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I am wondering if there are some other examples of Riemann non-integrable but Lebesgue integrable, besides the well-known Dirichlet function.

Thanks.

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Yuval's link shows the general criterion for bounded functions on a bounded interval, namely continuity almost everywhere. The Dirichlet function is a good example, but it is almost everywhere zero, and the zero function sure is Riemann integrable. If you consider the characteristic function of a Cantor set of positive measure, then it is not Riemann integrable because it is discontinuous at each point in this "fat" Cantor set, and you cannot modify the values on a set of measure 0 to fix this defect.

The same idea would work for any closed nowhere dense subset $E\subset[0,1]$ of positive measure, and you can appeal to the definition to see this. Because every subinterval of $[0,1]$ has nonempty intersection with $[0,1]\setminus E$, the lower Riemann sums of $\chi_E$ are all $0$. However, the upper Riemann sums are all bounded below by $m(E)\gt0$.

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    $\begingroup$ But why is the Dirichlet function not Riemann integrable? It is continuous at the irrationals, which have full measure. $\endgroup$ – t.b. Jan 21 '11 at 4:44
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    $\begingroup$ @THeo: There are variants named the same. For instance, some people call the characteristic function of the rationals as a Dirichlet function (which is probably what OP meant). $\endgroup$ – Aryabhata Jan 21 '11 at 4:47
  • $\begingroup$ @Theo,@Moron: Yes, I equate Dirichlet function and characteristic function of the rationals. $\endgroup$ – Jonas Meyer Jan 21 '11 at 4:49
  • $\begingroup$ @Jonas,@Moron: Thanks for clearing this up... $\endgroup$ – t.b. Jan 21 '11 at 4:53
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    $\begingroup$ @Theo: Are you're thinking of this function: en.wikipedia.org/wiki/Thomae%27s_function? $\endgroup$ – Hans Lundmark Jan 21 '11 at 11:30
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For me the "canonical" example is the characteristic function of the rational numbers in $[0,1]$. The upper integral is one, the lower integral is zero.

Edit: Jonas and Moron have informed me that some people call this example Dirichlet function (and I vaguely remember that we might have done so as well in our Analysis class).

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  • $\begingroup$ +1. This is the first one which comes to my mind when someone asks this question $\endgroup$ – user17762 Jan 21 '11 at 4:47
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Check the Wikipedia article on the Riemann integral for some ideas.

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  • $\begingroup$ I know the conclusion, but examples please! $\endgroup$ – Qiang Li Jan 21 '11 at 4:19
  • $\begingroup$ @Qiang Li: In that case, it would be nice if you included more information in your question. This is a very good answer to the question asked (rather than intended), especially as it asks "what functions or classes of functions are Riemann non-integrable but Lebesgue integrable?": This answer leads you to the class of bounded measurable functions on a bounded interval that are discontinuous on a set of positive measure as a partial answer. $\endgroup$ – Jonas Meyer Jan 21 '11 at 5:04

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