Measurable additive function is continuous
= Measurable additive function is continuous
Every Lebesgue-measurable additive function $f:\mathbb R^n\to\mathbb R^m$ is continuous. Small-value preimages are measurable; one of their integer dilates has positive measure, so the <Steinhaus theorem> puts a neighbourhood of zero in their difference set. Additivity then makes $f$ small on that neighbourhood.