Why the Dirichlet function is Lebesgue integrable?

The post introduces the concept of Lebesgue integration through measure theory, explaining how measures generalize notions like length, area, and volume. It defines key concepts such as $\sigma$-algebras, measures, measurable functions, and the Lebesgue integral. The Dirichlet function, which is $1$ on rational numbers and $0$ on irrationals, is used as an example. Although the function is nowhere continuous, it is simple and measurable. Since the set of rationals has Lebesgue measure zero, the integral of the Dirichlet function over $\mathbb{R}$ is zero, demonstrating how Lebesgue integration can handle functions that are not Riemann integrable.