About three possible approaches to integration with respect to valuations
by
Regina Tix
Fachbereich Mathematik, Technische Hochschule Darmstadt, Germany

Continuous valuations like they were introduced by J. D. Lawson can be seen as an order theoretical variant of probability and measure. They were used by C. Jones and G. Plotkin to model probabilistic non-determinism in computation. Therefore they look at the probabilistic powerdomain V(X) of all continuous valuations on a given topological space X. For these kind of models, integration becomes an essential tool to interpret various phenomena of programming languages with a probabilistic choice operator. We will present three different approaches to integration of lower semicontinuous functions with respect to valuations: The original approach by C. Jones and G. Plotkin, developed by O. Kirch, takes advantage of the fact that every lower semicontinuous function

f : X --> R+

R+   : = R+ \cup { \infty}

can be written as

f = Ú n in N  fn

where f_n are step functions with finite image, i.e.

fn = m å i=1  ri\chiUi

with r_i in R₊, and \chi_Ui are the characteristic functions of open subsets U_i of X.

The integral of a step function with respect to a continuous valuation \mu is defined by

ó õ X  ( m å i=1  ri\chiUi)d\mu: = m å i=1  ri\mu(Ui),

and hence

ó õ X  f d\mu: = Ú n in N  fn d\mu.

Another approach is via the Choquet-integral, namely

ó õ X  f d\mu: = ó õ \infty r=0  \mu(f-1(]r, \infty])) dr

, where the integral on the right side denotes the improper Riemann-integral on R₊. The third approach is due to R. Heckmann. He uses the universal property of the probabilistic powerdomain functor V to define

ó õ X  f d\mu: = id   (\mu o f-1),

where

id   : V( R+   ) --> R+

is given by the universal property. We will see that all these approaches yield the same notion of integral.

Date received: June 29, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-47.