Abstract
In the 1930's, Kolmogorov borrowed the axiomatic system of the Lebesgue measure as a foundation for what is now the standard theory of probability. The domain of the probability measure is assumed to possess the structure of a Boolean sigma-algebra, and the measure is assumed to be countably additive. The "expectation of a random variable" is developed as the integral of a measurable function. Around the same time as Kolmogorov's development, de Finetti introduced the notion of a "coherent" probability, consistent with the Lebesgue theory, but requiring neither countable additivity of the measure nor any sort of structure on its domain. In this thesis I present a theory of the integral, or expectation, with respect to this broader notion of a probability.