Jordan measure

In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped.

It turns out that for a set to have Jordan measure it should be well-behaved in a certain restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets. Historically speaking, the Jordan measure came first, towards the end of the nineteenth century. For historical reasons, the term Jordan measure is now well-established, despite the fact that it is not a true measure in its modern definition, since Jordan-measurable sets do not form a σ-algebra. For example, singleton sets ${\displaystyle \{x\}_{x\in \mathbb {R} }}$in ${\displaystyle \mathbb {R} }$ each have a Jordan measure of 0, while ${\displaystyle \mathbb {Q} \cap [0,1]}$, a countable union of them, is not Jordan-measurable. For this reason, some authors prefer to use the term Jordan content (see the article on content).

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