The concept of average is highly useful (and much maligned) concept in all of mathematics and in life. However, few people stop to think about what an average really \emph{is}. As it turns out, it is a very important theoretical concept in mathematics, and it isn't just something that helps one lie with statistics. It is really the heart of measure and integration theory. In this talk we'll learn how measure theory and integration unifies various different kinds of averages, and one big result: Jensen's inequality, and its applications to relating more exotic means to one another.