Abstract
The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy.
Such an approach was suggested and initiated by Segal in his pioneering article (Segal, Bull Am Math Soc 71:419-489, 1965). In our paper we follow his ideas from a different perspective, slightly more categorical, and strongly inspired by the point-free topology.
First, we develop a general (point-free) concept of measurability (extending the standard Lebesgue integration when applying to the classical sigma-algebra). Second (and here we have a major difference from the classical theory), we prove that every finite-additive function mu with values in [0,1] can be extended to a measure on an abstract sigma-algebra; this correspondence is functorial and yields uniqueness.
As an example we show that the Segal space can be characterized by completely canonical data. Furthermore, from our results it follows that a satisfactory point-free integration arises everywhere where we have a finite-additive probability function on a Boolean algebra.