[monadINtop]

You need measure theory for probability, economics, QFT and Physics, etc. And who is doing "all the math"? The vast majority of resarchers who "do math" are largely in PDEs and other fields that simply use the technology of math for "things that you find in reality" like engineering problems or machine learning and so forth. And most mathmaticians would agree that it is some of the most uninteresting and ugly kind of math.

Whereas the relative minority of people who study really abstract things like say k-theory or large cardinals in set theory are largely doing it out of interest in it's intrinsic beauty. And this is especially true for idk, some esoteric subfield of tropical geometry or modal logic or something, who's relevance to "things you find in reality" are completely orthogonal as to the motivations of those people who chose to spend their lives uncovering the truths within them.

Math research isn't about blindly marching from proof to proof by mechanical deduction with no conception of the larger picture like a uniform bubble spreading outwards, it is done by small communities of scholars who hack away at a specific nexus of interesting problems and structures for their own sake.

Sometimes, like with spin bundles or lie algebras or non-abelian geometry, yeah you can apply it to "real" problems, but that's not how the theory was developed, and as a theoretical physicist I will tell you that you will find no greater blindness to the underlying structure or ugliness in the use of the technology than those people that exclusively wield the technology against "real" problems, instead of appreciating it for its own sake.

[ajkjk]

Well it is the theory that underpins those at the moment, but that doesn't say much about the counterfactual where it isn't.

But I think I can say with confidence that none of those fields care about the fact that hitting a rational number out of the reals has probability 0. If they do something's wrong.

Edit: oh wow your reply got a lot longer after I responded

[afiori]

Measure theory is not about events of probability zero, it is about how to ignore them and prevent them from messing up results.

Let's say we play a game, we uniformly pick a random real between 0 and 1 and you win if it is rational, I win if it is not.

You can obviously see that it is unfair, but how do you prove it? You need a concept of integration that can easily ignore a dense set of discontinuities, Riemann integration is not going to give you any good results (in this case at the very best it would tell you that you win between 0% and 100% of the times, not very useful)

Measure theory and Lebesgue integration are a way to discard this noise.

Actually in measure theory you generally use L¹, L², etc spaces where functions are defined modulo null sets; that is the function that is 1 on rational and 0 on irrationals is considered to be the same function as the constant 0.

In measure theory on the Reals the value of a function at a specific point is generally considered irrelevant.

[ajkjk]

The difference is that I'm interested in a version of physics and economics that is not aware of the distinction between 'real' and 'rational'. Hence none of that should make a difference.

[afiori]

If you work with mostly continous and/or well behaved functions you do not need most of these.

I suspect that in both physics and economics you might end up using stochastic methods that use concepts and techniques similar to those of measure theory

[monadINtop]

then you are interested in a version of physics that doesn't exist, even though some might choose to supress the details at low level

[ajkjk]

Er? No I'm describing actual physics. Physics does not care about rational vs reals. Nothing is measured to infinite accuracy anyway.

[afiori]

One thing that is used a lot in physics are known as Dirac Deltas[0] that is, in very informal terms, the derivative of the function f(x) = 0 for negative x otherwise f(x) = 1.

Physics are very good at working with concepts and abstraction before any formal mathy justifications can be found, but the only way to formaly work with a dirac delta that makes sense formally is defining it in terms of measures

[0] https://en.wikipedia.org/wiki/Dirac_delta_function

[tsimionescu]

Even if accuracy is finite, the fact that, for example, circles aren't polygons is definitely relevant to physics. You might be able to get all of the relevant physics you need without the continuum by working with several disjoint sets of numbers (the rationals, the rationals-multiplied-by-pi, the rationals-multiplied-by-e, etc) but I'm not even sure of that.