# Otto M. Nikodym
**Otto M. Nikodym** (Otton Marcin Nikodym; 13 August 1887 – 4 May 1974) was a Polish mathematician whose name is attached to the Radon–Nikodym theorem of measure theory, the Nikodym set, and the Nikodym–Grothendieck boundedness theorem of functional analysis[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) • [2](https://ems.press/content/serial-article-files/9616). The set he constructed in 1927 remains an active object of research[3](https://terrytao.wordpress.com/2025/11/12/new-nikodym-set-constructions-over-finite-fields/).
| Key fact | Detail |
|---|---|
| Born / died | 13 August 1887, Zabłotów, Galicia, Austria-Hungary (now Zabolotiv, Ukraine); 4 May 1974, Utica, New York[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) |
| Doctorate | 26 June 1925, Warsaw University, thesis on A sets, formally advised by Wacław Sierpiński[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) |
| Signature result | Radon–Nikodym theorem: Radon proved it in 1913 for Rⁿ, Nikodym in 1930 for a σ-finite measure and an absolutely continuous countably additive set function[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) |
| 1930 paper | "Sur une généralisation des intégrales de M. J. Radon", *Fundamenta Mathematicae* 15, pp. 131–179[4](https://link.springer.com/rwe/10.1007/978-3-662-69359-9_710) |
| US career | Kenyon College, Gambier, Ohio, 1948–1965; U.S. citizen 29 June 1953[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) |
| Last book | *The Mathematical Apparatus for Quantum-Theories, based on the Theory of Boolean Lattices* (Springer-Verlag, 1966), almost a thousand pages[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) |
## Life and career
Nikodym was born in Zabłotów, in the Galician province of [Austria-Hungary](https://www.edgepedia.org/austria-hungary), and studied at the Universities of Lwów and Warsaw, and at the Sorbonne[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) • [5](https://history.aip.org/catalog/icos/305.html). He spent 1925–26 in Paris on a small Polish scholarship, received his doctorate from Warsaw University on 26 June 1925 for a Polish-language thesis on A sets (a class of point sets in the descriptive-set-theoretic hierarchy), formally advised by Sierpiński, and took his habilitation there on 21 June 1927[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/).
**Interwar Poland.** In March 1928 he transferred to the [Jagiellonian University](https://www.edgepedia.org/jagiellonian-university) in Kraków, and he also taught at the High Polytechnical School in Kraków[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) • [5](https://history.aip.org/catalog/icos/305.html). Between 1930 and 1945 in Warsaw he published 32 papers and four textbooks[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/). During the German occupation he and his wife Stanisława, herself a mathematician, held secret university classes at drastic personal risk, part of the underground education system that operated in occupied Poland[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/).
**Emigration.** At the end of 1946 the couple left Europe, going first to Belgium and then to France; from France they traveled to London for a conference, where William Transue of Kenyon College in Gambier, Ohio, offered them positions[6](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym_Stanislawa/). They sailed from [Southampton](https://www.edgepedia.org/southampton) on the Marine Flasher on 21 March 1948 and arrived in New York on 31 March[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/). Nikodym worked at Kenyon from 1948 to 1965 and became a U.S. citizen on 29 June 1953[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/). It is not known for certain why he emigrated, but those who knew him believed it was because he did not want to live in a Poland controlled by the Soviets[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/).
A first-hand account from a Kenyon student records him still teaching and doing research in the early 1960s, in his seventies, using Halmos's measure-theory text, a small stooped figure driven to work each morning by his wife[7](http://ordman.net/Edward/Nikodym.html). After retiring in 1966 he moved to [Utica, New York](https://www.edgepedia.org/utica-new-york), where his later research was sponsored in part by the Atomic Energy Commission and the [National Science Foundation](https://www.edgepedia.org/national-science-foundation)[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/). He suffered a stroke in 1971 and did not regain consciousness for the remaining years of his life[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/).
