So forgive me for a moment, as I am only comfortable with the VERY basics of Lebesgue measure, at the moment. I will be taking a course on it soon, but apart from the definitions and basic properties, I am unfamiliar.
I do know that the nonexistence of a nonmeasurable set requires the assumption that an inaccessible cardinal exists. Solovay's model assumes the existence of an inaccessible, and I believe it was Shelah that proved that this assumption is necessary. Therefore, it feels like a VERY important result to discuss anything related to measure theory. My questions are as follows:
- How important is the assumption that no nonmeasurable set exists to the study of measure theory as a whole? Is assuming an inaccessible cardinal exists in measure theory analogous to assuming choice in linear algebra or functional analysis?
- Why is this NEVER discussed in measure theory texts? I checked Tao, Rudin, Royden, Halmos, and Stein and Shakarchi, and none of them even MENTION this. I do know that a full discussion on Solovay's model requires some hefty set theory that might not be suitable for students reading the book (forcing, in particular), but why not just MENTION that it's required, without proof, even?
I would accept an answer of "the students aren't required to know advanced set theory," if it weren't for the fact that these aren't even mentioned without proof or further explanation. The interplay between AC, DC, "There exists an inaccessible cardinal", and "There exists a nonmeasurable set" is an exciting and interesting one, and one that I'm surprised is never brought up in introductory texts.
Best Answer
Measure theory deals with measurable sets. Whether or not these are all of them is irrelevant. Moreover, since standard foundation includes the Axiom of Choice (and it is, at the very least, an important concept worth understanding), it is outright a mistake to not have a discussion about non-measurable sets. But after you discuss that, measure theory is focused on measurable sets and measure algebras.
Solovay's model is one where for the Lebesgue measure, the measure algebra is $\mathcal P(\Bbb R)$. Beyond that is has very little to offer. Yes, every $f\colon\Bbb{R\to R}$ satisfying $f(x+y)=f(x)+f(y)$ is of the form $a\cdot x$, but really what we prove is that every measure solution is of that form, and in Solovay's model it just happens that every solution is measurable.
Not to mention that if you advance to studying finitely additive measures and reach things like Bartle integration, working in Solovay's model (or even Shelah's model where all sets have the Baire Property) takes out some of the interest, since $\ell^1$ is reflexive, and therefore there are no nontrivial finitely additive measures on $\Bbb N$.
So, why don't they mention it at all? I don't know. The measure theory course I took was based on Folland's "Real Analysis" book. I think that there is a mention of Solovay's result, albeit as a footnote. Maybe it is worth spending a couple of pages on this model, yes. But seeing how the Axiom of Choice is so ingrained in modern mathematics, and how measure theory is focused on measures and measure algebras anyway, the omission is quite understandable.