<p>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
$$
m(U \cup V) + m(U \cap V) = m(U) + m(V).
$$</p>\\
<p>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.</p>\\
<p>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.</p>\\
<p>I shall discuss the following results, proved in a draft paper "A
monad of valuation locales" available at <a href="http://www.cs.bham.ac.uk/~sjv/Riesz.pdf">http://www.cs.bham.ac.uk/~sjv/Riesz.pdf</a>:<ul>
<li> V is a strong monad, analogous to the Giry monad of measure theory.</li>
<li> There is a Riesz theorem that valuations are equivalent to linear
functionals on real-valued maps.</li>
<li> The monad is commutative: this is a categorical way of saying that
product valuations exist and there is a Fubini theorem.</li>
</ul></p>\\
<p>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).</p>