Solving this discontinuous integral using Lebesgue

Question

Not a duplicate look at $f(x)$ here!

Suppose we are to evaluate:

$$I = \int_{0}^{1} f(x) dx$$

Where

$$f(x)=\begin{cases}1 \space \text{if} \space x\space \text{is irrational}, & \newline 0 \space \text{if} \space x \space \text{is rational} \\ \end{cases}$$

How can I solve this using measure-theory or Lebesgue integrals?

Answer 1

HINT:

1. $\int_{[a,b]}1 dx=b-a$.

2. If $m(\{x:f(x)\neq g(x)\})=0$ then $\int f\,dm=\int g\,dm$.