In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. Each Vitali set is uncountable, and there are uncountably many Vitali sets. The proof of their existence depends on the axiom of choice.
Measurable sets Certain sets have a definite 'length' or 'mass'. For instance, the interval [ 0 , 1 ] {\displaystyle [0,1]} is deemed to have length 1 {\displaystyle 1} ; more generally, an interval [ a , b ] , a ≤ b {\displaystyle [a,b],a\leq b} , is deemed to have length b − a {\displaystyle b-a} . If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set [ 0 , 1 ] ∪ [ 2 , 3 ] {\displaystyle [0,1]\cup [2,3]} is composed of two intervals of length one, so we take its total length to be 2 {\displaystyle 2} . In terms of mass, we have two rods of mass 1 {\displaystyle 1} , so the total mass is 2 {\displaystyle 2} . There is a natural question here: if E {\displaystyle 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 numbers between 0 {\displaystyle 0} and 1 {\displaystyle 1} , given that the mass of the interval [ 0 , 1 ] {\displaystyle [0,1]} is 1 {\displaystyle 1} . The rationals are dense in the reals, so any value between and including 0 {\displaystyle 0} and 1 {\displaystyle 1} may appear reasonable. However, the closest generalization to mass must have the property of sigma additivity, which leads us to the Lebesgue measure. It assigns a measure of b − a {\displaystyle b-a} to the interval [ a , b ] {\displaystyle [a,b]} , but will assign a measure of 0 {\displaystyle 0} to the set of rational numbers because it is countable. 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.
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