Integral

this is the integral $\int_{\mathbb{R}} 1_{\mathbb{Q}}(x) \, d \mu(x)$ , where $1_{\mathbb{Q}}(x)$ is the indicator function on the reals that evaluates to 1 if $x$ is rational, and to $0$ otherwise. $\mu$ is Lebesgue measure. Then this integral is simply $\mu(\mathbb{Q}) = 0$ , since the rationals have zero Lebesgue measure.