In
mathematics, a Vitali set is an elementary example of a set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
s that is not
Lebesgue measurable
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 ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides w ...
, found by
Giuseppe Vitali
Giuseppe Vitali (26 August 1875 – 29 February 1932) was an Italian mathematician who worked in several branches of mathematical analysis. He gives his name to several entities in mathematics, most notably the Vitali set with which he was the fir ...
in 1905. The Vitali theorem is the
existence theorem
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ( ...
that there are such sets. There are
uncountably many Vitali sets, and their existence depends on the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. In 1970,
Robert Solovay
Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.
Biography
Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on ' ...
constructed a model of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...
without the axiom of choice where all sets of real numbers are Lebesgue measurable, assuming the existence of an
inaccessible cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of f ...
(see
Solovay model In the mathematical field of set theory, the Solovay model is a model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measur ...
).
Measurable sets
Certain sets have a definite 'length' or 'mass'. For instance, the
interval , 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
is deemed to have length 1; more generally, an interval
'a'', ''b'' ''a'' ≤ ''b'', is deemed to have length ''b'' − ''a''. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
∪
, 3
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
is composed of two intervals of length one, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2.
There is a natural question here: if ''E'' is an arbitrary subset of the real line, does it have a 'mass' or 'total length'? As an example, we might ask what is the mass of the set of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s, given that the mass of the interval
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
is 1. The rationals are
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
in the reals, so any value between and including 0 and 1 may appear reasonable.
However the closest generalization to mass is
sigma additivity
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivi ...
, which gives rise to the
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 ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides w ...
. It assigns a measure of ''b'' − ''a'' to the interval
'a'', ''b'' but will assign a measure of 0 to the set of rational numbers because it is
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
. Any set which has a well-defined Lebesgue measure is said to be "measurable", but the construction of the Lebesgue measure (for instance using
Carathéodory's extension theorem) does not make it obvious whether non-measurable sets exist. The answer to that question involves the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
.
Construction and proof
A Vitali set is a subset
of the
interval