Continuous Abelian Square

A continuous abelian square for a measurable function is an interval of positive length such that each color occupies equal Lebesgue measure in the two halves. Thus, writing for Lebesgue measure, the condition is

for every , ..., . This is the measure-theoretic analog of an abelian square in a word.

Salo et al. (2026) constructed a measurable function with three colors and no continuous abelian square. Every two-color measurable function has such a square, so three is the minimum. The construction refutes the case of Lasoń's conjecture (Salo et al. 2026). In contrast to four colors, the three-color square-avoiding colorings form a set of first category in the space of measurable colorings with the topology of local convergence in measure, identifying colorings that agree almost everywhere.