Finding lebesgue integral of piecewise function

Question

Consider the function $f(x)=\begin{cases} x & x\in \mathbb{Q}\cap [0,1] \\ -x & x\in [0,1]-\mathbb{Q} \\ \end{cases}$

How do you go about computing the following Lebesgue integral?

$$\int_0^{1}f(x)d\mu$$

I need to solve a similar problem, and though I found this on a different post from a year ago, the post dealt only with integrability and not how to solve it. My textbook does not have anything similar. Any help would be greatly appreciated.

Answer 1

• Prove the integral is defined (not difficult)
• Write $\int_{[0,1]} fd\mu = \int_{[0,1]\cap\mathbb{Q}} fd\mu + \int_{[0,1]\setminus\mathbb{Q}} fd\mu$, and observe that
  • The set $[0,1]\cap\mathbb{Q}$ has measure $\mu( [0,1]\cap\mathbb{Q}) = 0$; what can you say about the first integral?
  • Similarly, as $\mu \mathbb{Q} = 0$, what can you say about $\int_{[0,1]} (-x)\mu(dx)$? Use it for the second one, as $$\int_{[0,1]\setminus\mathbb{Q}} f d\mu = \int_{[0,1]} (-x) \mu(dx) - \int_{\mathbb{Q}} (-x) \mu(dx)$$