Jordan Measurable Region

Jordan Measurable Region

 

a region having a definite area or, what is the same thing, a definite plane measure in the sense of Jordan. The distinguishing property of a (Jordan) measurable region D is the possibility of including it “between” two polygons such that one of them is contained within the given region; the other, on the contrary, contains it; and the difference between their areas can be made arbitrarily small. In this case there exists only one number confined between the areas of all containing and contained polygons; and it is called the area of the (Jordan) measurable region D.

Some properties of measurable regions are (1) if a measurableregion D is contained within a measurable region D1, then thearea of D does not exceed the area of D1 (2) a region D con-sisting of two nonintersecting measurable regions D1 and D2 is measurable, and its area is equal to the sum of the regions D1 and D2; and (3) the common part of two measurable regions D1 and D2 is again a measurable region. In order that a region D be measurable, it is necessary and sufficient that its boundary have an area equal to zero; there exist regions that do not satisfythis condition and consequently are nonmeasurable (in the Jor-dan sense).

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