not Riemann integrable function

An example of a function that is not Riemann integrable but is Lebesgue integrable is the Dirichlet function (along the interval )

It is not Riemann integrable since inside any interval in the partition of the domain, there will be both and . So, the Riemann lower sum will be 0 and the upper sum will always be 1.

The Lebesgue integral doesn’t have any problems since the Dirichlet function is Borel measurable function. In fact, it’s a simple function, so its even easier to integrate.