Carathéodory's extension theorem

The Maßerweiterungssatz of Carathéodory is a set from the mathematical branch of measure theory. This phrase is used to extend the dimensions that are defined on rings amount to larger σ - algebras. This goes back to Constantin Carathéodory method can be attributed to the length determination of intervals, in particular the Lebesgue measure.

• 3.1 uniqueness
• 3.2 semirings

Wording of the sentence

It is a measure on a ring of sets amount of a base amount. Then there is a σ - algebra on comprehensive and an extension of a measure, so that is a complete measure space.

Construction

One defines by means of the given measure on the ring A defined on the entire power set of outer measure and therefrom by a suitable restriction is a measure on a σ - algebra. This construction will now be described in detail, and applied in parallel to the construction of the Lebesgue measure.

Measurements on rings

A lot of ring contains the empty set and with respect to finite unions and education of imbalances completed. A measure on a quantitative ring is a function of and, if pairwise disjoint sets are off, their union is again. Some authors also speak of a premeasure.

The standard example is the set of all finite unions of half-open intervals, always was. Such associations can be written as a disjoint union of such intervals always, and fixing, the length of such an interval is defined on a measure.

This generalizes easily to dimensions when one considers the amount of ring of all finite unions of n-dimensional intervals (box ), always was. Here you can also be restricted to disjoint unions and in such a case

Define, with the usual elementary geometric volume of a cuboid is. We call this sample the Lebesgue premeasure.

Construction of the outer measure

Let there be given a measure on a ring of sets amount of a base amount. Is for each subset

Said. Then on an outer measure. It can be shown that

For all and. The first property states that the preset level continues, the second that any amount of ground space is divided by any quantity of the given ring into two parts, which are additive with respect.

Transition to measurable quantities

The core construction is the definition of Carathéodory

To prove that this defines a σ - algebra, called the σ - algebra of measurable quantities, and that the restriction is a measure. Due to the above second nature of outer measure, and because the first is a continuation of. Finally, one shows that each set contains with outer measure 0, from which then yields the completeness of Maßraums.

Applying this construction to our example of the Lebesgue Prämaßes to, we obtain the Lebesgue measure on the Lebesgue σ - algebra. In this case, one can show that the Lebesgue σ - algebra is strictly larger than that of generated σ - algebra, which coincides with the borel σ - algebra between. However, the difference is not too large, because a lot of the Lebesgue σ - algebra differs only by a zero set of a Borel set, ie the Lebesgue σ - algebra is the completion of the borel between.

Comments

Unambiguity

As a consequence of the above theorem we obtain that every measure can be continued on the σ - algebra generated by the ring on a ring to a degree. One gets a uniqueness statement, if one additionally assumes that may be written as a countable union of sets of finite measure ring.

Half rings

Instead of quantity rings you can also start from the more general concept of the half ring. A measure or premeasure on a semi- ring is defined on rings, ie there is a lot of function such that and, if pairwise disjoint sets are off, their union is again.

To get into this situation to a Size extension, one first forms the ring generated which is equal to the set of all finite disjoint unions of sets from. If such a disjoint union, as is explained on the amount ring by fixing a measure. On the construction described above can then be applied.

The standard example is the semiring of all semi-open n-dimensional intervals (box )

With and it said measure of elementary geometric content. So The presented design results directly from the definition of the square output as the product of the side lengths to the Lebesgue measure. It can be generalized directly to general product dimensions.