Editing Set of uniqueness (section)

== Early research ==
The [[empty set]] is a set of uniqueness. This simply means that if a trigonometric series converges to zero ''everywhere'' then it is trivial. This was proved by [[Bernhard Riemann|Riemann]], using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using [[Toeplitz operator]]s. Later on, [[Georg Cantor]] generalized Riemann's techniques to show that any [[Countable set|countable]], [[closed set]] is a set of uniqueness, a discovery which led him to the development of [[set theory]]. [[Paul Cohen]], another innovator in set theory, started his career with a thesis on sets of uniqueness. 

As the theory of [[Lebesgue integration]] developed, it was assumed that any set of zero [[Measure (mathematics)|measure]] would be a set of uniqueness &mdash; in one dimension the locality principle for Fourier series shows that any set of positive measure is a set of multiplicity (in higher dimensions this is still an open question). This was disproved by [[Dmitrii Menshov|Dimitrii E. Menshov]] who in 1916 constructed an example of a set of multiplicity which has measure zero.