Clarification of some mathematical misunderstandings about Savage's foundation of statistics, 1954

It is proved that fine and tight comparative probability structures (where the set of events is assumed to be an algebra, not necessarily a sigma-algebra) have agreeing probability measures. Although this was often claimed in the literature, all proofs the author encountered are not valid for the general case, but only for sigma-algebras. Here the proof of Niiniluoto (1972, Annals of Mathematical Statistics) is supplemented. Furthermore an example is presented that reveals many misunderstandings in the literature. At the end a necessary and sufficient condition is given for comparative probability structures to have an almost agreeing probability measure.