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/sci/ - Science & Math

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8437856 No.8437856 [Reply] [Original]

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.

>> No.8438098

>>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?

>> No.8438101

>>8437856
spoiler: if the domain of integration is unbounded then there are riemann integrable functions which are not lebesgue integrable.

>> No.8438107

>>8438101
correct me if I'm wrong, but isn't it rather the Kurzweil-Henstock integral ?

>> No.8438128

>>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.

>> No.8438135

>>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.

>> No.8438167

>>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./

>> No.8438807
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8438807

>>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}

>> No.8438815

>>8437856
Because its analysis not topology/measure theory/whatever the flavour of the week is

>> No.8438818

>>8437856
Because the Lebesgue integral is taught in a Real Analysis course.

>> No.8439014

>>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.

>> No.8439145

>>8437856
WHY SO MANY RIEMANN THREADS IN THE LAST FEW DAYS?

>> No.8439153

>>8439145

it's probably the same brainlet making them

>> No.8439155

>>8439145
They become more accurate as their quantity approaches infinity

>> No.8439281

>>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.

>> No.8439325

>>8439281
The gauge integral is only defined for subsets of R^n. The lebesgue integral is defined for arbitrary measure spaces.

>> No.8439335

>>8438807
Good description, but I prefer to write such an S in terms of indicator functions.

>> No.8439347

>>8439014
Not enough epsilons to be called analysis imo