Measures and approximations using empirical structures

Suppose that N is a set of subsets of a finite set Q. N may be interpreted, for example, as a set of observed states as a result of an experiment or as a model of some theory. In a further step, the elements of N can be weighted by some set function f:N→R such as various kinds of uncertainty measures or probabilities. In this paper, we demonstrate the process of analysing data based on observed granules according to the model proposed by G. Gigerenzer, and argue the necessity of an error theory when passing from a theoretical model to empirical data. We also investigate changes that occur when using granule structures weaker than a Boolean algebra, and under which conditions results based on a granule set can be extended to the whole power set algebra. Our main examples are results of a test taken by a population of students, and their interpretation by various probability functions such as basic probability assignments and belief functions.