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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. 
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 xaxis into little parts. In Lebesgue integration we chop up the yaxis 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.  
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. 
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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