Extension To More Complicated Sets
Notice that a set which is a product of closed intervals,
is not a simple set, and neither is a ball. Thus, so far the set of Jordan measurable sets is still very limited. The key step is then defining a bounded set to be Jordan measurable if it is "well-approximated" by simple sets, exactly in the same way as a function is Riemann integrable if it is well-approximated by piecewise-constant functions.
Formally, for a bounded set B, define its inner Jordan measure as
and its outer measure as
where the infimum and supremum are taken over simple sets S. The set B is said to be Jordan measurable if the inner measure of B equals the outer measure. The common value of the two measures is then simply called the Jordan measure of B.
It turns out that all rectangles (open or closed), as well all balls, simplexes, etc., are Jordan measurable. Also, if one considers two continuous functions, the set of points between the graphs of those functions is Jordan measurable as long as that set is bounded and the common domain of the two functions is Jordan measurable. Any finite union and intersection of Jordan measurable sets is Jordan measurable, as well as the set difference of any two Jordan measurable sets. A compact set is not necessarily Jordan measurable. For example, the fat Cantor set is not. 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. Also, a bounded open set is not necessarily Jordan measurable. For example, the complement of the fat Cantor set (within the interval) is not. A bounded set is Jordan measurable if and only if its indicator function is Riemann-integrable.
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. From this it follows that a bounded set is Jordan measurable if and only if its boundary has Lebesgue measure zero. (Or equivalently, if the boundary has Jordan measure zero; the equivalence holds due to compactness of the boundary.)
Read more about this topic: Jordan Measure
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