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 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
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 .
A set is Jordan measurable if and only if is bounded and the boundary of is a Jordan null set.
A bounded set is Jordan-measurable if and only if is. Then also applies .
A compact set is a Lebesgue null set if and only if it is a Jordan null set.
Examples
The unit circle in is Jordan measurable because it is bounded and its edge is a Jordan null set.
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
Wolfgang Walter : Analysis (= basic knowledge of mathematics 4). 2nd volume. 2nd Edition. Springer, Berlin et al. 1991, ISBN 3-540-54566-2 , pp. 224-226.