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This is a follow-up to some of my comments to the OP and to the answer that @Conifold gave. A few days ago I purchased a copy of the book that I had mentioned, Measure and the Integral by Lebesgue (1966). I was correct that this book is where, many years ago (1980s, possibly even late 1970s), I had read about Lebesgue's fascination with this “paradox”. ...


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It depends on what counts as working "on it". His prior work under Hilbert was related to this area of geometry. Bolyai (1832) and Gerwein (1833) proved that polygons of equal area are equidecomposable, and Gauss urged a 3D extension in letters to Gerling mentioned by Hilbert. It was in the works in 1890-s. In 1896 Bricard reproved Gerling's 1844 result that ...


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