Real Analysis | Refinements of partitions.

We introduce the notion of a refinement of a partition, give an example, and prove a few results related to the lower and upper sum.
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[@qilejiang8697]

thank you very much

[@MClilypad]

Isn't your tumbnail wrong? Q is a refinement of P so L(Q) should be larger equal than L(P) right?

[@binadisenuthmi333]

Yeah it's wrong

[@jonaskoelker]

Here an mildly interesting idea: given two partitions, their union is a tighter fit than either input. [Oh, he got there.] That prompted some self-reflection. Here's the result: you might be an algebraist if 1. When given two lists of indicators that you might be an algebraist, you instinctively think about how to combine them. 2. When given two lists of indicators that you might be an algebraist, you instinctively think about how to combine them. [Repeat this for all non-negative powers of two, then for all finite sums of distinct non-negative powers of two.]

[@tomholroyd7519]

Interval orders are fascinating,. Here you've got an ordering of intervals with the same endpoints! It seems you could build a poset or lattice using refinement, and head off into multi-valued logic, non-boolean algebra, etc. What happens if the endpoints are [-inf, +inf]? I need to watch the previous video again

[@andelsonmelicio9886]

Que vídeo fascinante! Vendo de Angola 🇦🇴

[@punditgi]

I hope Michael will elaborate in another video on why the lemma on the common refinement at the end of this video is true

[@noahtaul]

Which lemma are you talking about? He only mentioned that the upper sum of Q is less than the upper sum of P2, but that’s almost exactly the same as the proof he wrote for the lower sums.

[@sophiaxiao5071]

Why do you prove the lemma using induction? My professor proved this by taking the infimum or supremum over a fixed partition. I honestly didn't understand my professor's proof.

[@luistorresferrer9038]

Awesome, thank u! 🤙

[@tomatrix7525]

Very good, i think a very basc visual aid would conpliment nicely though

[@goodplacetostop2973]

16:22

[@user-ii3du3xv2x]

omg that is great

[@user-qg6do2xn9t]

I am from Kazakhstan

[@kenichimori8533]

Partition is very largical scale of scala.bytes.

[@arvindsrinivasan424]

I love real analysis... but I was hoping for something about number theory partitions

[@hybmnzz2658]

Hilarious that his last video was about partitions but in number theory.

[@arvindsrinivasan424]

Lol, I know but.... you know like his usual style or some more in depth stuff/proofs

[@ajaykumarbarik4826]

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