Incoming links: Riemann integral
Lebesgue integral Updated 2025-07-16
"More complex and general" integral. Matches the Riemann integral for "simple functions", but also works for some "funkier" functions that Riemann does not work for.
Ciro Santilli sometimes wonders how much someone can gain from learning this besides the beauty of mathematics, since we can hand-wave a Lebesgue integral on almost anything that is of practical use. The beauty is good reason enough though.
Lebesgue integral of is complete but Riemann isn't Updated 2025-07-16
is:
- complete under the Lebesgue integral, this result is may be called the Riesz-Fischer theorem
- not complete under the Riemann integral: math.stackexchange.com/questions/397369/space-of-riemann-integrable-functions-not-complete
And then this is why quantum mechanics basically lives in : not being complete makes no sense physically, it would mean that you can get closer and closer to states that don't exist!
The missing link between basic and advanced Updated 2025-07-16
One of Ciro Santilli's strongest feeling in education is that material often falls in either of the two categories:
- hundreds of too basic popular science, e.g.:
- a 5 minute popular science video trying to explain quantum electrodynamics (an advanced subject) for someone who doesn't know what a Riemann integral is (a basic subject)
- a few full university courses that takes 20 hours to deliver the first punchline of the course
Ciro believes that there is often an important missing link between them, e.g.:
If we as a society are unable to provide this sweet Middle Way sweet-spot, it is unreasonable to expect that learners will ever have the motivation to advance, because it is just too boring! They are just more likely to go play video games instead.
It is Ciro's hope that OurBigBook.com will help to fill exactly that gap.
In Ciro's view, as of the 2020's this critical gap generally lies somewhere between the end of undergraduate studies, and at the start of postgraduate studies.
Let's take the gloves off more often, and give the full thing to interested students! Let students learn what they want to learn, and do that as soon as possible! Life is too short!
This problem is basically the knowledge version of the last mile problem. When we reach the end of graduate, there are enough directions of knowledge to go off into, that the probability that a great free tutorial exists is relatively low. Of course, as one approaches the realm of novel research, the branching is so wide that having perfect tutorials becomes impossible. Ciro's goal in life go push the last mile marker a bit further out.
Related:
- universityphysicstutorials.com/ by Adam Beatty mentions:
There are myriad resources for physics and maths. The Kahn Academy and Patrick JMT were the best for me. They really helped me out. The question is, what resources are there for the advanced undergraduate courses?
In this video, the noted chemist mentions how he managed to get into a chemistry research development before he even joined university, due to a somewhat exceptional situation. Section "The only reason for universities to exist should be the laboratories" also comes to mind. This is exactly the type of thing that Ciro Santilli wants to make much more widespread.
Stories of Ainan Celeste Cawley fighting to advance his kids education beyond school, and being forbidden to do so by a stupid educational system, also come to mind.
Bibliography: