Examples of Lebesgue-integrable, but not Riemann-integrable functions

Question

The standard example of this is the characteristic function of the rationals. However this is somewhat pathological as this function is zero almost everywhere. What are other examples that differ from a Riemann-integrable function on more than a set of measure zero?

Answer 1

Take an enumeration $\{x_1,x_2,\dots,x_n,\dots\}$ of the rationals in the interval $[0,1]$. For each $n \in \mathbb{N}$, define $$G_n=\left(x_n-\dfrac{1}{2^{n+2}},x_n+\dfrac{1}{2^{n+2}}\right).$$

Note that $\lambda(G_n)=\frac{1}{2^{n+1}}$.

Now define the set $$F=[0,1]\cap(\cup_{n=1}^\infty G_n)^c$$.

$F$ is clearly Lebesgue mensurable, and, because

$$\lambda\left(\bigcup_{n=1}^\infty G_n \right) \le \sum_{n=1}^{\infty} \lambda(G_n) = \sum_{n=1}^{\infty}\dfrac{1}{2^{n+1}}=\dfrac{1}{2}, $$

we have $$\lambda(F)>\dfrac{1}{2},$$

and $$\int \chi_F d\lambda>\dfrac{1}{2}.$$

$\chi_F$ is the charachteristic function of $F$.

However, $\chi_F$ is not Riemann integrable. Note that $F$ contains no interval, because it doesn't contain any rationals, so any interval will contain points that are not in $F$ $F$. Therefore, the minimum of $\chi_F$ in any interval will be $0$, and $$\underline{\int}\chi_F dx=0.$$