Thu, 06 Feb 2020
[ Previously: “Cases in which some statement S was considered to be proved, and later turned out to be false”. ]
In 1905, Henri Lebesgue claimed to have proved that if !!B!! is a subset of !!\Bbb R^2!! with the Borel property, then its projection onto a line (the !!x!!-axis, say) is a Borel subset of the line. This is false. The mistake was apparently noticed some years later by Andrei Souslin. In 1912 Souslin and Luzin defined an analytic set as the projection of a Borel set. All Borel sets are analytic, but, contrary to Lebesgue's claim, the converse is false. These sets are counterexamples to the plausible-seeming conjecture that all measurable sets are Borel.
I would like to track down more details about this. This Math Overflow post summarizes Lebesgue's error: