Probability via Expectation Measures

Since the seminal work of Kolmogorov, probability theory has been based on measure theory, where the central components are so-called probability measures, defined as measures with total mass equal to 1. In Kolmogorov’s theory, a probability measure is used to model an experiment with a single outcome that will belong to exactly one out of several disjoint sets. In this paper, we present a different basic model where an experiment results in a multiset, i.e. for each of the disjoint sets we get the number of observations in the set. This new framework is consistent with Kolmogorov’s theory, but the theory focuses on expected values rather than probabilities. We present examples from testing Goodness-of-Fit, Bayesian statistics, and quantum theory, where the shifted focus gives new insight or better performance. We also provide several new theorems that address some problems related to the change in focus.