Why is the Riemann/Darboux Integral still taught in undergrad analysis classes? The lebesgue integral is much more useful and is only slightly more complicated when first learning it.
Afterwards it is actually easier.
>>8437856
Would you mind teaching it to us now? I can at least visualize rectangles, what am I and the rest of us brainlets supposed to think when doing a Lebesgue integral?
>>8437856
spoiler: if the domain of integration is unbounded then there are riemann integrable functions which are not lebesgue integrable.
>>8438101
correct me if I'm wrong, but isn't it rather the Kurzweil-Henstock integral ?
>>8438107
This is why I hate giving human names to things, just call it an improper integral or an unbounded riemann integral. No one will know what you mean if you say Kurzweil-Henstock integral.
>>8438128
Here, let me help you:
https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral
It's not just an unbounded riemann integral btw.
>>8437856
Easier because you do not have to construct the lebesgue's mesure you faggot.
If you construct it properly, it is far more difficult than the Riemann/Darboux integral./
>>8438098
mu being the measure on our set M. For M subset R, you can just take mu to be the lebesgue measure which is intuitively simple.
Let mu be the lebesgue measure on R.
- Then mu( [a,b] ) = b - a.
- For general S subset R, mu(S) = inf { sum over i of mu(I_i) such that union over i of I_i contains S}
>>8437856
Because its analysis not topology/measure theory/whatever the flavour of the week is
>>8437856
Because the Lebesgue integral is taught in a Real Analysis course.
>>8438815
>Because its analysis not topology/measure theory
Usually like half of an undergrad analysis course is just sequential topology. And measure theory isn't something studied independently from analysis.
>>8437856
WHY SO MANY RIEMANN THREADS IN THE LAST FEW DAYS?
>>8439145
it's probably the same brainlet making them
>>8439145
They become more accurate as their quantity approaches infinity
>>8437856
> Complains about Riemann Integral
> Proposes teaching Lebesgue Integral
> No mention of gauge integral even though it allows far more functions to be integrable and is basically identical to the Riemann integral.
I'd insert a clever comment but I don't have the mental capacity to waste on such a scrub.
>>8439281
The gauge integral is only defined for subsets of R^n. The lebesgue integral is defined for arbitrary measure spaces.
>>8438807
Good description, but I prefer to write such an S in terms of indicator functions.
>>8439014
Not enough epsilons to be called analysis imo