## The Practical Impact of Set-Theoretic Axioms on Measure Theory

The set-theoretic evidence is that we could probably safely add axioms to make many more sets measurable. For example, we could add axioms that would make projective sets measurable.

I'm curious what would be the implications for working analysts of such a move. I can see two potential ways in which it could potentially have an impact:

• Currently, proving measurability of sets is a somewhat fussy activity. With the additional freedom provided by extra constructions, the existing theory would become much simpler.
• There are existing theories that are already straining at the limits of what can proved measurable in ZFC. These theories could be usefully extended.

I could also see that it potentially having no real impact. I'd be curious to hear which if any of these possibilities actually holds.

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A standard application of Martin's axiom en.wikipedia.org/wiki/Martin%27s_axiom is the existence of Banach limits satisfying some measurability conditions (medial means in the sense of Mokobodzki). See my answer to math.stackexchange.com/q/54554 for some details and references and part 4 of that answer for a basic sample application that might illustrate their power. – Theo Buehler Jan 1 at 23:12

Solovay's model already shows that the axiom of dependent choice (DC) is compatible with the assumption that all sets are Lebesgue measurable. As far as I am aware, DC suffices for essentially all applications that "working analysts" care about. If this is true, then the only practical impact of assuming that all sets are Lebesgue measurable is that you exchange slightly fussy proofs of measurability with slightly fussy proofs of results that currently invoke Hahn–Banach or other manifestations of AC, replacing AC with DC.

If I'm wrong and there are cases where DC isn't enough for "working analysts," I'd be curious to hear about it.

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I should point out a paper (from I think 1970 or so) by Garnir in which he excitingly proves automatic continuity for Banach spaces (and more) in Solovay's model. In this paper Garnir foresees a three forms of functional analysis, "Zornian" which work in ZFC; "Solovayian" which work in Solovay's model (and probably related models as well); and "agnostic" which is constructive type of proofs in ZF+DC without further assumptions. – Asaf Karagila Dec 31 at 20:02
And, 40 years later, functional analysts have not begun using solovayan analysis. – Gerald Edgar Dec 31 at 21:19
See my answer mathoverflow.net/questions/34863 for speculations about why analysis hasn't become Solovayian. – Andreas Blass Dec 31 at 21:38
Note that even if working analysts resist opting for DC over AC in practice, Solovay's result still shows that making more sets measurable won't let you "extend" to new theorems unless those extended theorems aren't provable using DC. So for arsmath's question it is still relevant to consider what results in analysis aren't provable with DC. – Timothy Chow Dec 31 at 22:28
I don't wish to turn my ignorance into undue vehemence, but apropos of Todd's remark, I have always felt that the category of Banach spaces becomes much nastier if the dual of $L^\infty$ is $L^1$. Closed subspaces of reflexive spaces are no longer guaranteed to be reflexive; something odd must be happening with Hahn-Banach (meaning that duals of certain classes of short exact sequences are now no longer short exact), one loses automatic recourse to the "embed into the double dual to take advantage of compactness in the weak-star topology" technique, and so on. – Yemon Choi Jan 1 at 4:11