## The Radon–Nikodym theorem
The theorem answers a basic question: when can one measure be recovered from another by integration? If ν is a finite measure absolutely continuous with respect to a σ-finite measure μ, meaning ν(A) = 0 whenever μ(A) = 0, then there is a measurable nonnegative function f such that ν(B) = ∫_B f dμ for every measurable B, and f is unique up to μ-null sets; the σ-finiteness assumption is necessary[8](https://encyclopediaofmath.org/wiki/Radon%E2%80%93Nikod%C3%BDm_theorem). The function f is the *Radon–Nikodym derivative*, written dν/dμ[2](https://ems.press/content/serial-article-files/9616). MathWorld states a common finite form: any finite complex measure absolutely continuous with respect to a σ-finite positive measure, such as [Lebesgue measure](https://www.edgepedia.org/lebesgue-measure), is given by the integral of some L¹ function[9](https://mathworld.wolfram.com/Radon-NikodymTheorem.html).
**The division of credit.** [Johann Radon](https://www.edgepedia.org/johann-radon) established the theorem in 1913 for Rⁿ, in "Theorie und Anwendungen der absolut additiven Mengenfunktionen" in the Sitzungsberichte of the Vienna Academy of Sciences, volume 112[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) • [4](https://link.springer.com/rwe/10.1007/978-3-662-69359-9_710). Nikodym's 1930 paper, "Sur une généralisation des intégrales de M. J. Radon", published in French in *Fundamenta Mathematicae* 15, pages 131–179, proved the general case of a σ-finite measure and an absolutely continuous countably additive set function[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/) • [10](https://eudml.org/doc/212339). The result is also known as the Lebesgue–Radon–Nikodym or Lebesgue–Nikodym theorem[2](https://ems.press/content/serial-article-files/9616).
Its centrality comes from what the derivative does. The theorem extends to signed, complex-valued, and finite-dimensional vector-valued measures[8](https://encyclopediaofmath.org/wiki/Radon%E2%80%93Nikod%C3%BDm_theorem). A [Banach space](https://www.edgepedia.org/banach-space) on which the conclusion always holds for vector-valued measures is said to have the *Radon–Nikodym property*, a notion that organizes a whole branch of vector-measure theory[8](https://encyclopediaofmath.org/wiki/Radon%E2%80%93Nikod%C3%BDm_theorem).
## Other mathematical work
**The Nikodym set.** In 1927 Nikodym showed how to produce a subset N of the unit square with area(N) = 1 such that for each point x ∈ N there is a line intersecting N in the single point x[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/). A related measure-zero formulation consists of sets containing a punctured line segment through every point of the square; these are close cousins of Kakeya (Besicovitch) sets, which contain a line segment in every direction[11](https://arxiv.org/pdf/2210.08320).
**Boundedness of finitely additive measures.** The Nikodym–Grothendieck boundedness theorem states that a simply bounded family of scalar bounded finitely additive measures on a σ-algebra is uniformly bounded; Nikodym's result from the 1930s was generalized about 30 years later by Alexandre Grothendieck[2](https://ems.press/content/serial-article-files/9616). A 1955 paper, "A theorem on infinite sequences of finitely additive real valued measures", in the *Rendiconti del Seminario Matematico della Università di Padova*, volume 24, pages 265–286, is part of the work behind the Nikodym boundedness and convergence theorems[12](https://www.numdam.org/item/RSMUP_1955__24__265_0/).
**Quantum logic.** His last book, *The Mathematical Apparatus for Quantum-Theories, based on the Theory of Boolean Lattices* (Springer-Verlag, 1966), contains on almost a thousand pages the mathematical formalism for quantum mechanics built on Boolean subalgebras of the logic of closed subspaces of a complex [Hilbert space](https://www.edgepedia.org/hilbert-space)[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/).
## By the numbers
The EMS survey counts more than 30 scientific papers before World War II and about 100 in total[2](https://ems.press/content/serial-article-files/9616); MacTutor gives 32 papers and four textbooks for 1930–45 in Warsaw and about 50 research papers after 1947[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/). The Zentralblatt für Mathematik database records his name in the titles of 783 papers, a count of eponymous citation that places him among the leading Polish mathematicians by that measure[2](https://ems.press/content/serial-article-files/9616).
