 Given a subset formula_1, with the length of interval formula_2 given by formula_3, the Lebesgue outer measure  formula_4 is defined as

The Lebesgue measure is defined on the Lebesgue "σ"-algebra, which is the collection of all sets formula_6 which satisfy the "Carathéodory criterion" which requires that for every formula_7,

For any set in the Lebesgue "σ"-algebra, its Lebesgue measure is given by its Lebesgue outer measure formula_9.Sets that are not included in the Lebesgue "σ"-algebra are not Lebesgue-measurable.Such sets do exist (e.g.