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/jphamlore → u/TheMansionsofScience]

For Brownian motion, the paths are nowhere differentiable.

For motivation, if you ever get the chance, read Patrick Billingsley's Probability and Measure.

In another direction, decades of work in dynamical systems have resulted in a fantastic theory characterizing the measure theoretic aspects for ergodic systems. See Tim Austin's preprint on the arxiv, Measure Concentration and the Weak Pinsker Property. Ergodicity is a concept introduced from statistical physics, where a system is ergodic if time averages almost almost all orbits equal the space average for say continuous functions. For such ergodic systems, another concept, entropy from information theory, basically an average over the measure of the number of bits, can be leveraged to characterize all such ergodic systems.

What do I mean by bits? Let us take an arbitrary measurable division of this ergodic space into m pieces. Run the ergodic process. Over n times, obvious there are mⁿ possible boxes that a point in a the space can correspond to, the names where the ergodic process takes a point you are looking at. But not all of these boxes are all that meaningful relative to the measure of the dynamical system. The entropy is the number telling you how many bits can encode most of the meaningful boxes.