A New Way to Measure Sets! (How to build a strictly monotone measure) #SoME2 #3b1b

*List of corrections/clarifications (including "clopen", "topologist", and "measure") below*

This is a summary of about a year or two of my own personal math research. It is not complete overview, nor does it constitute peer-reviewed research. I am a teacher by trade, not a professional mathematician. This is just a fun intro to my research to see if people are interested. I also do claim to have invented some of these ideas myself, but definitely don't claim that it hasn't been done before! There's a lot of research out there and I am not an "insider" to the fields I studied.

You can support me by sending a Venmo tip to the handle @MathTrain. Please don't donate unless you are able.

Check the pinned comment for an ongoing list of minor mistakes people have caught!

Thanks to the Summer of Math Exposition 2 #SoME2 for being the impetus for me making this passion project!

Correction list:
1. Whenever I say "clopen", I'm not referring to a set that is both closed and open (which is the correct use). I'm referring to half-open intervals, and I'm misusing terminology here. My mistake!

2. It's debatable what a topologist would actually say here, and it's not as cut-and-dry as I make it out to be in the video. Many have pointed out that topologists might say B is larger because it is open, while A is compact. Valid point, but dividing math up into subcategories isn't really the point of the video, so I chose to hand-wave this one and say "topology" since it's a broad topic in which every point matters.

3. The width of the interval at 5:30 should be 12, not 8. But this doesn't change the end result.

4. Technically, when I say "measure" I mean "polynomial-valued measure". An actual measure has codomain R, while I use codomain R[omega]. Also an actual measure is defined on a sigma-algebra, while this is only on a non-sigma algebra of sets. But that would've been a bit esoteric to include in this introduction. The actual algebra on which mu works is the algebra of definable sets in an o-minimal structure.

[MathTrain]

Some important corrections:
1. Whenever I say "clopen", I'm not referring to a set that is both closed and open (which is the correct use). I'm referring to half-open intervals, and I'm misusing terminology here. My mistake!
2. It's debatable what a topologist would actually say here, and it's not as cut-and-dry as I make it out to be in the video. Many have pointed out that topologists might say B is larger because it is open, while A is compact. Valid point, but dividing math up into subcategories isn't really the point of the video, so I chose to hand-wave this one and say "topology" since it's a broad topic in which every point matters.
3. The width of the interval at 5:30 should be 12, not 8. But this doesn't change the end result.
4. Technically, when I say "measure" I mean "polynomial-valued measure". An actual measure has codomain R, while I use codomain R[omega]. Also an actual measure is defined on a sigma-algebra, while this is only on a non-sigma algebra of sets. But that would've been a bit esoteric to include in this introduction.

[crazyspider]

you say in the video a "topologist view" but you really just meant a "set theorist view". containment is just a notion of set theory, and a lot of different subjects use set theory, including measure theory, so containment doesn't really have anything to do with topology in particular.

[andriesvancauwenberge]

This is the best #some2 video I have seen really love it. Keep up the great work. I would love to see more of this.

[oceannuclear]

Damn, you're out here making a video about your primary research at a level that is understandable by the (not so general) public. This is insanely cool!

[RuleAndLine]

Holy cow, I wasn't expecting original math research when I started watching this. Really interesting result!

[rcteg]

This was fantastically well made! I'm an undergrad in CS and have a hobbyists interest in math but no actual experience in analysis, measure theory, algebra, or topology, and you did a fantastic job of explaining this to a layman. Seriously well done and thanks for sharing, seeing this kind of math research done at an actually understandable level is super cool!

[harsinsinquin]

9:12 my first thought was “why not just use omega” and the second I said it you said it. It made me feel very delighted.

[justinofelipeimbert]

outstanding video, you introduced the question in a very simple way, so simple that people without any knowledge of measure theory or topology could be able to understand, your solution is very intuitive and you did a very good job maintaining the interest throughout the video.

again, outstanding video.

[ellismaddox]

Why would the large square not simply eat the small one?

[NixoticaTTV]

Hidden gem of a video, its really nice that we live in an age where people can communicate existing theorems or even laymanize their own theorems in a single digestible video. Really hope to see more :)

[spiffinn_music_lists]

I'm excited to get more information/content on this measure! Especially explanations for the points at the end of the video, and how to deal with and distinguish sets countably and uncountably infinite sets.

[azai.mp]

This is so cool! I did a project on nonstandard analysis in uni, and since then I've been wondering if measure theory would benefit from having more numbers to use as sizes. It's nice to see someone had the same idea and worked it out successfully

[DynestiGTI]

The use of ω, ω^2, etc. to label the progressively larger infinities reminds me of John Conway's *surreal numbers.* Also is this seriously the first time someone has created a strictly monotone measure that is independent of how the set is measured? I'm honestly surprised no one has thought about this before... good job!

[rishikaushik]

That's a very interesting system, it is more intuitive for me to visualize it slightly differently though. We can imagine each line segment as the complete length with 2 half points at each end, so ω^2 can be thought of as (the area + half parameter + 4 one-fourth points at each vertex), this also makes visualizing tiling areas easier. Any 2d shape we can deconstruct into - > the area + 0.5 parameter + (internal angle/360)th of a point at each sharp vertex.
We can even extend this to 3D with any space being broken into -> the volume + 0.5 surface area + (angle between faces/360)th of a segment at each sharp edge + (internal solid angle / complete sphere)th of a point at each vertex. In 3D the "density" of an edge can change continuously along its length if the angle between faces changes.

imagining like this every shape fits nicely

[ikhu]

Hey this is actually the coolest thing I've seen in a while. I *always* wanted to be able to consolidate how we could compare volumes of different dimensions while still preserving the information abt lower dimensions. Like adding a line to an are always gives (inf+1) lines so it just disappears but using these half-open intervals is an amazing way to do it, and how you can the get the span of the polynomial measure space from squaring/cubing the open interval.

[elidamon]

This is fascinating, It makes perfect sense to me to think of a shape owning a half share of its boundary, with its complement owning the other half share. But as others have already expressed, it seems like this would only work for finite collections of finite cells. I'm imagining problems with even some fairly mundane example sets. Like, how would you combine rational points with irrational points on a unit interval?

[captain-carre]

Very interesting. How would you measure infinite discrete sets with this measure ? Like the rationals on [0, 1]. If you start from the empty set and add all the points you get infinity. If you start from [0, 1] and remove the irrationals you get omega - infinity ?

Very interesting. For me, both squares were equal. After the video, I still think they're equal. But I really love this perfectly monotonic function you build. So clever and interesting.

[neoztar]

The -1/2 coefficient that you end up with for the boundaries is very reminiscent of the concept of "half-edges" from computational geometry where an edge between two vertices A and B can be broken up into two directed half-edges, one from A to B, and one from B to A.

So if we consider an arbitrary region within some larger tesselation to be intrinsically surrounded by half-edges, then it makes sense that when you cut it out to form its own shape, you have to either add in the missing half-boundary to get a closed set, or remove it to get an open set.

[jaafars.mahdawi]

Math is all about creativity and this video does just that! Keep it up, my friend!