Lebesgue measure

The Lebesgue measure [ ləbɛg ] ( after Henri Léon Lebesgue ) the measure in Euclidean space, the geometric objects to their content (length, area, volume, ...) is assigns.

Background

The Lebesgue measure is from the perspective of modern mathematics, the natural term for surface area and volume. This concept is the end product of a whole series of ideas that tried to grasp concepts such as area and volume mathematically exact. Only with the Lebesgue measure, this process can be considered complete. The Lebesgue measure assigns not only simple geometrical objects, but also much more general quantities, including all open and closed sets to a content. The existence of a non Lebesgue measurable amount (about the Vitali sets ) can be proved non- constructively using the axiom of choice. There are several sets that such a non- constructiveness is necessary in a certain sense, and this. , Under certain assumptions even a stronger way than many other basic sets of analysis

Definition

The Lebesgue - Borel measure on the Borel σ - algebra (also called Borel - Lebesgue measure, or just Borel measure ) is the unique measure with the property

Ie the degree to which intervals the length assigns ( in one-dimensional ) rectangles their area maps ( in two dimensions ), blocks their volume maps ( in three dimensions ), etc. for n-dimensional hypercube. This condition of the contents of any Borel sets is clearly defined. The Borel sets are called Borel - measurable or B- measurable. The Borel measure is bewegungsinvariant and normalized, but not completely. The existence of the Lebesgue - Borel measure was proved in the one-dimensional for the first time by Émile Borel in 1895, a more modern design over the Maßerweiterungssatz goes back to Constantin Carathéodory (1918).

The Lebesgue measure is the total amount which can be obtained from this measure, if one adds to all amounts that are between two Borel sets (), which have the same content, more precisely, and define so. The quantities for which the Lebesgue measure is defined in this way are called Lebesgue measurable ( or L- measurable) and form the Lebesgue - algebra.

B- measurable and L- measurable

It can be shown that the amount of L- measurable quantities is much larger than the amount of B- measurable quantities:

Where stands for cardinality and for the power set of a set.

Null sets

Sets whose Lebesgue measure is equal to 0 are called Lebesgue null sets. Countable point sets such as the set of rational numbers are always Lebesgue null sets. An example of an uncountable Lebesgue -null set is the Cantor. Applies a mathematical statement for an area with the exception of a Lebesgue -null set within the area, as they say: The statement is Lebesgue - almost everywhere.

Properties

The Lebesgue measure is the hair - measure on the locally compact topological group with the addition, therefore, the existence follows from the existence of the hair dimension. In particular, it is translationally, which means that the amount of a quantity does not change under translation. Moreover, it is invariant under reflections and rotations, so even bewegungsinvariant. The Lebesgue measure is infinite.

Construction of the Lebesgue measure

One possible definition of the Lebesgue measure is the construction of Carathéodory. Be the set of dyadic unit cell and the volume of, as these amounts only consist of products of intervals, defined simply to the volume as a product of the individual side lengths. is a semi- ring and a finite - premeasure. After Maßerweiterungssatz of Carathéodory it can be uniquely to a measure on the generated algebra, which is precisely the Borel sets, and so to continue receiving the Lebesgue - Borel measure.

Specifically, can the evidence as follows cause (the proof of the general Maßerweiterungssatzes goes into the essential points analog): For a given set A we define

Is a metric outer measure and defined on the entire power set of the set X is based. is not a measure in general. To get to a level, you have to go from the power set to a smaller amount of system as follows.

A lot of is - measurable if and only if:

(see measurability by Carathéodory ).

All quantities measurable with respect to σ - algebra from a form and on a measure, ie, is a measure.