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July 12th, 2018, 08:58 PM | #1 |
Senior Member Joined: Nov 2011 Posts: 248 Thanks: 3 | 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 | |
Senior Member Joined: Aug 2012 Posts: 2,156 Thanks: 630 |
Here is the difference in Henri Lebesgue's own words. Quote:
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 |
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 |
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 | |
Senior Member Joined: Oct 2009 Posts: 732 Thanks: 249 | Quote:
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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integration, lebesgue, riemann, rimman |
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