Lebesgue's theorem on Riemann integration

Question

Can someone provide me with some easy examples (or suggest me a book) to understand Lebesgue's theorem on Riemann integration.

Answer 1

That theorem says:

A bounded function $f\colon[a,b]\longrightarrow\Bbb R$ is Riemann-integrable if and only if the set$$\{x\in[a,b]\mid f\text{ is discontinuous at }x\}$$has Lebesgue measure $0$.

Here are two examples:

• If $f\colon[a,b]\longrightarrow\Bbb R$ is defined by$$f(x)=\begin{cases}1&\text{ if }x\in\Bbb Q\\0&\text{ otherwise,}\end{cases}$$then $f$ is discontinuous at every point of $[a,b]$. So, $f$ is not Riemann-integrable.
• If $f\colon[0,1]\longrightarrow\Bbb R$ is defined by$$f(x)=\begin{cases}\frac 1n&\text{ if }x=\frac mn&\text{ with }m\in\Bbb Z,\ n\in\Bbb N\text{, and }\gcd(m,n)=1\\0&\text{ otherwise,}\end{cases}$$then $f$ is discontinuous at the rationals and only at the rationals. Since the Lebesgue measure of $\Bbb Q\cap[0,1]$ is $0$, $f$ is Riemann-integrable (and its integral is $0$).