Intermezzo

Abstract

So far we have been dealing with countable sample spaces. The reason for this, as mentioned before, is that we can develop many interesting probabilistic notions without needing too many technical details. However, there are a number of intuitive probabilistic ideas which can not be studied using only countable sample spaces. Perhaps the most obvious kind of things which can not be captured in the countable setting are those involving an infinitely fine operation, such as choosing a random point from a line segment. In this intermezzo we see why this is so, and observe some problems that arise from this fact. After the intermezzo, we continue with the study of continuous probability theory, which does capture such infinitely fine operations.