Point-free measure theory using valuations

Point-free measure theory using valuations

Wed, 12/10/2011
16:00 		Steve Vickers (University of Birmingham) Analytic Topology in Mathematics and Computer Science Add to calendar L3
Point-free measure theory using valuations
In topos-valid point-free topology there is a good analogue of regular measures and associated measure theoretic concepts including integration. It is expressed in terms of valuations, essentially measures restricted to the opens. A valuation $ m $ is $ 0 $ on the empty set and Scott continuous, as well as satisfying the modular law
$$ m(U \cup V) + m(U \cap V) = m(U) + m(V). $$

Of course, that begs the question of why one would want to work with topos-valid point-free topology, but I'll give some general justification regarding fibrewise topology of bundles and a more specific example from recent topos work on quantum foundations.
The focus of the talk is the valuation locale, an analogue of hyperspaces: if $ X $ is a point-free space (locale) then its valuation locale $ VX $ is a point-free space whose points are the valuations on $ X $. It was developed by Heckmann, by Coquand and Spitters, and by myself out of the probabilistic powerdomain of Jones and Plotkin.
I shall discuss the following results, proved in a draft paper "A monad of valuation locales" available at http://www.cs.bham.ac.uk/~sjv/Riesz.pdf:
  • V is a strong monad, analogous to the Giry monad of measure theory.
  • There is a Riesz theorem that valuations are equivalent to linear functionals on real-valued maps.
  • The monad is commutative: this is a categorical way of saying that product valuations exist and there is a Fubini theorem.

The technical core is an analysis of simple maps to the reals. They can be used to approximate more general maps, and provide a means to reducing the calculations to finitary algebra. In particular the free commutative monoid $ M(L) $ over a distributive lattice $ L $, subject to certain relations including ones deriving from the modular law, can be got as a tensor product in a semilattice sense of $ L $ with the natural numbers. It also satisfies the Principle of Inclusion and Exclusion (in a form presented without subtraction).