Check out Terry Tao’s measure-theory book, starting with ‘let us try to formalise some of the intuition for measure discussed earlier’ on page 18, through to ‘it turns out that the Jordan concept of measurability is not quite adequate, and must be extended to the more general notion of Lebesgue measurability, with the corresponding notion of Lebesgue measure that extends Jordan measure’ on p. 18.
I’ve understood for some time that there’s a notion of “non-measurable set”, and that you want your definition of ‘measure’ to preserve certain intuitive ideas — e.g., that taking an object and moving it a few feet doesn’t change its measure. I didn’t understand that there was any connection between non-measurability and the axiom of choice. Tao’s words here are some of the first that have properly oriented me toward the problem that we’re trying to solve, and the origins of that problem to begin with.
My partner is taking a biostatistics course, which is reminding me of how much I loved this stuff at CMU. I’m inclined to find a course in measure theory around here. We have a university or two.