  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 \cite{segal}. 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.
