Measure, Integration & Real Analysis · Problem 2B.30

Measures — Sheldon Axler

Mathematics — Measure Theory · measurable-functions

Problem

Show that

\lim_{j \to \infty}\left(\lim_{k \to \infty}\left(\cos(j!\pi x)\right)^{2k}\right) = \begin{cases} 1 & \text{if } x \text{ is rational,} \\ 0 & \text{if } x \text{ is irrational} \end{cases}

for every

x \in \mathbf{R}

. [This example is due to Henri Lebesgue.]