Defining the Most Generalized, Natural Extension of the Expected Value on Measurable Functions

Abstract

In this paper, we will extend the expected value of the function w.r.t the uniform probability measure on the Caratheodory extension to a larger class of functions, since the set of all functions with infinite or undefined expected values may form a prevalent subset of the set of all measurable functions. Before we get to the specific problem (or main question) of the paper, we will outline some preliminary definitions. We then will define a precise main question that will attempt to offer a unique solution and we'll offer a partial solution to the question. Along the way, we will ask a series of questions that will clarify our understanding of the paper.