I heard that one of the drawback of Riemann integral is;it is very laborious to find Riemann integral in higher dimensions e.g Calculating Riemann integral of bounded function f : D ⊂ R 2 → R ,where D is compact domain of R 2 .We can find it but it will require more work,and technically it is much difficult.As we know in lebesgue integration,we do partition of range rather than domain as we do in Riemann integration,and therefore lebesgue integration is believe to be easier to apply in case of higher dimensions rather than Riemann integral.But to calculate integral of f : D ⊂ R 2 → R is same as double integral and I never saw any lebesgue approach to double or triple integral.Can we actually find double integral via lebegue's technique of integration?Is this technique easier as observed by me?If it is so then why don't I see lebesgue's technique to calculate double integral

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I heard that one of the drawback of Riemann integral is;it is very laborious to find Riemann integral in higher dimensions e.g Calculating Riemann integral of bounded function   f  :  D  ⊂            R      2        →      R  ,where   D is compact domain of             R      2      .We can find it but it will require more work,and technically it is much difficult.As we know in lebesgue integration,we do partition of range rather than domain as we do in Riemann integration,and therefore lebesgue integration is believe to be easier to apply in case of higher dimensions rather than Riemann integral.But to calculate integral of   f  :  D  ⊂            R      2        →      R   is same as double integral and I never saw any lebesgue approach to double or triple integral.Can we actually find double integral via lebegue&#039;s technique of integration?Is this technique easier as observed by me?If it is so then why don&#039;t I see lebesgue&#039;s technique to calculate double integral

arhaitategr · Accepted Answer

We usually use change of variables theorem and Fubini/Tonelli theorem to convert integrals over             R        n   to iterated integrals. These are all theorems from measure theory. While they can be stated and proved for the Riemann integral, they will be less general than the corresponding statements for the Lebesgue integral, and the proofs aren&#039;t that much easier.

I heard that one of the drawback of Riemann integral is;it is very laborious to find Riemann integra

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