5.5 Area of regions in the plane

It would be desirable to be able to assign to each subset $S$ of the Cartesian plane $R^2$ a nonnegative real number $A\left(S\right)$ called its area. We would insist based on our intuition that (i) if $S$ is a rectangle with sides of length $L$ and $W$ then the number $A\left(S\right)$ should be $LW,$ so that this abstract notion of area would generalize our intuitively fundamental one. We would also insist that (ii) if $S$ were the union of two disjoint parts, $S=S_1\cup S_2,$ then $A\left(S\right)$ should be $A\left(S_1\right)+A\left(S_2\right).$ (We were taught in high school plane geometry that the whole is the sum of its parts.) In fact, even if $S$ were the union of an infinite number of disjoint parts, $S={\cup }_{n=1}^{\infty }{S}_n$ with $S_i\cap S_j=\varnothing$ if $i\ne j,$ we would insist that (iii) $A\left(S\right)={\sum }_{n=1}^{\infty }A\left(S_n\right).$

The search for such a definition of area for every subset of $R^2$ motivated much of modern mathematics. Whether or not such an assignment exists is intimately related to subtle questions in basic set theory,e.g., the Axiom of Choice and the Continuum Hypothesis . Most mathematical analysts assume that the Axiom of Choice holds, and as a result of that assumption,it has been shown that there can be no assignment $S\to A\left(S\right)$ satisfying the above three requirements. Conversely, if one does not assume that the Axiom of Choiceholds, then it has also been shown that it is perfectly consistent to assume as a basic axiom that such an assignment $S\to A\left(S\right)$ does exist. We will not pursue these subtle points here, leaving them to a course in Set Theory or Measure Theory.However, Here's a statement of the Axiom of Choice, and we invite the reader to think about how reasonable it seems.

AXIOM OF CHOICE Let $\mathcal{S}$ be a collection of sets. Then there exists a set $A$ that contains exactly one element out of each of the sets $S$ in $\mathcal{S}.$

The difficulty mathematicians encountered in trying to define area turned out to be involved withdefining $A\left(S\right)$ for every subset $S\in R^2.$ To avoid this difficulty, we will restrict our attention here to certain “ reasonable” subsets $S.$ Of course, we certainly want these sets to include the rectangles and all other common geometric sets.

By a (open) rectangle we will mean a set $R=(a,b)\times (c,d)$ in $R^2.$ That is, $R=\left\{\right(x,y):a<x<b\text{and}c<y<d\}.$ The analogous definition of a closed rectangle $[a,b]\times [c,d]$ should be clear: $[a,b]\times [c,d]=\left\{\right(x,y):a\le x\le b,c\le y\le d\}.$

By the area of a (open or closed) rectangle $R=(a,b)\times (c,d)$ or $[a,b]\times [c,d]$ we mean the number $A\left(R\right)=(b-a)(d-c).$ .

The fundamental notion behind our definition of the area of a set $S$ is this. If an open rectangle $R=(a,b)\times (c,d)$ is a subset of $S,$ then the area $A\left(S\right)$ surely should be greater than or equal to $A\left(R\right)=(b-a)(d-c).$ And, if $S$ contains the disjoint union of several open rectangles, then the area of $S$ should be greater than or equal to the sum of their areas.

We now specify precisely for which sets we will define the area. Let $[a,b]$ be a fixed closed bounded interval in $R$ and let $l$ and $u$ be two continuous real-valued functions on $[a,b]$ for which $l\left(x\right)<u\left(x\right)$ for all $x\in (a,b).$

Given $[a,b],l,$ and $u$ as in the above, let $S$ be the set of all pairs $(x,y)\in R^2,$ for which $a<x<b$ and $l\left(x\right)<y<u\left(x\right).$ Then $S$ is called an open geometric set. If we replace the $<$ signs with $\le$ signs, i.e., if $S$ is the set of all $(x,y)$ such that $a\le x\le b,$ and $l\left(x\right)\le y\le u\left(x\right),$ then $S$ is called a closed geometric set. In either case, we say that $S$ is bounded on the left and right by the vertical line segments $\left\{\right(a,y):l(a)\le y\le u(a\left)\right\}$ and $\left\{\right(b,y):l(b)\le y\le u(b\left)\right\},$ and it is bounded below by the graph of the function $l$ and bounded above by the graph of the function $u.$ We call the union of these four bounding curves the boundary of $S,$ and denote it by $C_S.$

If the bounding functions $u$ and $l$ of a geometric set $S$ are smooth or piecewise smooth functions, we will call $S$ a smooth or piecewise smooth geometric set.