Compute the following Lebesgue integral $\int_{[0,\pi/2]}fd\mu$

Question

The following problem is from the book "Theorems and problems in Functional Analysis" by A. A. Kirillov and A.D. Gvishiani:

Compute $\int_{[0,\pi/2]}fd\mu$ when $$f(x)=\begin{cases}\sin(x)\,\,\,\,\,\,\text{ if $\cos(x)$ is rational}\\ \sin^2(x)\,\,\,\,\,\,\text{ if $\cos(x)$ is irrational} \end{cases}$$

My reasoning is:

Let $D=\{x\in[0,\pi/2]:\cos(x) $ is rational$\}$ then $\mu(D)=0$

Therefore $$\int_0^{\pi/2}fd\mu=\int_0^{\pi/2}\sin^2(x)dx=\frac12\int_0^{\pi/2}1-\cos(2x)dx=\frac{\pi}{4}$$

Is it right? Somehow I am not quite certain about my solution. Any help would be highly appreciated. Thanks in advance.

Answer 1

Your proof is essentially correct, you are just missing a rigorous proof the fact that $D$ is a Lebesgue-nullset.

To see this, observe that $x \mapsto \cos(x)$ is a strictly monotone decreasing function on the interval $[0, \pi/2]$. Therefore, for each rational number $r \in \mathbb{Q}$ we have at most one $x \in [0, \pi/2]$ that satisfies $\cos(x) = r$. But this means that $D$ is (at most) countable an therefore a Lebesgue-nullset.