In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Nikodym set is a subset of the unit square in
with complement of
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
zero (i.e. with an area of 1), such that, given any point in the set, there is a straight line that only intersects the set at that point. The existence of a Nikodym set was first proved by
Otto Nikodym in 1927. Subsequently, constructions were found of Nikodym sets having continuum many exceptional lines for each point, and
Kenneth Falconer found analogues in higher dimensions.
Nikodym sets are closely related to
Kakeya set
In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensiona ...
s (also known as Besicovitch sets).
The existence of Nikodym sets is sometimes compared with the
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then ...
. There is, however, an important difference between the two: the Banach–Tarski paradox relies on non-measurable sets.
Mathematicians have also researched Nikodym sets over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s (as opposed to
).
References
{{reflist
Measure theory
Paradoxes