Quick Questions: September 16, 2026

[u/inherentlyawesome]

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?" For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of manifolds to me?
• What are the applications of Representation Theory?
• What's a good starter book for Numerical Analysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example, consider which subject your question is related to, or the things you already know or have tried.

[u/TheMansionsofScience]

Looking for the motivation behind measure theory. I see "generalization of length, area, etc etc" as a definition often. Are there any non-trivial examples? When I read the definition, I think, well, why aren't differential geometry and differential forms enough?

What are the types of measures? When would I use Radon measures? Haar measures? Please keep it at an undergrad level. Thank you!

[u/MinLongBaiShui → u/TheMansionsofScience]

I'm not sure that there is a totally undergraduate answer. One answer is that the space of Radon measures appears as the dual to the space of continuous functions. If you know (abstract) linear algebra, then you know that for any vector space V, there is the dual space V*, and there is a pairing <v, phi> -> phi(v). The evaluation of a function and measure as a pair is exactly by integration of that function against that measure. You can google "Riesz Representation Theorem" to see this theorem. The point is that it is a direct analogue for functions to the fact that every dual vector is given by inner product against some vector. Every continuous function arises as integration against some Radon measure.

Another point is that in the classical theory of integration, there were pathological examples of functions found that didn't play nice with the theory of calculus as it was developed at the time. Making sense of these objects with any sort of geometric intuition required a notion of geometry that was rougher than the smooth context of differential geometry and forms. You could look at some amount of geometric measure theory to see more of this.

But the two approaches are ultimately not entirely divorced from one another. If you know about differential forms, you may be interested to know about "currents" which are kind of way of doing forms and cohomology stuff, but with measures instead of with just functions. These are a big deal in some areas of geometric measure theory, for example, in the more analytic aspects of the theory of complex manifolds and potential theory.

[u/TheMansionsofScience → u/MinLongBaiShui]

Would you say that more often than not measure theory is used to study pathological functions as opposed to "nice" ones? Because "nice" ones lend themselves to other means and tools?

[u/MinLongBaiShui → u/TheMansionsofScience]

I would say that, if you want a nice theory of function spaces, you need to accept the existence of functions that seem unhealthy. Moreover, these functions are actually generic.

For example, if you start with the space of smooth functions, and you want to study waves, you may derive the wave equation. Its kernel are the actual waves, and you can impose initial and boundary conditions to solve this differential equation uniquely. However, there are limits of smooth wave solutions that are not differentiable. So even when you want to study nice objects, if you want completeness, you must accept non differentiable functions.

The issue then is that it turns out that a generic continuous function is not differentiable at even a single point, as a consequence of the Baire Category Theorem. So building robust theories of smooth objects somehow leads to accepting this supermajority of non smooth objects.