A foundation for statistics

Probability theory has had an almost unquestioned foundation since Kolmogorov (1933) [111]. Probability is a (normalized) measure on an algebra of events i.e. subsets of an arbitrary set. The probability of an event may be a result in a theory about any part of the real world. It may be an assumption that the beliefs or knowledge of agents can be expressed as total preorders on a system of subsets of events. Theorem 9 page 98 then gives probability as a representation of this relation. Under the assumptions of this representation theorem the two assumptions — a total preorder on an algebra or a probability measure on this algebra — are therefore equivalent.