Early Research
The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. This was proved by Riemann, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators. Later on, Cantor generalized Riemann's techniques to show that any countable, closed set is a set of uniqueness, a discovery which led him to the development of set theory. Interestingly, Paul Cohen, another great 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 would be a set of uniqueness — 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 D. E. Menshov who in 1916 constructed an example of a set of multiplicity which has measure zero.
Read more about this topic: Set Of Uniqueness
Famous quotes containing the words early and/or research:
““Never hug and kiss your children! Mother love may make your children’s infancy unhappy and prevent them from pursuing a career or getting married!” That’s total hogwash, of course. But it shows on extreme example of what state-of-the-art “scientific” parenting was supposed to be in early twentieth-century America. After all, that was the heyday of efficiency experts, time-and-motion studies, and the like.”
—Lawrence Kutner (20th century)
“Our science has become terrible, our research dangerous, our findings deadly. We physicists have to make peace with reality. Reality is not as strong as we are. We will ruin reality.”
—Friedrich Dürrenmatt (1921–1990)