Jordan measure

Historically speaking, the Jordan measure came first, towards the end of the nineteenth century.

The common value of the two measures is then simply called the Jordan measure of B.

Jordan measure of "simple sets"

That is indeed true, but only if one replaces the Jordan measure with the Lebesgue measure.

One defines the Jordan measure of such a rectangle to be the product of the lengths of the intervals:

In mathematical analysis, Jordan measure (or Jordan content) is an area measure that predates measure theory.

(Or equivalently, if the boundary has Jordan measure zero; the equivalence holds due to compactness of the boundary.)

The SVC is an example of a compact set that is not Jordan measurable, see Jordan measure.

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.

One can show that this definition of the Jordan measure of S is independent of the representation of S as a finite union of disjoint rectangles.

The Lebesgue measure of a set is the same as its Jordan measure as long as that set has a Jordan measure.

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure and Young measure.

At the same time he started his graduate studies at the Sorbonne, where he learned about Émile Borel's work on the incipient measure theory and Camille Jordan's work on the Jordan measure.

One cannot define the Jordan measure of S as simply the sum of the measures of the individual rectangles, because such a representation of S is far from unique, and there could be significant overlaps between the rectangles.

Its inner Jordan measure vanishes, since its complement is dense; however, its outer Jordan measure does not vanish, since it cannot be less than (in fact, is equal to) its Lebesgue measure.

Luckily, any such simple set S can be rewritten as a union of another finite family of rectangles, rectangles which this time are mutually disjoint, and then one defines the Jordan measure m(S) as the sum of measures of the disjoint rectangles.

Also, the Lebesgue measure, unlike the Jordan measure, is a true measure, that is, any countable union of Lebesgue measurable sets is Lebesgue measurable, whereas countable unions of Jordan measurable sets need not be Jordan measurable.

Equivalently, for a bounded set B the inner Jordan measure of B is the Lebesgue measure of the interior of B and the outer Jordan measure is the Lebesgue measure of the closure.