## How it compares with Radon and the Polish school
The theorem's name divides priority cleanly: Radon's 1913 special case on Rⁿ, Nikodym's 1930 general proof[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/). Nikodym's connection to the Polish school ran long and deep. In 1916 in Kraków he met [Stefan Banach](https://www.edgepedia.org/stefan-banach) and Witold Wilkosz after overhearing a conversation about the Lebesgue integral in the Planty park[1](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/). Nikodym's career split between Kraków, Warsaw and, after 1948, a small liberal-arts college in Ohio; the Zentralblatt count of 783 title mentions places him among the leading Polish mathematicians by that measure[2](https://ems.press/content/serial-article-files/9616).
## Recent developments
The Nikodym set has returned to the research frontier. In November 2025 [Terence Tao](https://www.edgepedia.org/terence-tao) announced new Nikodym set constructions over finite fields, noting that Nikodym sets are close cousins of Kakeya sets and that applying a random projective transformation to a Nikodym set yields most of a Kakeya set, so any lower bound on Kakeya sets transfers to Nikodym sets[3](https://terrytao.wordpress.com/2025/11/12/new-nikodym-set-constructions-over-finite-fields/). A 2026 arXiv preprint studies generalizations of planar Nikodym-type sets, defined as measure-zero Borel sets in the plane containing a punctured line segment through each point of the unit square, building on Falconer's 1985 work and the negative results of Bourgain (1986) and Marstrand (1987)[13](https://arxiv.org/html/2609.39217).
The finite-field formulation is now standard in the polynomial-method toolkit. In [Larry Guth](https://www.edgepedia.org/larry-guth)'s MIT lecture notes, for a finite field F with q elements, a set N ⊂ Fⁿ is a generalized Nikodym set if for each point x ∈ Fⁿ there is a line L(x) containing x with |L(x) ∩ N| ≥ q/2; the trivial example is the whole space Fⁿ[14](https://math.mit.edu/~lguth/PolyMethod/lect3.pdf). The 2022 preprint literature restates the classical Euclidean definition, a set A such that for every x there is a line ℓ through x with A ∩ ℓ containing a unit line segment, and credits the existence to Nikodym's 1927 paper[11](https://arxiv.org/pdf/2210.08320).
## References
1. [Otton Nikodym (1887–1974), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym/)
2. [Banach and Nikodym on the... (EMS Press article)](https://ems.press/content/serial-article-files/9616)
3. [New Nikodym set constructions over finite fields, Terence Tao's blog (November 2025)](https://terrytao.wordpress.com/2025/11/12/new-nikodym-set-constructions-over-finite-fields/)
4. [The Radon–Nikodým Theorem, Springer encyclopedia entry](https://link.springer.com/rwe/10.1007/978-3-662-69359-9_710)
5. [Otton Martin Nikodym papers, 1925–1981, Niels Bohr Library & Archives, AIP](https://history.aip.org/catalog/icos/305.html)
6. [Stanisława Nikodym (1897–1988), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Nikodym_Stanislawa/)
7. [Edward Ordman's reminiscence of Otton Nikodym at Kenyon College](http://ordman.net/Edward/Nikodym.html)
8. [Radon–Nikodým theorem, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Radon%E2%80%93Nikod%C3%BDm_theorem)
9. [Radon–Nikodym Theorem, Wolfram MathWorld](https://mathworld.wolfram.com/Radon-NikodymTheorem.html)
10. [Nikodym, Otton. "Sur une généralisation des intégrales de M. J. Radon", EUDML record](https://eudml.org/doc/212339)
11. [arXiv preprint on Nikodym sets (2022)](https://arxiv.org/pdf/2210.08320)
12. [O.M. Nikodým, Rendiconti del Seminario Matematico della Università di Padova 24 (1955), Numdam](https://www.numdam.org/item/RSMUP_1955__24__265_0/)
13. [On planar Nikodym-type sets, arXiv preprint](https://arxiv.org/html/2609.39217)
14. [The Finite-Field Nikodym and Kakeya Problems, Larry Guth, MIT polynomial method lecture notes](https://math.mit.edu/~lguth/PolyMethod/lect3.pdf)
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