Not riemann integrable => not lebesgue integrable ?

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A function that is not Riemann integrable does not necessarily imply it is also not Lebesgue integrable. For instance, the characteristic function of the rationals is Lebesgue integrable but not Riemann integrable. Functions can be Lebesgue integrable even if they are not bounded, as shown with the example of 1/x on (0,1]. The discussion highlights that Lebesgue integrability is a broader concept, allowing for certain functions to be integrable under Lebesgue's framework while failing Riemann's criteria. Overall, the relationship between Riemann and Lebesgue integrability is nuanced, with specific conditions determining integrability in each case.
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you know that if f is riemann integrable then it is also lebesgue integrable. if it were true also that f not riemann integrable implied f not lebesgue integrable, then the two notions of integrability would be the same. then no one would bother with lebesgue integrals since they would not give anything new.
 

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