Comparing the Sizes of Sets in Different Ways (An Introduction to Lebesgue Measure)

Just like we can measure "how much" we have of an everyday object in different ways (mass, volume, length, etc) we can measure "how much set" we have in many different ways.

(as well as previews, behind-the-scenes, and more content).

Music by Sakis Strigas:

My current equipment for making videos (affiliate links):

[EpicMathTime]

"A wizard is never late, nor is he early, he arrives precisely when he means to." 😂

Hope you guys liked the new video! As you can see, I've made quite a few updates/changes to some visuals and style of the videos. Feel free to let me know how you felt about these changes, as I'm still making adjustments.



Thank you so much for being a part of this community!

Jon

[duncanw]

For all those who need it: The infimum of a set is the largest lower bound; i.e. for the set {2, 3, 4} the infimum is 2 because it is the greatest number less than or equal to every element in the set. It is not simply the smallest element in the set, because if we have the set {1/n, n ∈ N} has no least element, but has infimum 0.

[haminatmiyaxwen]

Learning about the Lebesgue Measure while Fear Inoculum plays in the background just made the video go from great to perfect!

[samhollins]

I remember watching your videos when you had 500 subs thinking "I can't believe this guy only has 500 subs."

Now I'm watching at 8000 and your quality is still just as ahead of your sub count as it was back then.

_I can't believe this guy only has 8000 subs._

You are destined to be huge. I'm serious. Don't stop. You're a wonder.

[TheNiTeMaR]

Doing measure theory this year, thanks for the head start! Your content is formatted brilliantly.

[caladbolg]

Cardinality is a common go-to when math folk try to popularize math. Some people get wowed by the counter-intuitive ideas like N and Z being "the same size", but sometimes (esp online) some laymen see it as sophistry (Z is obviously bigger!). And the response from mathematicians is usually not the best.

So I'm really glad you pointed out that cardinality is not THE way to compare sets, just one of possible tools. In fact, no useful notion of set size extends finite size comparison in a completely intuitive manner.

Another cool one is natural density for subsets of N. Here there are "as many" evens as odds, and that's "half" of naturals.

[OtiumAbscondita]

I'm the number one fan, and that's why I watch, like and comment on your videos before they go public!

[TheBlundertker]

Am I really going to be the first one to comment about Fear Inoculum coming on halfway through?


No offense Jon, but that was the best part! XD


Great vid BTW

[masontdoyle]

I do undergraduate research in measure theory and optimal transportation. I was super excited to see this video drop! Great explanation of everything!

[ozzyfromspace]

That’s my first time seeing someone prove that the Lebesgue measure of the Natural numbers is zero. Thank you, this video was epic! I honestly smiled pretty hard through it! 😁✨🙌🏽🎊

[diegovera]

This is so fantastic ! so easy to understand thank you so

[destroctiveblade]

So a fellow mathematician who's also a tool fan huh, I think we would get along quite well, and it just happens that I am studying the lebesgue measure this semester so thank you this wqs useful

[freyachandra]

Loved this video! I just love your creativity :)

[AdoptedWartortle]

This was cool! Great way to get back into the math mood for back to school.

[victor]

You should do an analysis course, or metric spaces, or topology, really any uni math course would be awesome. Love your videos, you explain things perfectly!

[ldbspg]

I love abstract/theoretical measure theory!(I dislike applied math, so NOT probability) Thank you so much! I definitely think you should upload more measure theory videos(Jordan, Hausdorff, etc.)!

[konstantinkostov]

Mixing tool with measure theory, I see you are a man of culture. 😎

[perappelgren]

Maybe I missed it, but what can be said of the intersection of two members of an open interval family? Are they tacitly understood to be disjoint?

Most important: Thanks 🙏 once again for a brilliant video!

[tristanlovsin]

Love the vid, tool being in it only makes it better :D

[jimvouts]

Thnx so much for putting TOOL on the video