Jordan measure

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The Jordan measure is a term from measure theory . This goes back to Marie Ennemond Camille Jordan , who developed it in 1890 based on the work of Giuseppe Peano . With the Jordan measure one can assign restricted subsets of one content and get an integral term which is analogous to the Riemann integral term .

definition

A set (with a blue border) is approximated once by subsets (like the set with a green border) and once by supersets (like the set with a purple border) .

It denotes for

the half-open -dimensional hyper-rectangle and

the set of all such hyper-rectangles. Alternatively, half-open intervals of the shape can be used for definition . Be further

the set of all finite unions of pairwise disjoint hyperrectangles.

It further denotes the content that goes through for everyone with for everyone

and is defined.

Let the inner content of a bounded set A be

its external content is

A set is called Jordan-measurable or squarable if it is bounded and .

The Jordan measure of a Jordan measurable set is given by.

If it holds for a bounded , then Jordan is measurable and is called the Jordan null set .

properties

  1. The Jordan measure is a content and not -additive , that is, countable associations of Jordan-measurable sets need not necessarily be Jordan-measurable (see also example 2). Therefore the Jordan measure is not a measure in the sense of measure theory .
  2. If Jordan is measurable, then Lebesgue is also measurable , and it applies . Here referred to the Lebesgue measure of .
  3. A set is Jordan measurable if and only if is bounded and the boundary of is a Jordan null set.
  4. A bounded set is Jordan-measurable if and only if is. Then also applies .
  5. A compact set is a Lebesgue null set if and only if it is a Jordan null set.

Examples

  1. The unit circle in is Jordan measurable because it is bounded and its edge is a Jordan null set.
  2. The amount cannot be measured by Jordan. Because for a lot of true and for any amount applies whence follows. Applies to everyone . Due to the additivity of the Lebesgue measure, the following applies . is therefore a Lebesgue null set. can be represented as a countable union of the rational numbers in , where each of the sets is Jordan measurable. Since it is not Jordan-measurable, it follows that the Jordan-measurable sets do not form a σ-algebra . The example shows that the Jordan measure (on the Jordan measurable sets) is not a measure.

literature

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