Measure, Integration & Real Analysis · Problem 1B.1

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integrable-thomae-function

Problem

Define

f : [0, 1] \to \mathbf{R}

as follows:

f(a) = \begin{cases} 0 & \text{if } a \text{ is irrational,} \\ \frac{1}{n} & \text{if } a \text{ is rational and } n \text{ is the smallest positive integer such that } a = \frac{m}{n} \text{ for some integer } m. \end{cases}

Show that

f

is Riemann integrable and compute

\displaystyle\int_0^1 f