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Let be a closed geometric set in the plane. If is a real-valued function on we would like to define what it means for to be “integrable” and then what the “integral” of is. To do this, we will simply mimic our development for integration of functions on a closed interval
So, what should be a “step function” in this context? That is, what should is a “partition” of be in this context? Presumably a step function is going to be a function that is constant on the “elements” of a partition.Our idea is to replace the subintervals determined by a partition of the interval by geometric subsets of the geometric set
The overlap of two geometric sets and is defined to be the interior of their intersection. and are called nonoverlapping if this overlap is the empty set.
A partition of a closed geometric set in is a finite collection of nonoverlapping closed geometric sets for which i.e., the union of the 's is all of the geometric set
The open subsets are called the elements of the partition.
A step function on the closed geometric set is a real-valued function on for which there exists a partition of such that for all i.e., is constant on each element of the partition
REMARK One example of a partition of a geometric set, though not at all the most general kind, is the following.Suppose the geometric set is determined by the interval and the two bounding functions and Let be a partition of the interval We make a partition of by constructing vertical lines at the points from to Then is the geometric set determined by the interval and the two bounding functions and that are the restrictions of and to the interval
A step function is constant on the open geometric sets that form the elements of some partition. We say nothing about the values of on the “boundaries” of these geometric sets. For a step function on an interval we do not worry about the finitely many values of at the endpoints of the subintervals. However, in the plane, we are ignoring the values on the boundaries, which are infinite sets. As a consequence, a step function on a geometric set may very well have an infinite range,and may not even be a bounded function, unlike the case for a step function on an interval.The idea is that the boundaries of geometric sets are “negligible” sets as far as area is concerned, so that the values of a function on these boundaries shouldn't affect the integral (average value) of the function.
Before continuing our development of the integral of functions in the plane, we digress to present an analog of [link] to functions that are continuous on a closed geometric set.
Let be a continuous real-valued function whose domain is a closed geometric set Then there exists a sequence of step functions on that converges uniformly to
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