Measurable solutions of Cauchy's equation

Remarks

The proof reduces the claim to boundedness on an interval. By Lusin's theorem, $f$ is bounded on some compact set $K$ of positive measure. By the Steinhaus theorem, the difference set $K - K$ contains an open interval around $0$. Each point $x = a - b$ of that interval with $a, b \in K$ has $f(x) = f(a) - f(b)$, so $f$ is bounded on this interval. Then Darboux's theorem on additive functions gives $f(x) = cx$. Measurability cannot be dropped: a Hamel basis gives additive functions that are not linear, and such functions are not Lebesgue measurable. This is one of the regularity conditions that settle Cauchy's functional equation.

Going deeper

For a student

Measurability is weaker than continuity, monotonicity or local boundedness, so this result contains the other classical criteria. In practice, any additive function that comes from an explicit formula, a limit of continuous functions or a probability model is measurable, so it is linear. The same argument works for additive maps between $\R^n$ and $\R^m$. More generally, a Haar measurable homomorphism between second countable locally compact groups is continuous. The key tools, Lusin's theorem and the Steinhaus theorem, are standard in measure theory.