras of Graham and Jones, but are more convenient to work with. So we
define power Kegelspitzen, and consider free algebras, functional
representations, and predicate transformers. To do so we make use of previous
work on domain-theoretic cones (d-cones), with the bridge between the two of
them being provided by a free d-cone construction on Kegelspitzen.
