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 $[0,1]$)

It is not Riemann integrable since inside any interval in the partition of the domain, there will be both $x_0\in \mathbb{Q}$ and $x_1\in \mathbb{R}\backslash \mathbb{Q}$. 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.

$\int fd\mu =1\cdot \mu ([0,1]\cap \mathbb{Q})+0\cdot \mu ([0,1]\backslash \mathbb{Q})=1\cdot 0+0\cdot 1=0$