Borel Sets, Measures and All That

Abstract

Stochastic coalgebraic logic works in a general probabilistic setting, in contrast to some more traditionally oriented approaches that use finite or countably infinite state spaces. Since the universe of discourse is no longer finite, one needs to look at properties of general measurable spaces; these spaces have a rich structure which needs to be exploited, so we will deal with measurable sets and maps, and with σ-algebras and their generators. Here the π-λ-Theorem is of particular prominence. But sometimes a measurable structure is not rich enough for some properties, so we will specialize this structure to the Borel sets of a topological space. We will quickly see that it is the class of Polish spaces which attracts most attention, together with the class of analytic spaces, i.e., the continuous images of Polish spaces. We give the basic properties of these spaces in Sections 1.3 and 1.4.