My Math Forum Lebesgue integration - Riemann integration

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 July 12th, 2018, 07:58 PM #1 Senior Member   Joined: Nov 2011 Posts: 224 Thanks: 2 Lebesgue integration - Riemann integration What the differences between Lebesgue integration and Riemann integration? Last edited by skipjack; July 13th, 2018 at 05:48 AM.
July 12th, 2018, 08:41 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 08:47 PM.

 July 13th, 2018, 03:11 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 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 05:49 AM.
July 13th, 2018, 09:01 AM   #4
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Quote:
 Originally Posted by Country Boy 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.

 Tags integration, lebesgue, riemann, rimman

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