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

Writing $\mathcal{L}$ for the collection of Lebesgue measurable sets, saying that $\mathcal{L}$ is a sigma-algebra means that $\mathbb{R}\in \mathcal{L}$, that $\mathbb{R}\setminus A\in \mathcal{L}$ whenever $A\in \mathcal{L}$, and that ${\bigcup }_{n=1}^{\infty }{A}_n\in \mathcal{L}$ whenever every $A_n\in \mathcal{L}$. These properties also imply closure under countable intersections and differences.

A real-valued function $f$ is a Lebesgue measurable function precisely when the inverse image of every Borel set is in $\mathcal{L}$. Equivalently, it suffices to test the half-lines:The equivalence follows because these half-lines generate the Borel sigma-algebra, and inverse images preserve complements and countable unions.

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.