LebesgueOuterMaße

Lebesgue outer measure is a concept in measure theory, a branch of mathematics that deals with the measurement of sets. It was introduced by Henri Lebesgue in the early 20th century as part of his development of the Lebesgue integral, which is a more general and powerful tool than the Riemann integral.

The Lebesgue outer measure is defined for any subset of the real line or, more generally, any metric space. It is a function that assigns a non-negative real number or infinity to each subset, representing its "size" or "measure". The outer measure is defined using the concept of covering the set with intervals (in the case of the real line) or open balls (in the case of a general metric space).

The definition of the Lebesgue outer measure involves taking the infimum of the sum of the lengths (or diameters) of all possible coverings of the set. This infimum is then taken over all possible coverings, which ensures that the outer measure is well-defined for any subset.

One of the key properties of the Lebesgue outer measure is that it is monotone: if A is a subset of B, then the outer measure of A is less than or equal to the outer measure of B. Another important property is that the outer measure of the union of a countable collection of sets is less than or equal to the sum of the outer measures of the individual sets.

The Lebesgue outer measure is not a measure in the strict sense, because it does not satisfy the property that the measure of a countable union of disjoint sets is equal to the sum of the measures of the individual sets. However, it can be used to define a measure, known as the Lebesgue measure, which does satisfy this property for a certain class of sets, known as Lebesgue measurable sets.