Consider .

Note that , and

Thus , and is improper Riemann integrable.

However note that

which diverges as (harmonic series).

Thus is not Lebesgue integrable on .

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# sin(1/x)/x is improper Riemann Integrable but not Lebesgue Integrable on (0,1]

Consider .

Note that , and

Thus , and is improper Riemann integrable.

However note that

which diverges as (harmonic series).

Thus is not Lebesgue integrable on .

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