Lebesgue's premeasure

The Lebesgue content and the Lebesgue premeasure are two closely related set functions of measure theory , a branch of mathematics . Based on these terms, the Lebesgue measure is constructed, which in turn provides the Lebesgue integral , which is a generalization of the Riemann integral .

Lebesgue content

Given the half-ring of half-open intervals. Then the content is called ${\displaystyle {\mathcal {I}}:=\{(a,b]\mid a,b\in \mathbb {R} ,a\leq b\}}$

${\displaystyle \lambda ^{*}\colon {\mathcal {I}}\mapsto [0,\infty ],\,\lambda ^{*}((a,b])=b-a}$

the Lebesgue content. It is σ-finite .

He lets himself onto the ring created by the half ring ${\displaystyle {\mathcal {I}}}$

${\displaystyle {\mathcal {S}}:=\left\{\left.\bigcup _{j=1}^{n}I_{j}\,\right|\,I_{1},\dotsc ,I_{n}\in {\mathcal {I}},I_{j}\,{\text{ paarweise disjunkt }}\right\}}$

continue through

${\displaystyle \lambda '(I)=\sum _{i=1}^{n}\lambda ^{*}(I_{i}),\ {\text{ wenn }}\,I=\biguplus _{i=1}^{n}I_{i}}$.

Here is and . ${\displaystyle I\in {\mathcal {S}}}$ ${\displaystyle I_{i}\in {\mathcal {I}}}$

Lebesgue's premeasure

In fact, the Lebesgue content is already a premeasure , so it is σ-additive . Therefore applies

${\displaystyle \lambda \left(\bigcup _{i=1}^{\infty }I_{i}\right)=\sum _{i=1}^{\infty }\lambda (I_{i})}$,

if the pairs are disjoint and holds. This can be seen as follows: Given an interval ${\displaystyle I_{i}}$ ${\displaystyle \bigcup _{i=1}^{\infty }I_{i}\in {\mathcal {I}}}$

${\displaystyle \bigcup _{i=1}^{\infty }I_{i}=:I\in {\mathcal {I}}}$.

Then this interval contains the closure of an interval and each of the intervals is itself contained within another interval. So is ${\displaystyle {\overline {H}}}$ ${\displaystyle I_{i}}$ ${\displaystyle J_{i}^{\circ }}$

${\displaystyle {\overline {H}}\subset \bigcup _{i=1}^{\infty }J_{i}^{\circ }}$.

But since it is compact , there is a finite partial coverage an . Now, however, all relevant intervals can be selected so that ${\displaystyle {\overline {H}}}$ ${\displaystyle J_{i}^{\circ }}$

${\displaystyle (1-\epsilon )\lambda (I)\leq \lambda (H){\text{ und }}\lambda (J_{i})\leq (1+\epsilon )\lambda (I_{i})}$

applies. With the finiteness of the coverage, the σ-subadditivity of . However, this is equivalent to σ-additivity for content, so the content is a premeasure. The continuation to the generated ring works in the same way as above; the σ-finiteness is also retained. ${\displaystyle \lambda }$

As an alternative to the direct evidence of the σ-additivity indicated here, the Lebesgue premise can also be viewed as a special case of the Lebesgue-Stieltjes premise and the σ-additivity can be derived from the σ-additivity of the Lebesgue-Stieltjes premise.

Higher dimensional case

If you choose the as the basic amount , then the n-dimensional Lebesgue premise can be found on the half-ring of the half-open cuboid ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle {\mathcal {I}}^{n}:=\{(a,b]\mid a,b\in \mathbb {R} ^{n},a<b\}}$

define, being here

${\displaystyle (a,b]:=\{(x_{1},\dots ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}\leq b_{1},\dots ,a_{n}<x_{n}\leq b_{n}\}}$

means. One then sets as Lebesgue's premeasure

${\displaystyle \lambda ^{n}((a,b]):=\prod _{i=1}^{n}(b_{i}-a_{i})}$

This is exactly the elementary geometric volume of a cuboid, namely the product of the side lengths, in the two-dimensional case it is the area of a rectangle. Both the verification of the σ-additivity and the continuation on the ring generated by run analogously to the one-dimensional case. ${\displaystyle {\mathcal {I}}^{n}}$

comment

• The notation with and used here serves for better differentiation, mostly content as well as pre-measure and measure are only designated with , regardless of the quantity system they are defined on. ${\displaystyle \lambda ^{*},\lambda '}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$
• When constructing the Lebesgue measure , other set half rings than those used here are occasionally used. Examples would be the dyadic unit cells , the half-open intervals closed on the left, the half-open intervals closed on the left with rational corner points or others. The half rings generally do not match, but the σ-algebras generated by these half rings are identical. Each choice of the half-rings then yields the same Lebesgue measure.
• For the n-dimensional case it can be shown that the n-dimensional Lebesgue measure constructed from this case is exactly the n-fold product measure of the one-dimensional Lebesgue measure. The result is therefore independent of the type of construction.
• Since the premeasure is σ-finite, the continuation to the Lebesgue measure according to the measure extension theorem of Carathéodory is unambiguous. The exact construction of the continuation can also be found there .
• Both the Lebesgue content and the Lebesgue premeasure are special cases of Stieltjes'schen content and Lebesgue Stieltjes Prämaßes with . ${\displaystyle F(x)=x}$

literature

• Jürgen Elstrodt: Measure and integration theory . 6th edition. Springer, Berlin / Heidelberg / New York 2009, ISBN 978-3-540-89727-9 .