Past exam of the integration course of the University of Oxford / 2023 / 1 / a / ii / Solution

The density of the rational numbers gives a countable way to separate two unequal values:Indeed, $f(x)<g(x)$ holds exactly when some rational number lies strictly between them. Every set inside the union is a Lebesgue measurable set, by the definition of a measurable function and the closure properties of a sigma-algebra. Since $\mathbb{Q}$ is countable, the union is measurable.