The basic order properties, as well as some metric and algebraic properties, are studied of the set of finitely additive transition functions on an arbitrary measurable space, as endowed with the structure of an ordered normed algebra, and some connections are revealed with the classical spaces of linear operators, vector measures, and measurable vector-valued functions. In particular, the question is examined of splitting the space of transition functions into the sum of the subspaces of countably additive and purely finitely additive transition functions.

Keywords:transition function, purely finitely additive measure, lifting of a measure space, vector measure, measurable vector-valued function, ordered vector space, vector lattice, Riesz space, K-space, Banach lattice, ordered Banach algebra.