Function Riemann integrable that are not Lebesgue integrable.

Question

I had an exam today and one of my question was: Give a function $f$ that is Riemann integrable but not Lebesgue integrable

How is it possible ? I always thought that Riemann $\implies $ Lebesgue, isn't it ?

Answer 1

A function $f$ is Lebesgue integrable if and only if $f^+=0\vee f$ and $f^-=0\wedge f$ are integrable. So if $f\geq 0$ is Riemann integrable then it will be also Lebesgue integrable. But if $f$ has not a constant sign, then $f$ can be Riemann integrable but not Lebesgue integrable as $f(x)=\frac{\sin(x)}{x}$ on $[0,\infty [$. And this is because $\int f^+=\int f^-=+\infty$. Therefore $\int f=\int f^+-\int f^-$ is not defined.

Answer 2

You may like to visit http://www.math.vanderbilt.edu/~schectex/ccc/gauge/ for an excellent introduction to the different integrals. I am reproducing an image from that website for your quick reference.