Vitali Set - Measurable Sets

Measurable Sets

Certain sets have a definite 'length' or 'mass'. For instance, the interval is deemed to have length 1; more generally, an interval, 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 ∪ 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 numbers, given that the mass of the interval is 1. The rationals are dense in the reals, so any non negative value may appear reasonable.

However the closest generalization to mass is sigma additivity, which gives rise to the Lebesgue measure. It assigns a measure of b − a to the interval, but will assign a measure of 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 there exist non-measurable sets. The answer to that question involves the axiom of choice.