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July 12th, 2018, 08:58 PM   #1
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Lebesgue integration - Riemann integration

What the differences between Lebesgue integration and Riemann integration?

Last edited by skipjack; July 13th, 2018 at 06:48 AM.
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July 12th, 2018, 09:41 PM   #2
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Here is the difference in Henri Lebesgue's own words.


Quote:
Originally Posted by Henri Lebesgue
I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.
https://en.wikipedia.org/wiki/Lebesgue_integration

Or as my professor said: In Riemann integration we chop up the x-axis into little parts. In Lebesgue integration we chop up the y-axis into little parts.

Or, in Riemann integration we approximate a given function with step functions. In Lebesgue integration we approximate a given function with simple functions.

Last edited by Maschke; July 12th, 2018 at 09:47 PM.
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July 13th, 2018, 04:11 AM   #3
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Riemann integration requires that the region of integration be an interval or a union of intervals. Lesbesque integration only requires that the region of integration be a union of measurable sets.

Last edited by skipjack; July 13th, 2018 at 06:49 AM.
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July 13th, 2018, 10:01 AM   #4
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Quote:
Originally Posted by Country Boy View Post
Lesbesque integration only requires that the region of integration be a union of measurable sets.
Any countable union of measurable sets is a measurable set. So Lebesgue integration requires that the domain of integration is a measurable set.

And we can extend Riemann integration to handle more complex domains, but that isn't used much in the single variable case, but a lot in multiple dimensions.
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