Outer measure

Outer Dimension is a term from the mathematical branch measure theory, which was introduced in 1914 by Constantin Carathéodory. An external measure is a set function of the power set of a set in the interval which satisfies the following axioms:

• " Monotony "
• " Subadditivity "

The name of outer measure is modeled on the terms inner and outer measure, which were used by Borel and Lebesgue. The theory of Carathéodory does not use inner dimension and simplifies the basic evidence considerably.

• 3.1 Examples

Construction of an outer measure

Be any quantity system and a picture with. If, for all:

Then on an outer measure.

Metric outer measure

A metric outer dimension is an outer dimension of a metric space with the additional feature:

For all sets A and B with a positive distance, ie.

Example

• To construct the Lebesgue measure a metric outer measure is being used.

Measurability by Carathéodory

Be an outer measure on the power set of a set. A set is measurable with respect to or short - measurable if

It should be noted that the term measurability in the measure theory has two meanings, one being measurable relative to a measuring chamber, and on the other with respect to a measurable by Carathéodory outer dimension.

This notion of measurability comes from Constantin Carathéodory.

Examples

• Be a lot with

• Are - measurable.
• Null sets are measurable: Be with. Then is - measurable.

σ - algebra of measurable quantities -

Is an outer dimension, it is the amount

A σ - algebra and a full measure.